Nathalie Domede

Luisa Pena

Nicolas Fady

N Domede Orcid

Historical review of lighthouse design under wind load: the Ile Vierge lighthouse

Keywords: lighthouse, wind load, masonry structures, diagnosis

The purpose of the study presented here is to describe the design method used by engineers in the 19 th and early 20 th centuries to size lighthouses structures, and to compare them with current European standards. The historical information presented here is derived from two sources in particular: the archives of the Ile Vierge lighthouse, which was built at the northern tip of Brittany between 1897 and 1902, and the scientific journals, books and courses available at that time. Firstly, the article describes the structure of the Ile Vierge lighthouse which is the tallest stone lighthouse in Europe. The wind load acting on lighthouses and chimneys was formulated in France by L. Fresnel in 1825. Fresnel's work is compared to WJ M. Rankine's theory used in the UK. The safety coefficients applied with respect to the stability of the towers in France and the UK in the 19 th century are compared. On the occasion of the construction of the Ile Vierge lighthouse, the masonry strength under wind loads led to discussion among engineers of the French lighthouse authorities. They also worried about the risk of excessive oscillations of the towers, which could cause a dysfunction of the lamp. Thirdly, the standard wind action defined in Eurocode 1 is applied to the Ile Vierge lighthouse. Calculation results are given. A comparative analysis of the ancient and modern methods is presented. Overall, this historical research shows that the designers of lighthouses in the 19 th century had the same objectives as today: to build strong, durable structures in which deformations under loads are compatible with serviceability.

INTRODUCTION

This paper is part of civil engineering research on the structural diagnosis of the masonry lighthouses in Brittany (western France). Most of them were built between 1830 and the early 20th century. Some of these buildings present disorders such as visible cracks on the walls and corrosion on the metal parts, which call their stability and durability into question.

There are about 150 tall lighthouses in France, including about thirty at sea. The vital role of these constructions in safe navigation, and their cultural and heritage value mean that these structures must be maintained in service as long as possible. To help achieve this, a diagnosis methodology is being developed on the Ile Vierge lighthouse described at the beginning of this paper. The first step of this methodology concerns historical research on masonry towers. This research makes sense today because, while research works on the optical apparatus of lighthouses are extremely numerous and detailed, those on the civil engineering of lighthouses are rare.

The historical and mechanical research presented here focuses on the method used by 19 th century engineers to calculate the masonry towers under wind action. The historical information presented is derived from the archives of the Ile Vierge lighthouse and from scientific journals, books and courses available at that time and during the previous 140 years.

The construction of the Ile Vierge lighthouse, a building of exceptional height for the time, raised questions and caused debate among the engineers of the French lighthouse authority (Service des Phares et Balises).

Their letters, available in the departmental archives of Finistère (the western end of Brittany), testify to this. They discussed the wind velocity and wind pressure on the structure, the criteria for sizing the tower, and the choice of materials to ensure its durability. They shared the feedback from the lighthouses already built in France, but not only in France. The assumptions and calculation methods used by engineers in England for the design of lighthouses and chimneys were also consulted and analysed. It was only after a long and fruitful comparative analysis of the practices of both, which lasted from June 1896 to March 1897, that the project for the construction of the Ile Vierge lighthouse was definitively adopted, in its third version, and that construction could start. It is a synthesis of this analysis that is presented in this article.

Firstly, a synthetic description of the Ile Vierge lighthouse is presented. Secondly, the article focuses on the calculation methods for the wind action on lighthouse towers and chimneys applied by 19 th century engineers in France and England. Then, the wind action defined by the current European standards of Eurocode 1 (EC 1) is applied to the Ile Vierge lighthouse. A comparative analysis of the old and modern calculation methods completes the study.

THE ILE VIERGE LIGHTHOUSE

The Ile Vierge is an island located 1.5 km from the northern coast of Brittany, the most westerly region of France (Fig. 1), at the entrance of the Channel. This maritime corridor between France and England hosts exceptionally dense maritime traffic as it is the unavoidable passage between the Atlantic Ocean and the North Sea. This is why an impressive number of lighthouses have been built along the Channel coast, both on the French side and on the English side (about 100 tall lighthouses in total).

On the Ile Vierge, two lighthouses were built, the first in 1845, the second between 1897 and 1902 (Fig. 2). It is the latter lighthouse that is the subject of this paper. Its focal plane had to rise 75 m above the highest tides, in accordance with the Ministerial decision of August 22, 1896 [A1], so the tower had to be 82 m high (84.50 m at the top of the lantern), 10 m more than the highest French lighthouse of the time [1]. Bourdelles. The architecture was clearly the work of engineers concerned with the functionality of the lighthouse, its robustness, its durability and its cost, as evidenced by the letters they exchanged during the year 1896. (a)

Figure 1 -Location of Ile Vierge

The optical apparatus Originally, the optical apparatus consisted of optics each having an Aüer burner [A5] and two 3.55 m high Fresnel lenses [START_REF] Goffic | Le plus haut phare de France[END_REF] (Fig. 3). The optical apparatus was supported by a base resting in a mercury bath to limit friction. The lantern protecting the optical apparatus was a cylindrical metal structure of 5 m inner diameter, glazed in its vertical part (4.20 m high), and topped by a dome. This structure is anchored in a 50 cm thick cylindrical wall made of kersantite (local magmatic rock) stone masonry, called the "basement wall" (Fig. 3). This perimeter wall delimits a cylindrical room (3.40 m high and 5 m of inside diameter) that houses the machine of the optical apparatus. Its floor is metallic.

(b)

The tower

The tower is a stone masonry structure with a circular cross section (15 m at the bottom, 7 m minimum at level 65.80. Fig. 4), like about half of the tall French lighthouses, the others being of square or octagonal section [START_REF] Allard | Les phares, histoire, construction, éclairage[END_REF].

The square shape, more affected by wind load, was restricted to low-rise towers. The internal void space of the towers is almost always cylindrical [START_REF] Quinette De Rochemont | Cours de travaux maritimes[END_REF]. In the case of the Ile Vierge lighthouse, it has a constant diameter (5 m) and contains the 395 stone steps giving access to the service room of the lighthouse keeper (Fig. 4).

The entire structure between the 0.00 and 73.42 levels is in granite (inner side) and kersantite masonry (exterior side of the tower and the entire basement wall). The inner side of the wall of the tower below the service room is lined with a brick wall covered with blue opaline plates (Fig. 5). Under the Apparatus room is the service room of the lighthouse keeper (who was housed in the old, smaller lighthouse of the Ile Vierge). The floor of this room, built in March 1900, is in reinforced cement following the Hennebique technique, patented 8 years earlier. Two types of stone were used: granite and kersanton. They were already used together in the Pierres Noires lighthouse (circular tower 28 m high completed in 1871) and the Eckmühl lighthouse (square tower 65 m high completed in 1897), also located at the western end of Brittany [START_REF] Chauris | Construction d'un phare en mer d'Iroise : les pierres Noires (1866-1872)[END_REF]. The internal facing of the cylinder, not exposed to weathering, was built with granite rubble extracted from the ground of the island near the lighthouse [A1] [A7] (Fig. 6). The kersanton stones were used for the outer facing of the tower and the entire basement wall of the lantern at the top of the tower.

Figure 6 -Extraction of granite at the foot of the Ile Vierge lighthouse [A8]

Kersantite is a grey or greenish magmatic rock found in western Brittany. It looks like granite but is not. The kersantite used for the Ile Vierge lighthouse came from the quarries of Logona, close to Brest [1] [A8]. This rock was highly appreciated for the construction of lighthouses in Brittany and even beyond, as far as the Ile de Ré (in the Baleines lighthouse) [START_REF] Chauris | Construction d'un phare en mer d'Iroise : les pierres Noires (1866-1872)[END_REF]. This rock was preferred to the local granite, despite its higher cost, because of its ability to form sharp edges and its great resistance to weathering, as evidenced by the many churches built with this rock in the region [A7]. The choice of kersanton for the external face of the lighthouse and granite for the inner face was not made lightly: it ensured the resistance of the construction to harsh weather conditions by putting the most durable stone on the most exposed wall facings.

(d) The mortar

The mortars were made on site, using non-salted water transported to the island (no fresh water source) with sand from the northern beaches of the island of Wrach, close to the Ile Vierge [A8], and Portland cement (cement to sand ratio = 400 kg.m-3) [A5]. Cement was preferred to lime after long discussions among the engineers, stemming from the finding that all the tall towers of the time such as those of the Calais lighthouse (55 m high, 1848) and the Coubre lighthouse (64m high, 1895), built with hydraulic lime mortar, are split according to a diametrical plane perpendicular to the direction of the strongest winds" [A2]. In addition, the engineers did not want a repeat of the problems that occurred on the Penmarch Lighthouse for which "when the construction progressed well, it sometimes seemed prudent to slow it down because the masonry was not hard enough" [A9].

Thus, cement was preferred in the Ile Vierge, as in the Eckmühl lighthouse, to increase the stiffness of the masonry tower and avoid ovalization under wind loads, especially at early ages of the mortar. This choice was also supported by the engineer L. Bourdelles with regard to the dynamic action of the wind, as will be seen later (section 3.3).

