Gilles Dolfo
email: gilles.dolfo@wanadoo.fr

Jacques Vigué

Daniel Lhuillier

Daniel Lhuillier Stokes

Stokes force on a cylinder in the presence of fluid confinement

INTRODUCTION

In 1851, Stokes [1] calculated the force exerted by the surrounding fluid on a sphere and on a cylinder in oscillating motion, in the limit of a vanishing Reynolds number. In the case of the sphere, Stokes obtained closedform expressions of the stream function and of the force while, in the case of the cylinder, he was not able to express the stream function in closed form, because it involves modified Bessel functions which had not already been defined (following Watson [START_REF] Watson | The relations between modified Bessel functions and their derivatives are given in chapter 3 and their asymptotic expansions in chapter 7[END_REF], these functions were defined by Basset in 1886). He was nevertheless able to give asymptotic approximations of the force in the two limits a δ and a δ (where a is the cylinder radius and δ is the viscous penetration depth) and the exact expression of the force was given in 1963 by Stuart [START_REF] Stuart | Laminar Boundary Layers[END_REF].

In the same paper [1], the effect of fluid confinement was treated in the case of a sphere oscillating in a larger sphere but it was impossible to treat the similar problem of a cylinder of radius a oscillating in a larger cylinder of radius b, because a closed-form expression of the stream function was not available. This problem has been treated in 1976 by Chen, Wambsganss, and Jendrzejczyk [START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF] who expressed the fluid stream function and the force exerted by the fluid on the oscillating cylinder, using modified Bessel functions.

Because the problem of the oscillating cylinder is not treated in detail in most textbooks, we have chosen a tutorial point of view and we start our note by following Stokes' derivation as closely as possible. We then review the papers which have calculated the fluid confinement effect for the cylinder by various methods. We develop an exact calculation and, using asymptotic approximations of the modified Bessel functions, we give an expansion of the force in powers of the ratio δ/a, the expansion coefficients being functions of the ratio γ = a/b of the cylinder radii.

NAVIER-STOKES EQUATIONS AND THEIR SOLUTION

Navier-Stokes equations and the stream function ψ

The starting point is the Navier-Stokes equations relating the pressure p and velocity v

∇p = η∆v -ρ dv dt + (v • ∇) v (1)
∇ • v = 0. ( 2
)
Here ρ is the fluid density assumed to be constant and η its viscosity. Stokes neglected the non-linear term (v • ∇) v because he assumed a small enough velocity: the problem then has an analytic solution. This approximation is good if the Reynolds number Re is very small, Re 1. We consider a cylinder of radius a oscillating in a cylinder of radius b. When the inner cylinder is at rest, the axes of both cylinders coincide with the z-axis, and the oscillation is along the x-axis, with the inner cylinder center at x(t) = x 0 cos (ωt). The problem is restricted to 2 dimensions. Noting u and v the velocity components, eqs. (1,[START_REF] Watson | The relations between modified Bessel functions and their derivatives are given in chapter 3 and their asymptotic expansions in chapter 7[END_REF] are projected on the xand y-axes

∂p ∂x = η ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 -ρ ∂u ∂t (3) ∂p ∂y = η ∂ 2 v ∂x 2 + ∂ 2 v ∂y 2 -ρ ∂v ∂t (4) ∂u ∂x + ∂v ∂y = 0. (5)
Equation ( 5) proves that dψ defined by

dψ = udy -vdx (6)
is an exact differential. ψ is the stream function and its dimension is the product of a length by a velocity.

Elimination of the pressure p and its expression as a function of ψ By derivation of eq. ( 3) with respect to y and eq. ( 4) with respect to x, one gets two expressions of ∂ 2 p/∂x∂y. We write that these expressions are equal and we thus eliminate the pressure p. We replace u by u = ∂ψ/∂y and v by v = -∂ψ/∂x and we get

∂ 2 ∂x 2 + ∂ 2 ∂y 2 - 1 ν ∂ ∂t ∂ 2 ∂x 2 + ∂ 2 ∂y 2 ψ = 0, (7)
where ν = η/ρ is the kinematic viscosity. The general solution of this equation is

ψ = ψ 1 + ψ 2 (8)
with ψ 1 and ψ 2 solutions of the following equations

∂ 2 ∂x 2 + ∂ 2 ∂y 2 ψ 1 = 0 (9) ∂ 2 ∂x 2 + ∂ 2 ∂y 2 - 1 ν ∂ ∂t ψ 2 = 0 (10)
Using equations [START_REF] Stuart | Laminar Boundary Layers[END_REF][START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF] and ( 6), the pressure p is expressed as a function of

ψ dp = ∂p ∂x dx + ∂p ∂y dy = η dx ∂ ∂y -dy ∂ ∂x ∂ 2 ∂x 2 + ∂ 2 ∂y 2 - 1 ν