(e)

The foundations Geotechnical surveys carried out on the Ile Vierge before the construction started had identified a good granitic soil [A7]. The solution for the foundation works described in the technical specifications [A8] was therefore simply to sit the masonries directly on the granitic soil after excavating about 1 m of ground. Unfortunately, the excavation at the location of the tower brought to light a poor quality sand zone (4 m wide and about 2 m deep), in the axis of the lighthouse, that had to be excavated (which explains the particular form of the foundations in Fig. 5). There was no metal anchorage in the rock substratum.

(f) The construction works

During the 4 years of work, all the food for the workers, and all the animals (horses), materials, tools, and equipment, were delivered by boat, with the exception of granite, which was taken from the island itself [A10].

The materials arrived by boat (Fig. 7), were transported to the lighthouse by wagons pulled on rails by horses, then raised to the work platform using a winch (Fig. 8).

Figure 7 -Loading and unloading on the Ile Vierge [4]

Barracks for the masons had been built on the island by the masonry company [START_REF] Goffic | Le plus haut phare de France[END_REF] but the ordinary workers returned to the continent every evening. The working conditions on the tower were difficult, even though construction was stopped during the winter and when storms occurred. Le Goffic, a journalist and writer of the time visited the work site when the central stairs consisted of about 200 steps [START_REF] Goffic | Le plus haut phare de France[END_REF] (Fig. 8). He was impressed by the height already reached and the risks of falling from the work platform. Fortunately, there were no accidents on the Ile Vierge during the construction of the lighthouse. Those whose work is going to be discussed in this paper do not figure in this list. They were engineers who, during the 19 th century, designed buildings such as lighthouses, towers, and chimneys, very exposed to the wind. To ensure the stability, resistance and durability of these buildings, the engineers had to make assumptions about the wind pressure on the structures, perform calculations, and take responsibility for the results. Here are their names:

(i) John Smeaton

John Smeaton, (1724 -1792) is considered as the founder of the civil engineering profession in Great Britain. He built the Eddystone lighthouse (1756-1759) while developing a hydraulic lime mortar able to harden underwater and so was the first to understand how to constitute a hydraulic lime. On May 31, 1759, John Smeaton read the results of numerous experiments conducted to improve the efficiency of windmills [START_REF] Smeaton | Experimental enquiry concerning the natural powers of wind and water to turn mills and other machines[END_REF] in front of the Royal Society of London. In particular, this work presents a table (Fig. 9) that links the wind velocity with its pressure measured on a plate perpendicular to the flow, in accordance with a scale commonly used at the time to denote the wind intensity in everyday language.

Figure 9 -Wind velocity and load according to Smeaton [9]

Smeaton took care to point out that the speed values exceeding 50 miles per hour (80 km.h-1) were not as reliable as the lower values because they were obtained by extrapolation of the measurements made with mean winds. This extrapolation was obtained using equation ( 1). Transformed into current international units (N, m, s), the coefficient is equal to 1.2 which is about the density of air [START_REF] Cunha | Le phare de l'Ile Vierge[END_REF]. It should be noted, however, that Smeaton never referred to the work of Daniel Bernoulli.

= 0.0022 With in psf and in ft.s-1 (1)

= 1.2 With in N.m-2 and in m.s-1 (2)

(ii) Augustin Fresnel

In 1825, on the occasion of the construction of the Belle-Ile lighthouse, Augustin Fresnel studied the wind action on a tower. The method he used in his calculations was reported by his younger brother, Leonor Fresnel, in 1831 [START_REF] Fresnel | Mémoire sur la stabilité du phare de Belle-île en mer[END_REF]. Augustin Fresnel assumed the wind limit velocity to be 50 m.s-1 which "exceeds by one ninth the velocity of the strongest hurricanes mentioned in the yearly report of the French Bureau of Longitudes in the year 1818". Referring to the experiments reported by Smeaton 66 years earlier, he states that the maximum wind pressure to be considered on the Belle-Ile lighthouse equals 2750N.m-2 (this is actually the load for 48 m.s-1).

(iii) Léonor Fresnel

In 1831, Léonor Fresnel validated the wind pressure value chosen by his older brother Augustin, by comparing it with the notes of Navier published in the 1819 edition of Bélidor's Architecture Hydraulique [START_REF] Fresnel | Mémoire sur la stabilité du phare de Belle-île en mer[END_REF]. He extended the calculations and established a method to verify the stability of lighthouses under wind load. The principle, in the case of a square tower, is the following: the wind moment is calculated at the 0.000 level by applying 2750N.m-2 to one of the faces of the tower from bottom to top. He admitted that this pressure "can be seen as greater than the maximum effort of our hurricanes". The self-weight moment, which L. Fresnel calls "moment of resistance", is obtained by multiplying the total weight of the tower by the half-width of its base. He called the ratio of these two moments "absolute stability". He acknowledged that, in his calculation, he ignored the deflection of the top of the tower, but said that this calculation (which is now called second order calculation) was not useful because the phenomenon was negligible.

This method is transposed to the case of a circular section tower by assuming that "the wind action on a vertical cylinder is reduced to 2/3 of what it would be on the meridian section of the same cylinder". So, the total wind force (in Newtons) on a cylindrical tower according to L. Fresnel is given in equation (3) with the height and diameter in metres, and =48m.s-1.

= (1.2 ) (3)
Léonor Fresnel applied this method to several existing chimneys and lighthouses that had demonstrated stability under wind and storms for a long time (in particular the 76 m high Genoa tower in Italy built in 1543, the French Cordouan lighthouse built in 1611, and industrial chimneys in brick), and finally wrote that stability is ensured if the "absolute stability" is at least equal to 4 [A11].

(iv) William John Macquorn Rankine

According to Guilly in 1876, "the English may be the only ones involved in defining stability rules for tall masonry chimneys" [START_REF] Gouilly | Théorie sur la stabilité des hautes cheminées en maçonnerie[END_REF]. Actually, Rankine, in A Manuel of Applied Mechanics published in 1858 [START_REF] Macquorn Rankine | A manual of applied mechanics[END_REF] dedicated 4 pages to towers and chimneys.

For Rankine, as for Fresnel and Smeaton, the wind action on the construction was horizontal, uniform and only in the direction of the wind. It concerned square and round chimneys. Rankine said that the maximum wind pressure on a flat surface was 55 psf (pounds per square foot, or 2680N.m-2) corresponding to a velocity of 47 m.s-1. He based this assumption, on the one hand, on the measurements carried out by means of anemometers and, on the other hand, on the feedback on existing chimneys. He did not refer to Smeaton's works. To pass from a square tower to a round tower, he chose the coefficient 1 ⁄ 2 and not the 2 ⁄ 3 value used by Léonor Fresnel. So, the total wind force (F in Newtons) on a cylindrical tower according to Rankine is given in equation ( 4) with the height and diameter in metres, and =47m.s-1.

= (1.2 ) (4)
To verify the stability of the cylindrical towers, Rankine kept the same safety coefficient as Léonor Fresnel, i.e. 4, but used only 3 for the square towers [START_REF] Macquorn Rankine | A manual of applied mechanics[END_REF].

(v) Gustave Eiffel

The wind action taken into account by Eiffel [START_REF] Eiffel | Mémoire présenté à l'appui du projet définitif du viaduc de Garabit[END_REF] when calculating the Garabit viaduct, built in 1880 in France, was 2700N.m -2 when the bridge was not loaded with trains and 1500N.m-2 otherwise. For the famous tower he built in Paris (1889), he chose to vary the wind pressure from 2000N.m-2 on the ground to 4000N.m-2 at the top of the tower. In his calculation notes for the 300 m tower [14], Eiffel gave an equation similar to (2) as "generally accepted", and which is the one proposed by Smeaton 130 years earlier. He discussed the validity of this equation with respect to the measurements carried out on his tower, and proposed a coefficient 0.9 instead of 1.2 for the calculation of the metal frame.

(vi) First standardization in France

The first French Standard on wind action to be taken into account on structures was the "Circulaire du Ministre des Travaux Publics" dated August 29, 1891 [START_REF][END_REF]. It added wind action to the previous texts of 1858 and 1877, which considered only gravity actions. This circular imposed exactly the assumptions that Eiffel had applied for the calculation of the Garabit bridge 10 years earlier. The calculation assumptions of neighbouring European countries, at the same time, were similar, as Schueremans points out [START_REF] Schueremans | On the evolution in design and calculation of steel structures over the 19 th century in Belgium, France and England[END_REF].

(b) Dynamic wind action

The vibration of lighthouses and chimneys under wind action was known by the 19 th century engineers. It was taught by De Rochemont in 1897 in the Ecole des Ponts et Chaussées that the wind action could determine oscillations able to harm the working of optical apparatus [START_REF] Quinette De Rochemont | Cours de travaux maritimes[END_REF].

Fresnel noted [START_REF] Fresnel | Mémoire sur la stabilité du phare de Belle-île en mer[END_REF] that the towers "experience in storms similar oscillations to the swinging of trees". These oscillations result from the "elasticity of the building and the intermittent wind." He evaluated the period of these oscillations at 2 seconds. On the towers he inspected, he observed no damage that could come from these oscillations and concluded that the oscillations caused by storms could not compromise the lighthouses, if of good construction, i.e. if their "absolute stability" was greater than 4.

In 1864, [START_REF] Reynaud | Mémoire sur l'éclairage et le balisage des côtes de France[END_REF] Léonce Reynaud, Director of the Service des Phares et Balises from 1846 to 1878 warned against the effect of oscillations, which had indeed been observed in most lighthouses over 40 m. He observed that, because of the oscillations of the tower, the liquid contained in vases overflowed, the motor weights of the optical machine hit the walls, and a few people suffered effects similar to those on a rolling boat. Reynaud specifies that oscillations do not cause a failure of the towers, nor any "spall in the edges of the stones, opening of joints, cracks on wall facings".

However, he recommends sizing the towers with a safety coefficient of 5 under a wind action of 2750N.m-2 on a flat surface normal to the wind direction (instead of 4 for Fresnel) to prevent these oscillations. It has nothing to do with the resistance of the masonry structures but can affect the correct functioning of the optical machines and the comfort of the keepers.

To sum up, the dynamic effects on tall towers were known to 19 th -century engineers but, without any available formulation or calculation model, they used safety factors between 3 and 5 in their designs. On the occasion of the construction of the Ile Vierge lighthouse, the question of the oscillations was the subject of debates, as will be seen in next section.

WIND ON ILE VIERGE LIGHTHOUSE (a) Calculations in 1897

The stability of the Ile Vierge lighthouse under wind load was calculated by the engineer De Joly, of the Service des Phares et Balises, according to the approach of Léonor Fresnel.

This calculation was checked in 1897 [A11] by the engineer Bourdelles, Director of the Service des Phares et Balises. In Paris on June 29, 1896, De Joly wrote that he was not at all anxious about the stability of the tower after comparing the calculation assumptions with those used by Fresnel and Rankine [A2]. The stability scheme attached to the project [A11] established that the tower was not exposed to any tension stresses, that the maximum compression stress due to the wind load and the self-weight of the masonry (calculated with a density of 2400kg.m -3 ) did not exceed 1.05 MPa, and that the stress on the ground was less than 0.7 MPa, which remains below the limit set by the engineers (1.2MPa [A5]). The "absolute stability" defined by Fresnel and calculated by Bourdelles for the Ile Vierge lighthouse was 10.8 "which should offer all desirable guarantees, especially in comparison with the coefficients of the tallest existing lighthouses".

However, Bourdelles worried about the effect of oscillations that could hamper the operation of the optical apparatus. He contested the opinion that "the dynamic effects of gusts can be calculated by a static model"[A12]. He recalled that "tall lighthouses experience oscillations that have a magnitude large enough to paralyze the optical apparatus during storms."

Based on his observations of existing lighthouses, Bourdelles attributed the good behaviour of the towers under dynamic wind load to the "good quality of the material, which increases their stiffness". For the Ile Vierge lighthouse, he therefore agreed with the use of Portland cement rather than lime, and insisted on "the care that must be taken in the construction of the masonry especially in the upper part of the tower", even voicing the thought that it would be appropriate to "increase the cement ratio in the mortars in this part " [A12]. In response, the engineer Considère, in charge of the work on site, supported this point of view.

At the end of the construction of the Ile Vierge lighthouse, it was decided, at the request of Ribière, Head Engineer of the Service des Phares et Balises, to measure the oscillations of the tower using the oscillograph developed by Richard in 1904 (Fig. 10). A scientific article of 1905 [START_REF] Bonnin | Les oscillations des tours de phare[END_REF] reported that "at the new Ile Vierge lighthouse, the highest in the world and whose tower has a very high stiffness, the oscillograph revealed no perceptible movement in the strongest storms". The period measured thanks to this new equipment was 0.775 seconds [START_REF] Ribière | Oscillations des tours des phares[END_REF]. It was therefore accepted by the engineers that "by giving great stiffness to the towers, the oscillations, even in the strongest storms, can be made imperceptible, as at the Ile Vierge lighthouse".

Figure 10 -Richard's oscillograph [START_REF] Bonnin | Les oscillations des tours de phare[END_REF].

To sum up, in the 19 th century, it was considered that the ratio between the wind pressure (in N.m-2) on a square plate and the square of the wind velocity (in m.s-1) was equal to 1.2 . A coefficient of 1 ⁄ 2 (for Rankine) or 2 ⁄ 3 (for Fresnel) was applied to obtain the wind pressure on a cylindrical tower, whatever its height, diameter and materials. The maximum wind velocity taken into account was 47 m.s-1. In the case of the Ile Vierge lighthouse, 80 m high and with a mean diameter of 9.60 m, the total wind force equalled 1408 kN for Fresnel and 1029 kN for Rankine, while the wind velocity was similar. A safety coefficient of 4 was applied for the verification of stability in order to prevent the optical apparatus from being stopped in case of storms. The design of the tower was therefore linked, not to the masonry strength, but to the proper functioning of the optical apparatus.

(b) Eurocode 1

Reynolds' experiments in 1883 that highlighted the laminar-turbulent transition were not immediately applied to the calculation of lighthouses and chimneys. In 1921, Custodis, the first manufacturer of industrial chimneys in the USA, wrote that the determination of the wind pressure on a chimney remained a controversial subject, because this pressure could only be determined by empirical measurements and no theoretical law on the air flow impacting an object was known [START_REF] Custodis | Custodis chimneys. Radial Brick Chimneys[END_REF]. Therefore, the method used at the beginning of the 20 th century, as in the previous century, to determine the wind force for any cylinder was to apply a correction coefficient to the wind pressure on a flat surface, according to Equation (2) resulting from Smeaton's measurements.

However, as a result of the development of aviation, aerodynamics knowledge developed considerably between 1890 and 1920. Wind tunnel tests developed throughout Europe. Meetings of the most advanced scientists in this field were organized from 1921 in order to standardize their experimental methods and the formulation of equations [START_REF] Knight | Standardization and aerodynamics[END_REF].

Since this work was completed, in 1923, the wind pressure on a structure has been obtained by applying a dimensionless coefficient " " to the dynamic pressure of the wind q, equal to , where is the air density (1.225 kg.m-3) and the wind velocity far upstream of the structure. The standards provide a method for determining . These methods are derived from experimental research that has revealed the parameters involved in the wind pressure on a structure and what was hidden in the Fresnel and Rankine coefficients.

In Eurocode 1-4 [START_REF]NF EN 1991-1-4 and NA[END_REF], the wind force acting on a structure may be determined directly by using expression:

= . . ! . " (5)
Where is the structural factor, is the force coefficient for the structure, ! is the peak velocity pressure and " is the reference area of the structure. To simplify the calculation, the windows and openings in the lighthouse have been neglected. The calculation was carried out with = 80 m and = 9.60 m (mean value). Here, =1 and " = .

(i) Peak velocity pressure

The basic wind velocity corresponding to an average return period of 50 years in the Finistère is # = 26%. & ' and so the basic pressure is # = # = 414). % ' . The mean velocity is * = + # = 40.7 %. & ' where = 1.565 and + = 1. The mean pressure is * = * = 1014). % ' .

The peak velocity pressure formula is ! = .1 + 70 1 2 * where the turbulence intensity is 0 1 (80%) = 0.103 and the standard deviation of velocity 0 1 . * = 4%. & ' . Finally, the peak velocity pressure is ! = 1750). % ' with a peak velocity ! = 53.5%. & ' .

(ii) Force coefficient

For a finite circular cylinder, Eurocode 1 is calculated as (6) (EC1-1-1 Expression 7.19).

= ,+ . (6)
,+ is the force coefficient of cylinders without free-end flow. It depends on the Reynolds number 6 (here 3.10 7 ) and on the equivalent roughness 7 ⁄ . Here 7 = 1mm (granite masonry equivalent to rough concrete). Hence ,+ = 0.9.

The end-effect factor is a function of solidity ratio ( =1) versus slenderness = 2 ⁄ (EC1 -French National Annex). Here = 17, = 0.75 (EC1 Figure 7.36).

Finally

=0.67.

The total wind force in direction of the wind, applied on the Ile Vierge lighthouse is 900kN. For comparison, :; (in Newton) can be expressed by formula [START_REF] Chauris | Construction d'un phare en mer d'Iroise : les pierres Noires (1866-1872)[END_REF], with the height and diameter in metres, and =53.5m.s-1.

:; = 0.41 [START_REF] Chauris | Construction d'un phare en mer d'Iroise : les pierres Noires (1866-1872)[END_REF] This is the static wind action. With EC1 (Annex E), structural vibrations may occur if the frequency of vortexshedding is the same as a natural frequency of the structure. The effect of vortex shedding need not be investigated when the critical wind velocity is higher than 1.25 times the mean wind velocity, which is true in the case of the Ile Vierge lighthouse.

(c) Comparison

Table 1 shows a synthesis of the equations of Smeaton, Fresnel, Rankine, Eiffel and EC1, and the equivalent drag coefficient for cylinders calculated as < = = > ? @AB (SI units. " = ). In both cases, the wind action W is increased by 50% (J = 1.5), which gives an action of 1.5 x 900 = 1350 kN, a value between those of Fresnel and Rankine. In addition, a reduction factor J L, = 0.9 is applied for unfavourable permanent action so as the self-weight here. The stabilizing moment was calculated by De Joly in 1897 taking account of a masonry density equal to 2400kg.m -3 while the real value of masonry density is around 2600kg.m - 3 (granite and kersanton cut stone masonry). So, with regard to the stability of the tower, the old and current calculations present similar values and lead to the same conclusion: no risk of failure.

Moreover, the value of the critical wind velocity in regards of vortex shedding, calculated with EC1, highlights that the Ile Vierge lighthouse does not present a risk of vibratory phenomena, which had been demonstrated by the measurements carried out at the end of its construction using Richard's oscillograph. Actually, the dimensions of the lighthouse were chosen by the engineers for this purpose: to prevent any oscillation that would lead to its shutdown. In today's language: to ensure serviceability.

CONCLUSION

The research presented here highlights the working methods of 19 th century engineers involved in the design and construction of lighthouses in France. They chose the materials and the dimensions of the towers with great care in order to guarantee good progress of the construction, as well as the safety and durability of the structure. The construction of the Ile Vierge lighthouse during the last years of the 19 th century gained from the feedback and the knowledge acquired by the French and English engineers over the century.

With regard to the wind action, the calculation method remained the same throughout the 19 th century. It was based, on the one hand, on the wind pressure measurements presented by Smeaton in England in 1759 and, on the other hand, on the application of a shape coefficient to take the cylindrical shape of the structure into account. Engineers such as the Fresnel brothers, Rankine and Eiffel kept the wind velocity and pressure values presented by Smeaton in 1759, as did all French engineers trained at the Ecole des Ponts et Chaussées in France, and, in particular, the Service des Phares et Balises engineers. The construction of lighthouses and towers throughout the 19 th century finally provided a practical validation, at the scale of the structure, of experimental research carried out in the 18 th century on a reduced sample. The first standards, in France as in England, came into being at the end of the 19 th century, and formalized the wind load that all these engineers were already used to employing.

Research carried out in the 20 th century provided a better understanding of the parameters involved in the wind action. All the lighthouses were designed with high safety coefficients (10 in the case of the Ile Vierge lighthouse) which led to oversizing of the towers. The aim of the engineers was to prevent oscillations appearing that could cause a dysfunction of the optical apparatus. In other words, their goal was to ensure serviceability in all types of weather.

Overall, this historical research shows that the designers of lighthouses in the 19 th century had the same objectives as engineers today: to build resistant, sustainable structures in which deformations under loads remain compatible with serviceability.

FINISTÈRE DEPARTMENTAL ARCHIVES, FRANCE

Figure 2 -

 2 Figure 2 -Ile Vierge lighthouse. (a) Left: aerial view of the island -(b) Right: aerial view of the lantern room and the balcony around its basement wall [A3] [A4].

Figure 3 -

 3 Figure 3 -Ile Vierge lighthouse. The optical apparatus. Left: the original lens [2] -Right: the current lens. The levels are in metres, the 0.00 reference level being the ground at the foot of the tower.

Figure 4 -Figure 5 -

 45 Figure 4 -Plan of the tower [A6]

Figure 8 -

 8 Figure 8 -Construction of the Ile Vierge lighthouse[START_REF] Goffic | Le plus haut phare de France[END_REF]

  The Ile Vierge lighthouse thus became the most powerful lighthouse of the 60 major lighthouses existing on the French coast of the Atlantic Ocean when it was first lit on March 1, 1902,[START_REF] Cunha | Le phare de l'Ile Vierge[END_REF] and it remains the highest stone lighthouse in Europe today. The University of North Carolina, USA, classifies it as "the tallest traditional masonry lighthouse ever built" [3]. It has been listed as a French historical monument since May 23, 2011.

The geometry of the lighthouse was designed by A. Considère and G. Pigeaud, both engineers of the Ecole des Ponts et Chaussées, working for the French lighthouse authority (Service des Phares et Balises) led by L.

  Smeaton (1724Smeaton ( -1792)), AugustinFresnel (1788Fresnel ( -1827) ) and Leonor Fresnel (1786 -1869), Rankine (1820 -1872) andEiffel (1832Eiffel ( -1923)). Their advances in the understanding of wind action were taught at the Ecole des Ponts et Chaussées in France throughout the 19 th century, as part of the "maritime works" course, and implemented by the successive engineers of the Service des Phares et Balises, all trained at this school. They form the basis of the first standard on wind action on structures, established at the end of the 19 th century.

Table 1 -

 1 Comparison of drag coefficientThe total wind forces on the Ile Vierge lighthouse, calculated with Fresnel, Rankine and EC1 are 1408kN, 1029kN, 900kN, respectively. Looking at these figures, one could think that Fresnel and Rankine made more conservative calculations than EC1. However, it is necessary to remind that, with Eurocode, two Ultimate Limit States are to be considered: the EQU Equilibrium Limit State and the STR Strength Limit State.

		Year	Shape	C DEFGHI [SI]	<
	Smeaton	1759	Flat plate			= 1.2
	Fresnel	1831	Cylinder	=	2 3	(1.2 )	1.3
	Rankine	1858	Cylinder	=	1 2	(1.2 )	1.0
	Eiffel	1900	Eiffel tower			= 0.9
	Eurocode 1	1991	Cylinder	=		. . ! . "	0.67

  A1. 25S art. 233. Projet C.3. Phare de l'ile vierge. Modification de l'éclairage. Reconstruction de la tour. Rapport de l'ingénieur ordinaire. Signé Pigeaud le 8 janvier 1897. A2. 25S art. 233 Courrier signé de l'ingénieur ordinaire, G de Joly, Paris, le 29 juin 1896. A3. 36FI68 Photo Archives du Finistère, Photothèque, photo aérienne de l'ile Vierge A4. IC 27 -2012 Photo Archives du Photothèque, photo aérienne de la tête du phare de l'ile Vierge. A5. 25S art. 233. Courrier « transformation du phare de l'ile vierge. Exécution de la décision ministérielle du 22 avril 1896 », signé de l'ingénieur ordinaire, G de Joly, Paris, le 13 juin 1896. A6. 25S art. 231. Elévation et coupe de la tour en maçonnerie. A7. 2FI195/54 Photo de l'extraction du granite sur l'ile vierge. Archives du Finistère, Photothèque --sans date. A8. 25S art. 235. Cahier des charges dressé par l'ingénieur ordinaire à Brest le 8 janvier 1897. A9. 25S art. 233. Rapport de l'ingénieur en chef, M. Considère Quimper, le 03 novembre 1896. A10. 25S art. 233. Projet n°3. A.3. Phare de l'ile vierge. Modification de l'éclairage. Reconstruction de la tour. Détail estimatif des travaux. Signé Pigeaud, Brest le 8 janvier 1897. A11. 25S art. 233. Extrait du rapport de l'inspecteur général, directeur du service des phares et balises en date du 13 février 1897. Stabilité de la tour. A12. Ministère des travaux publics, Phares et Balises, Finistère, phare de l'ile vierge. Reconstruction. Courrier de M. Bourdelles à M. Considère, Paris le 17 mars 1897.


D Bouchet

D Cao

R Carminati

Y De Wilde

V Krachmalnicoff

Long-Range Plasmon Assisted Energy Transfer Between Fluorescent Emitters

Keywords: numbers: 78.67.-n, 42.25.Bs, 33.50.-j

We demonstrate plasmon assisted energy transfer between fluorophores located at distances up to 7 µm on the top of a thin silver film. Thanks to the strong confinement and large propagation length of surface plasmon polaritons, the range of the energy transfer is almost two orders of magnitude larger than the values reported in the literature so far. The parameters driving the energy transfer range are thoroughly characterized and are in very good agreement with theoretically expected values.

Coupling of fluorescent emitters has been a topic of major interest in research over the past decades. The relevance of this subject stems from its many possible exploitations. In the domains of chemical-physics and life science, short-range dipole-dipole interaction (Förster Resonant Energy Transfer, FRET) between fluorophores has been widely employed to study phenomena occurring over distances of the order of 10 nm [1][START_REF] Medintz | FRET -Förster Resonance Energy Transfer: From Theory to Applications[END_REF][START_REF] Ha | [END_REF].

Coupling between fluorophores can be tailored by shaping their electromagnetic environment. Together with the emergence of nanophotonics, several studies about the influence of a nanostructured environment on fluorescent emission have come to light [4]. In particular, it has been demonstrated that metallic structures can lead to the control of the rate, the intensity and the directionality of fluorescent emission [5,6], and single photon generation of single plasmons has been observed and studied [7][8][9]. Plasmonic structures are attractive because, due to the high confinement of plasmon modes, they can ensure a very efficient coupling between the emitter and the environment, as demonstrated by the observation of high Purcell factors [10,11]. Furthermore, plasmons can propagate over distances going from a few microns to several millimeters, depending on the material and the wavelength.

While the interaction between plasmons and one fluorescent emitter has already been widely studied, the influence of a plasmonic environment on the coupling between different fluorophores has been explored only in very specific situations involving FRET [12][13][14][15][16][17][18][19][20][21]. In this regime, dipole-dipole interaction restricts energy transfer to distances of a few nanometers. In this Letter, we study the interaction between fluorophores in a plasmonic environment from a different perspective. We take benefit of the confined propagation of surface plasmons polaritons (SPP) along the interface between a metal and a dielectric to realize the transfer of energy between fluorophores on distances which are much larger than what would be allowed by FRET. We report on the observation of plasmon assisted energy transfer between ensembles of fluo-rescent molecules lying on top of a continuous silver film over distances up to 7 µm. The first ensemble (donor) is excited by a laser and then decays by launching a SPP propagating on the top surface of the metallic layer. The energy carried by the plasmon is then absorbed by the second ensemble of fluorophores (acceptor) embedded in a thin polymer layer deposited on top of the metal. Fluorescence photons emitted by the acceptor are the signature of the plasmon assisted energy transfer between the donor and the acceptor. On the basis of a simple theoretical model we estimated the energy transfer efficiency to be enhanced by the presence of the plasmonic film by up to a factor of 30.

The sample, shown in Fig. 1a, is made of a 50 nm thick silver layer deposited on a glass coverslip and then covered by a layer of SiO 2 (thickness 10 nm) which protects the metal against oxidation and prevents quenching of the fluorophores by the metal. An aqueous solution containing polyvinyl alcohol (PVA), acceptor molecules (Atto665, AttoTec Gmbh) and donor polystyrene fluorescent beads (Red FluoSpheres, 100 nm diameter, Invitrogen) was then spin-coated on the silica layer. The parameters of the spin-coating process and the concentration of acceptors and donors in the initial solution are chosen in order to have a few isolated donor beads, surrounded by a homogeneous layer of acceptors, about 50 nm thick. The distance between two close-by donors is such that a single donor bead can be addressed easily with the excitation laser. Care has been taken that the donor emission spectrum and the acceptor absorption spectrum have a good overlap, which is needed for the energy transfer to occur. Moreover, since the aim is to realize the transfer via a SPP, the spectral overlap of the donor-acceptor pair has been chosen such that it occurs in a region of the spectrum for which the plasmon can propagate over distances up to several microns.

In the experiment, we excite the donor bead with a supercontinuum pulsed laser (Fianium SC450, repetition rate 10 MHz), filtered at 532 nm (filter bandwidth 20 nm). The excitation laser, as shown in Fig. 1, is focused on the bead over a spot of 1.5 µm by means of a micro-lensed fiber (Nanonics Imaging Ltd., working distance 4 µm), mounted on a three axis piezoelectric nanopositioner. Efficient collection of acceptor fluorescence is ensured by an oil immersion microscope objective (×100, N.A.=1.4), located below the sample (see Fig. 1b). Fluorescence photons are then filtered by a dichroic mirror and detected either by a fibered spectrometer (Acton SP2300, Princeton Instruments, fiber core diameter 50 µm) or, after passing through a confocal pinhole (diameter 50 µm), by an avalanche photodiode (MPD PDM-series) in order to perform decay rate measurements (acquisition board PicoQuant Hydra-Harp400). In either case, the signal which is measured arises from a 500 nm sized region of the sample which is optically conjugated with the detection system. The use of the microlensed fiber allows us to separate the excitation and the detection regions by a very well controlled distance d going up to several microns. Indeed, the position of the microscope objective is fixed while both the sample and the microlensed fiber can be translated. To measure the acceptor fluorescence as a function of d, we move the donor and the microlensed fiber together by a distance d with respect to the microscope objective. Fluorescence of the donor collected through the microlensed fiber is monitored on an avalanche photodiode and maximized for each distance. An EM-CCD camera allow us to record wide-field images of the sample. Fluorescence emission spectra of acceptors only on silver and of a donor only on silver are reported in Fig. 1c in red and blue respectively. Moreover, in order to check the influence on the donor of the presence of acceptors in its proximity, we compared the fluorescence decay of the donor on silver in the presence and in the absence of the acceptors, when the excitation spot coincides with the fluorescence collection region (i.e. d = 0 µm in Fig. 1b). This measurement is reported in Fig. 1d and shows that the decay rate of the donor in the presence of the acceptors (red data) is larger than without acceptors (blue data) by a factor of 1.4 (see [23] for complementary information). This effect is due to the dipole-dipole interaction between the donor and the acceptors surrounding the bead, which is responsible for a short-range energy transfer (FRET) occurring between molecules located at distances of the order of ten nanometers.

In order to characterize the propagation of the plasmon launched by the donor on the silver film, we monitored the radiative losses of the plasmon for several distances d up to 6 µm. The sample used is made as the one previously described except that the acceptor molecules are absent in the PVA layer. Experimental results are shown in Fig. 2. The wavelength dependence of the dielectric constant of silver induces a dependence of the propagation length of the plasmon on the wavelength. Since longer wavelengths of the spectrum propagate over longer distances, the plasmon spectrum is red-shifted during the propagation. This appears clearly in Fig. 2a that shows, in blue, the spectrum of the donor at d = 0 µm and, in black, the spectrum of plasmonic radiative losses detected at d = 5 µm. In order to confirm the origin of the observed red-shift, we fit the experimental spectra, for several distances d, with a function given by the product between the spectrum of the donor on silver at d = 0 µm and the decreasing exponential decay behavior of the plasmon propagating along the interface between a semi-infinite silver layer and a semi-infinite PVA layer (ǫ = 2.25). The dependence of the dielectric constant of the silver film on the wavelength is taken into account [22], on the basis of the Drude model with five interband transitions. The result of the fit for d = 5 µm is the red curve plotted in Fig. 2a, which is in very good agreement with the experimental data. We have verified that a good agreement is found when fitting the data for distances up to 6 µm. At larger distances the signal to noise ratio in the spectroscopic measurements is of the order of unity which prevents one to perform any reliable analysis of the data. The spectrally averaged propagation distance of the SPPs excited by the donor, l D , can be estimated from the experimental spectra recorded as a function of the distance d. Fig. 2b shows a plot of the total intensity of plasmonic radiative losses as a function of d together with a fit (solid line in Fig. 2b) using the function A/d exp(-d/l D ), where A is a normalization constant.

The fit results in a propagation length of 3.4 ± 0.1 µm. This is slightly smaller than the theoretically expected value l th D = 3.8 µm estimated by calculating the average of the SPP propagation length over all the donor fluorescence spectrum range, weighted by the fluorescence intensity at each wavelength. The slight difference is likely due to the theoretical estimation in which losses due to surface roughness are not accounted for.

After characterizing the propagation of the plasmon launched by the donor, we studied the occurrence of the energy transfer between the plasmon and the acceptors embedded in the PVA film. For that purpose, we measured the spectrum at different distances d from the donor position on the sample sketched in Fig. 1a. To have a thorough understanding of the measurements, the various processes that could contribute to the signal collected at distance d from the donor must all be appropriately considered. First, due to the finite size of the excitation laser spot (diameter of 1.5 µm) acceptors located in the proximity of the donor can be directly excited by the laser. This effect has been minimized by blue detuning the excitation laser with respect to the accep- tor absorption spectrum, but it is still present and acts as a background that has to be subtracted from our measurements. Second, acceptors that are directly excited by the laser, may also excite a SPP. As in the case of the plasmon launched by the donor, radiative losses by the plasmon launched by the acceptors can be detected at a distance d. Their spectrum will overlap with the acceptor fluorescence peak, by artificially increasing the signal to be measured. Third, since acceptors are present in the vicinity of the donor, short range energy transfer (FRET) occurs. Acceptors excited by this mechanism can decay by launching a SPP with a spectrum overlapping with the acceptor fluorescence spectrum, that can be detected at a distance d due to radiative losses. Since the contribution of the FRET signal is negligible with respect to the signal emitted by the bead, we neglected this effect.

In order to account for the contribution of the first and the second processes, we measured, for each distance d, the signal produced by the acceptors due to direct laser excitation, in the absence of the donor. This is done by shifting the sample such that only acceptors are excited by the laser. This background signal is subtracted from the signal measured in the presence of the donor. The result is shown in Fig. 3 (black data) for several distances d. The peak centered at 610 nm is due to radiative losses of the SPP launched by the donor, while the peak centered at 695 nm is the fluorescence of the acceptors excited by the plasmon. The presence of the second peak is the unambiguous signature of the occurrence of energy transfer through the SPP excitation. For comparison, a control sample made of donors and acceptors on a bare glass substrate shows only short range energy transfer (FRET). The curve shown in red is a fit to the experimental data by means of a function that is the sum of the spectrum of the plasmon excited by the donor as measured on the sample without acceptors (see Fig. 2a) and of the spectrum of acceptor fluorescence on silver (shown in Fig. 1c). The fit function has only two free parameters that are normalization constants. The good agreement between the fit result and the experimental data reinforces the conclusions. A second proof of the observation of plasmon mediated energy transfer over distances of up to 7 µm is given by the graph shown in red in Fig. 4, where we report the integral of the energy transfer spectra, after subtraction of the plasmon losses signal, as a function of the distance from the donor. As done in Fig. 2b, data are fitted with a decreasing exponential function to take into account the exponential decay of the plasmon propagation as a function of the distance. This allows us to measure the energy transfer range l et = 5.4 ± 0.9 µm. We checked that, for different donors on the same sample and for different acceptor concentrations, the energy transfer range is consistent with the data shown in the manuscript.

At first sight, it might be surprising to measure l et > l D . However this can be understood by taking into ac-Intensity (a.u.) count the wavelength dependence of the plasmon energy absorbed by the acceptors. Indeed, if we measure l D by taking into account only the spectral region around the maximum of the acceptor absorption spectrum (i.e. 635 nm ≤ λ ≤ 680 nm), we find a propagation length of 5.3 ± 1 µm. This result agrees with the value of l et . Data are plotted in blue in Fig. 4 together with the fit result (blue solid lines). This is a further evidence of the fact that acceptors are excited via the SPP launched by the donor.

The energy transfer rate is defined as the enhancement of the donor decay rate induced by the presence of the acceptors. In the present realization of the experiment, a reliable measurement of the energy transfer rate at long distances is not possible, due to the occurrence of FRET between the donors and the surrounding acceptors. Therefore, we theoretically estimated the plasmon assisted energy transfer efficiency by calculating the Green function at the interface between two semi-infinite media (PVA and silver). As it is reported in the supplemental material, the presence of the metallic film induces an enhancement of up to a factor of 30 with respect to the energy transfer occurring in a homogeneous medium (n = 1.5) [23]. In conclusion, we have demonstrated the occurrence of SPP assisted energy transfer between fluorescent emitters located at distances of up to 7 µm on the top of a thin silver film. This observation was supported by the measurement of the propagation distance of the SPP on silver, measured on an independent sample and by a very good agreement with the theoretically expected value.

The transfer range which we demonstrate is 60 times larger than the only value reported in the literature so far on plasmon mediated energy transfer between fluorescent emitters [24]. In [24], the authors studied the energy transfer between two layers of molecules located on both sides of a silver film, a configuration that differs from the one studied in this Letter. The maximum thickness of the film, fixing the maximum distance of the energy transfer, was 120 nm.

This work is the first step towards the observation of a plasmon assisted coupling between fluorophores, and opens rich perspectives. Indeed, a more efficient, long-range energy transfer can be reached by shaping the plasmonic support. Energy transfer efficiency can be enhanced by several orders of magnitude by using structured plasmonic supports, such as hybrid plasmondielectric waveguides [25] and could allow the demonstration of efficient long-range energy transfer between single photon emitters. The realization of optimized substrates for the observation of the energy transfer would open rich perspectives in several domains that are concerned by inter-molecular coupling, spanning from biology to physics. In particular, the use of recently developed techniques for the accurate nanopositioning of fluorescent emitters in the near-field of nanostructured samples [26,27] will be required for studies of optimized structures towards the realization of devices.

FIG. 1 .

 1 FIG.1. a) Sketch of the sample. b) Scheme of the experimental setup. MLF: micro-lensed fiber, D: donor, A: acceptor. c) Fluorescence spectrum of the donor only (blue) and of the acceptor only (red) on the top of a 50 nm thick silver film with a 10 nm silica spacer. d) Fluorescence decay rate of a donor bead on the silver film in the presence (red) and in the absence of the acceptors (blue).

FIG. 2 .

 2 FIG. 2. a) Normalized spectra of the radiative losses of the plasmon launched by the donor due to scattering by surface roughness. Blue and black curves: experimental data for d = 0 and d = 5 µm respectively. Red curve: fit to the experimental data at d = 5 µm taking into account the dependence, on the wavelength, of the dielectric constant of silver. b) Scattered intensity as a function of the distance d. Black points: experimental data, black solid line: fit to the experimental data (see text for details). Error bars have been calculated by taking into account the statistical error on the entries in each bin of the spectrum histogram.

FIG. 3 .

 3 FIG.3. Black points: Spectra measured at a distance d from the donor ranging from 2 µm to 5 µm, with an integration time of 200 s. The red curve is the result of a fit of the experimental data with a function which is the sum of the acceptor fluorescence spectrum and the plasmon spectrum.

FIG. 4 .

 4 FIG.4. Red: Intensity of the acceptor fluorescence as a function of the distance d from the donor (dots) and expected theoretical dependence (solid line). Blue: Intensity, as a function of the distance d, of the plasmon launched by the donor integrated over the region of the spectrum that overlaps with the acceptor absorption, i.e. 635 nm ≤ λ ≤ 680 nm, (dots) and expected theoretical dependence (solid line). Error bars have been calculated by taking into account the statistical error on the entries in each bin of the spectrum histogram.

The authors thank E. Fort and A. Cazé for stimulating discussions, S. Gresillon and O. Loison for helping in sample preparation and A. Souilah for technical support. This work was supported by LABEX WIFI (Laboratory of Excellence ANR-10-LABX-24) within the French Program Investments for the Future under reference ANR-10-IDEX-0001-02 PSL*, by the Region Ile-de-France in the framework of DIM Nano-K and by the Programme Emergences 2015 of the City of Paris.


Guillaume Brysbaert

Ralf Blossey

Marc F Lensink
email: marc.lensink@univ-lille.fr

The inclusion of water molecules in residue interaction networks identifies additional central residues

Keywords: water, solvation, protein surface, protein complex, residue interaction network, centrality analysis, central residues

The relevance of water molecules for the recognition and the interaction of biomolecules is widely appreciated. In this paper we address the role that water molecules associated to protein complexes play for the functional relevance of residues by considering their residue interaction networks (RINs). These are commonly defined on the basis of the amino acid composition of the proteins themselves, disregarding the solvation state of the protein. We determine properties of the RINs of two protein complexes, colicin E2/Im2 and barnase/barstar, with and without associated water molecules, using a previously developed methodology and its associated application RINspector. We find that the inclusion of water molecules in RINs leads to an increase in the number of central residues which adds a novel mechanism to the relevance of water molecules for protein function.

INTRODUCTION

Water plays an undeniably important role in all biological processes [START_REF] Bellissent-Funel | Water determines the structure and dynamics of proteins[END_REF]). Besides giving rise to phenomena such as the hydrophobic effect, which leads to self-assembly of nonpolar or amphipathic molecules such as lipids [START_REF] Chandler | Interfaces and the driving force of hydrophobic assembly[END_REF]), water molecules also solvate protein structures. Their polarized nature allows them to bond to both positively as well as negatively charged molecules or molecular groups on the surface of the protein. At the short range this leads to hydrogen bonding and the formation of a hydration network around the protein. For both simple polypeptides and proteins, this network has been shown to exhibit slower dynamics and stronger hydrogen bonding [START_REF] Czapiewski | Structural properties of hydration shell around various conformations of simple polypeptides[END_REF]; [START_REF] Dutta | Water dynamics at protein-protein interfaces: molecular dynamics study of virus-host receptor complexes[END_REF]), resulting in slower exchange with bulk water. For enzymes in particular, the network exhibited coupled protein-water motion near the active site and allowed exposure of said active site to bulk solvent [START_REF] Nakasako | Large-scale networks of hydration water molecules around bovine beta-trypsin revealed by cryogenic X-ray crystal structure analysis[END_REF]; [START_REF] Grossman | Correlated structural kinetics and retarded solvent dynamics at the metalloprotease active site[END_REF]; [START_REF] Xu | Water's potential role: Insights from studies of the p53 core domain[END_REF]).

It can therefore be hypothesized that water plays an active role in channeling substrate to the designated binding pocket once it reaches the protein's surface. An active role for water has indeed been established in substrate binding to protein O-fucosyltransferase 2, where the binding mechanism involves a dynamic network of watermediated interactions [START_REF] Valero-Gonz Ález | A proactive role of water molecules in acceptor recognition by protein O-fucosyltransferase 2[END_REF]). Also for non-enzymatic substrate binding water can play an essential role, as evidenced by the crystal structure of the bacterial adhesin FimH, where a water molecule is found completely buried between the ligand and its binding pocket, engaging in hydrogen bonding to three protein residues and a hydroxyl group of the mannose ligand [START_REF] Wellens | Intervening with urinary tract infections using anti-adhesives based on the crystal structure of the FimH-oligomannose-3 complex[END_REF]; [START_REF] Touaibia | Sites for dynamic protein-carbohydrate interactions of O-and C-linked mannosides on the E. coli FimH adhesin[END_REF]).

At larger distances, the water molecules are producing an average and fluctuating electrostatic field, which shields any long-range interaction. The dynamic properties of water then result in so-called electrostatic screening and effectively reduce the electrostatic potential of charged particles, including proteins [START_REF] Schutz | What are the dielectric "constants" of proteins and how to validate electrostatic models?[END_REF]), which has a strong modulating effect on their initial interaction. However, water has been shown to play an active role also in the association of enzyme/inhibitor protein pairs, exemplified by the high-affinity barnase/barstar complex and known for its strong electrostatic binding [START_REF] Buckle | Protein-protein recognition: crystal structural analysis of a barnase-barstar complex at 2.0-å resolution[END_REF]). A detailed investigation by molecular dynamics simulation has shown that the interaction is characterized by a water-rich binding funnel, where anisotropically oriented water molecules participate in the association between both protein partners [START_REF] Ahmad | Adhesive water networks facilitate binding of protein interfaces[END_REF]). The final binding pose, as represented by the crystal structure, shows a considerable number of water molecules at the interface, with a third of them being completely buried.

The important role of water in biomolecular recognition is by now widely acknowledged (Ben-Naim ( 2002)), and protein-protein and protein-ligand interfaces provide a rich catalogue of examples. As listed above, these range from specifically placed water molecules in binding pockets to networks of more or less mobile interface waters, which are more often than not actively participating in the binding mechanism and/or thermodynamics [START_REF] Li | Water at biomolecular binding interfaces[END_REF]; [START_REF] Ahmad | Adhesive water networks facilitate binding of protein interfaces[END_REF]; [START_REF] Dutta | Water dynamics at protein-protein interfaces: molecular dynamics study of virus-host receptor complexes[END_REF]; [START_REF] Chong | Dynamics of hydration water plays a key role in determining the binding thermodynamics of protein complexes[END_REF]). The more specific the positioning of individual water molecules becomes, the more relevant will be their role in the precise functioning of proteins and their complexes. The computational prediction of water positions has already been an important issue in proteinligand interaction [START_REF] Boobbyer | New hydrogen-bond potentials for use in determining energetically favorable binding sites on molecules of known structure[END_REF] 2016)), but also the modeling of the hydrogen-bonded network around proteins in light of protein-protein interaction has seen an increased interest [START_REF] Jiang | A "solvated rotamer" approach to modeling watermediated hydrogen bonds at protein-protein interfaces[END_REF]; [START_REF] Bui | WATGEN: an algorithm for modeling water networks at protein-protein interfaces[END_REF]; [START_REF] Copie | On the ability of molecular dynamics simulation and continuum electrostatics to treat interfacial water molecules in protein-protein complexes[END_REF]). More recently, the accurate prediction of the hydrogen-bonded network in the interface region has been the topic of several CAPRI blind prediction trials [START_REF] Lensink | Blind prediction of interfacial water positions in CAPRI[END_REF][START_REF] Lensink | Modeling protein-protein and protein-peptide complexes: CAPRI 6th edition[END_REF]).

Considering the fact that the water molecules around proteins and at the interface of protein complexes form a network of more or less immobile entities, we can in fact integrate them in a graph-based representation of the underlying protein structure(s). The approach we use here is that of Residue Interaction Networks (RINs), where a node represents a residue and an edge a detected interaction. Depicting protein residues as a network, we can then consider immobile water molecules as an extension of the latter. The use of a network-based approach as a representation of a protein tertiary structure allows the application of graph analysis tools such as centrality analyses. Residues identified by such an approach have been shown to carry importance for function or fold [START_REF] Amitai | Network analysis of protein structures identifies functional residues[END_REF]; [START_REF] Greene | Protein structure networks[END_REF][START_REF] Del Sol | Residues crucial for maintaining short paths in network communication mediate signaling in proteins[END_REF]). The question we address in this paper is how the inclusion of protein-associated water molecules affects the functional importance of protein residues. By employing a tool we recently developed (Brysbaert et al. (2018a,b)), we consider the residue interaction networks of exemplary protein complexes under inclusion of surface and interfacial water molecules. Specifically, we determine what effect the water molecules, when considered as nodes in these networks, have on centrality calculations of protein residues, i.e., whether and in what way they modify the results obtained for the waterless proteins, in order to gain additional insight into the biological role they play for the relevance of protein residues.

METHODS

Structures

Our analyses are based on the colicin E2 DNase/Im2 complex (PDB ID: 3U43) and on the barnase/barstar complex (PDB ID: 1BRS, chains B,E). These structures contain positions of water molecules identified by X-ray diffraction that we considered in the generation of Residue Interaction Networks (RINs). The complexes were visualized with UCSF Chimera 1.11.2 [START_REF] Pettersen | UCSF chimera -a visualization system for exploratory research and analysis[END_REF]). We defined residues at the interface as those that have any atom within 7 Å of any atom of the other chain. We deliberately decided to be more stringent than the oftenapplied CAPRI definition of 10 Å [START_REF] Lensink | Blind predictions of protein interfaces by docking calculations in capri[END_REF]; [START_REF] Lensink | Blind prediction of interfacial water positions in CAPRI[END_REF]) to avoid considering residues too far from the surface where the water molecules are essentially located and to limit the inclusion in the interface of water molecules that are not shared between the two chains. We then consider a water molecule to be included in the interface if it is connected to at least one interface residue in the RIN (see below).

Residue Interaction Networks

We generated three residue interaction networks (RINs) for both complexes mentioned above: one without water molecules, one with water molecules and one with water molecules only at the interface (as defined above, i.e. only water molecules that are connected to residues at the interface are conserved). Residue Interaction Networks are defined here as contact networks were nodes are residues or water molecules and edges are detected contacts.

They were generated with an in-house C program considering a residue-residue contact when the distance between any atom pair of both residues was found below 5 Å, while water-water and water-residue contacts were detected for a distance below 3.5 Å. The networks were visualized with Cytoscape 3.6.1 [START_REF] Shannon | Cytoscape: a software environment for integrated models of biomolecular interaction networks[END_REF]). We also created RINs for these structures by considering identical distance thresholds for all contacts (water and residues), namely 3.5 Å.

Centrality Analyses

We computed Residue Centrality Analyses (RCAs) and Betweenness Centrality Analyses (BCAs) on all the RINs with the RINspector app Brysbaert et al. (2018a) for Cytoscape using its automation functionalities [START_REF] Brysbaert | Comparing protein structures with rinspector automation in cytoscape[END_REF]. Each analysis produces a Z-score for each residue. We considered as central a residue with a Z-score ≥ 2. RCA and BCA are both based on shortest path lengths calculations in the network. While RCA evaluates the effect of each node removal on the average shortest path length of the network, BCA evaluates how often a node is crossed by shortest paths. For RCA, the higher the Z-score, the higher the effect on the average shortest path length, i.e. the higher the effect on the communications inside the network/structure. For BCA, the higher the Z-score, the more the node is crossed by shortest paths compared to the other nodes, i.e. the higher the control this node has on the interactions of other nodes in the network.

RESULTS AND DISCUSSION

We first discuss the results we have obtained for the colicin E2 DNase/Im2 complex, depicted in Figure 1. The

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molecules. But those water molecules do seem to have an effect on the centrality of the protein residues, as the right-hand network shows a significant number of additional central residues (existing ones are not shown in the image). The additional ones are summarized in Table 1. The Table presents the central residues that are added in the 'wet' vs. 'dry' network, with residues at the protein-protein interface shown in blue. We find that the inclusion of water molecules in the generation of the network leads to an increase in the number of central residues, both for RCA as well as BCA. In particular, when focusing on the interface of the complex and considering the union of the two methods, we obtain eight additional central residues (see Table caption for details), fairly evenly distributed over both chains.

In [START_REF] Lensink | Blind prediction of interfacial water positions in CAPRI[END_REF]; [START_REF] Copie | On the ability of molecular dynamics simulation and continuum electrostatics to treat interfacial water molecules in protein-protein complexes[END_REF]), it was shown that Arg98.B contacts Glu30.A and Asn34.A through a water-mediated contact and Asp33.A via a buried water molecule. Here, we see that while Glu30.A was the only central residue found in the centrality analysis performed on the 'dry' RIN, Arg98.B and Asp33.A are found in addition to Glu30.A in the 'wet' centrality analysis. Likewise, Tyr55.A has a water-mediated contact with Gln70.B, but only Tyr55.A was found as central without water, while with water we also identify Gln70.B. In addition, we find in the 'wet' RIN that a water-mediated contact Arg24.A/Lys81.B is found as central while it is not identified at all in the 'dry' RIN.

Only three water molecules have a relevant Z-score. All three are located at the bottom of the protein complex, far from the interface. They are only found in RCA and their score is high only because they are on the only path to connect to other water molecules. They are located near three central residues that are lost in the 'wet' vs. 'dry' calculation and probably represent a shift in weight in this region. The lost central residues are His92.A, His93.A and His94.A, which are all three part of the His-tag of chain A. We therefore consider them irrelevant and removed them from the Table .    The fact that water molecules have low Z-scores is likely due to their presence at the surface of the molecule and therefore in the periphery of the network. Consequently, these nodes are not crucial for maintaining shortest paths between other nodes in the network. Even at the interface between the two molecules, contacts between the residues of the two molecules are sufficiently numerous to maintain shortest paths, resulting in Z-scores of water molecules below 2.

In order to assess the importance of interface waters vs. all water molecules, we then created the same type of RIN, but added only the interfacial water molecules, defined as those molecules that possess at least one link with an interface residue. The centrality analyses produced a subset of the previously found additional interface residues, with two additional ones: Gln92.B and Glu97.B. With respect to the additional residues from the 'wet' calculation however, now four residues are lost: Asp33.A, Asp62.A, Arg57.A and Arg24.A.

Adding only interfacial water molecules to the network adds significant weight to the network in this region. Its implementation can be argued by considering the fact that interfacial waters would typically be the least mobile ones and are hence the first candidates to extend the local interaction network. However, in our point of view it is of not more than anecdotal interest; those waters are found in a tightly bound region of the system and add few direct links between both protein chains. Those links that are added either result in a shift of central residues or provide a certain redundancy to the network that might result in the elimination of residues that are otherwise considered central.

We further wanted to investigate the variation of the Z-scores between the 'wet' and 'dry' RINs. We therefore calculated ∆Z = Z wet -Z dry for all residue nodes. The mean values are µ(∆Z allresidues ) = 0.5071 with σ = 0.869 for the RCA calculation, and µ = 0.704 (σ = 0.453) for the BCA calculation. The Z-scores of the residues are globally shifted to higher values when adding water. This increase is compensated by the low Z-score values observed for water nodes for which we have µ(Z wet ) = -0.366 (σ = 0.521) for RCA and µ(Z wet ) = -0.509 (σ = 0.121) for This is a provisional file, not the final typeset article BCA. Focusing on the added central residues (excluding water molecules since they are absent from the 'dry' RIN), Supplementary Table S1 presents their ∆Z values and the comparison to the RCA and BCA mean values µ(∆Z allresidues ). The table shows that the central residues added by the 'wet' RIN exhibit ∆Z values that are significantly higher than the average ones (by 1.20 and 0.62 standard deviation for RCA and BCA, resp.), meaning that this observation of new central residues is not only due to the global shift of Z-scores of residues compensating for low-Z water nodes.

The addition of water molecules to the RIN leads to very few water molecules that show a relevant Z-score, none of them located at the protein-protein interaction interface. Since this effect could be due to the fact that we considered a maximum residue-residue contact distance of 5 Å, whereas for water contact detection it was set to 3.5 Å, we additionally performed the same analysis after generating additional 'wet' RINs using a 3.5 Å overall contact distance. However, those results gave no water molecules with a relevant Z-score, excluding the distance threshold as a determining factor for the detection of water molecules as central nodes in the network.

Finally, we wanted to see the impact of removing water molecules at the interface on the centrality of residues.

We considered only those at the interface, because some of them are buried compared to all the other ones that are exposed to the solvent and peripheral in the RINs. Because Z-scores are very low for these molecules (≤ 0.3), we decided to focus on the degree of the water nodes. Starting from the 'wet' RIN, we removed water nodes in three ways: (i) only the single one with maximum degree (8 edges), (ii) the ones with a degree ≥ 5 (13 water nodes), (iii) the 3 water molecules that were considered as conserved in the protein family of the structure [START_REF] Wojdyla | Structure of the ultra-high affinity colicin e2 dnase-im2 complex[END_REF]). In all cases, the central residues were exactly the same as the ones with all the water nodes, both for RCA as well as BCA, denoting a global effect of water on the network rather than a specific effect of some water molecules.

To confirm our results, we performed the same analysis with the barnase/barstar complex (PDB ID: 1BRS, chains B and E). The results are shown in Supplementary Figure 1 and Supplementary Table S2. Also here, we find that the inclusion of water nodes in the network adds central residues. This time, only one residue is lost, incidentally not located at the interface. When focusing on the interface between the two chains, we again do not find central water nodes, even when applying the same distance criterion for residue contacts and water contacts. As for the E2/Im2 complex, we created a third RIN considering only the water molecules at the interface. Centrality analyses gave again a subset of the residues listed in Supplementary Table S2, when comparing to the 'dry' RIN, with only one additional residue found, Arg59.B, while Asn58.B is lost. We also compared the shift of Z-scores for the added central residues, which once again are higher, with a mean of µ = 1.328 for RCA compared to µ = 0.274 (σ = 0.738), µ = 0.764 for BCA, compared to µ = 0.517 (σ = 0.434), concluding that also here the shift in Z-scores is significant.

In order to confirm the relevance of considering water molecules for RIN generation and centrality analyses, we have focused on the additionally identified central residues. We compared our findings to the literature and listed the discussed central residues in Table 2. For the E2/Im2 complex, we identified Asp33 of Im2, located at the interface. [START_REF] Wojdyla | Structure of the ultra-high affinity colicin e2 dnase-im2 complex[END_REF] have shown that the specificity at the interface is organized by Asp33 in helix II of Im2, that these interactions involve water molecules and that a mutation into alanine destabilizes the complex [START_REF] Li | Highly discriminating proteinprotein interaction specificities in the context of a conserved binding energy hotspot[END_REF]). In addition, the two hotspot residues Arg38 and Tyr55 of Im2, already identified as central in the absence of water, show both an increased Z-score when adding water. For the barnase/barstar complex, two of the additionally identified residues are Trp44 (barnase) and Asp35 (barstar). [START_REF] Wang | How optimal are the binding energetics of barnase and barstar?[END_REF] have shown by computational analysis that the mutation of Trp44 into Tyr, Glu or Asp would have a favorable effect on electrostatic binding free energies compared to wild-type. Opposing to this, the mutation of Asp35 into Ala was found to destabilize the complex [START_REF] Buckle | Protein-protein recognition: crystal structural analysis of a barnase-barstar complex at 2.0-å resolution[END_REF]; [START_REF] Schreiber | Energetics of protein-protein interactions: analysis of the barnase-barstar interface by single mutations and double mutant cycles[END_REF]; [START_REF] Wang | How optimal are the binding energetics of barnase and barstar?[END_REF]). [START_REF] Hartley | Directed mutagenesis and barnase-barstar recognition[END_REF]) showed that a mutation of Asp35 into Lys produces no detectable activity and it is involved in the mutual recognition of the two proteins. Analogous to this, the Lys27Ala and Arg87Ala mutations (both on barnase) have been shown to destabilize

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Inclusion of water molecules in residue interaction networks the complex [START_REF] Buckle | Protein-protein recognition: crystal structural analysis of a barnase-barstar complex at 2.0-å resolution[END_REF]). Both residues were already identified in centrality analyses performed in absence of water molecules, but their Z-scores increase upon inclusion of water molecules. All the central residues found from the literature as key residues for the function of the complexes are retrieved by BCA when considering water, while RCA finds only part of them. Generally, in disconnected networks RCA is expected to give better results than BCA since the latter is more sensitive to breaks in the network. This trend is counteracted by the inclusion of water molecules.

CONCLUSION

It has been shown that water plays an important role in the binding of macromolecules, mainly due to its electrostatic nature, resulting in a lowered dynamics of "surface-bound" water molecules. In this paper we have shown that the inclusion of such water molecules in the generation of residue interaction networks increases the number of central residues, without a significant loss of any. Our results for two enzyme/inhibitor protein complexes (colicin E2/Im2 and barnase/barstar) are based on the use of two, residue and betweenness, centrality measures. This finding is promising because, as generally with central residues, these additional residues might carry importance for the function or structural stability of the system or interacting molecules. It is also reassuring from the point of view of the RIN methodology, since the majority of central residues are also found when including water molecules in the network; they are hence robustly defined.

Water molecules themselves are found to exhibit low Z-scores, also those located in the interface region of the protein complex, with only few exceptions, while protein residues show a globally positive shift in Z-score values.

While the effect of the included waters on the centralities is not specific at the level of single water molecules, the added central nodes are significant and not due to a global shift of Z-scores of protein residue centralities. These modifications of the centrality scores indicate that certain functionally relevant protein residues may be missed in the RIN analysis by only considering 'dry' RINs, i.e. excluding water molecules (like Asp33 of Im2 in the E2/Im2 complex or Asp35 of barstar in the barnase/barstar complex). The low significance of the water molecules themselves might indicate at a relative unimportance for an "exact" positioning of water molecules. It might therefore be of interest to extend the methodology of RIN generation and centrality analyses, which was here based exclusively on waters from crystal structures, to predicted waters [START_REF] Jeszen Öi | Mobility-based prediction of hydration structures of protein surfaces[END_REF]), allowing also its application to modeled structures. Table 1. List of added central residues when considering water in the residue interaction network compared to without water, for E2/Im2 complex (chain A = Im2; chain B = E2). A residue is considered as central when its Z-score ≥ 2 for RCA (Residue Centrality Analysis) or BCA (Betweenness Centrality Analysis). Union and intersection of resulting sets of RCA and BCA are also shown. The union excludes the residues that were added in RCA but already present in BCA without water (e.g. Glu30.A) and those that were added in BCA but already present in RCA without water. Residues are ordered following descending Z-scores. The complete Z-score value changes for the centrality measures RCA and BCA for the complex with and without water are listed in Supplementary Table S1. Residues at the interface of the complex are shown in blue.

TABLES AND FIGURES

  ; Schymkowitz et al. (2005); Huggins and Tidor (2011); Wang et al. (2011); Ross et al. (2012); de Ruyck et al. (2016); Jeszen öi et al. (

Figure

  Figure shows the structure and networks that lie at the basis of the calculations. Both the network with ('wet', right) and without ('dry', middle) water are shown. Central nodes are colored from yellow to red, following their Z score. Do note that the right-hand image shows the differential network, i.e. only additional central nodes are shown, and that the left-hand structure depicts these additional central nodes in the same colors. Non-central interface residues are colored in blue. All residues can be seen in the structure image on the left, in their context with the surrounding water molecules that form an integral part of the network. It can be seen from the networks in the Figure that their organization is globally similar, although the 'wet' RIN is significantly more dense due to the presence of water

Figure 1 .

 1 Figure 1. Structure and RINs of the colicin E2 DNase-Im2 complex. For all three images, the residues at the interface between the two chains (see Methods) are colored in blue, and central residues are colored in a gradient from yellow (Z-score ≥ 2) to red (Z-score ≥ 4). Left -Cartoon representation of the E2/Im2 complex (PDB ID: 3U43), with chain B (E2) located at the top and chain A (Im2) at the bottom; E2 is depicted in a darker shade while Im2 is lighter. Water molecules are shown as small red spheres. Central residues are drawn in stick representation; the depicted central residues are the additional ones that are highlighted in the right-hand network. Middle -Residue Interaction Network generated from the structure, excluding water molecules. Right -Differential Residue Interaction Network of the 'wet' vs. 'dry' network (see main text), highlighting only the additional central residues (also listed in Table 1/Union).

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CONFLICT OF INTEREST STATEMENT

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AUTHOR CONTRIBUTIONS

RB and MFL conceived the design of the study; GB performed the analysis; all authors discussed and interpreted the results; all authors wrote the manuscript.

SUPPLEMENTAL DATA

The supplementary material contains Supplementary Figure 1 and Supplementary Tables 1 and2. This is a provisional file, not the final typeset article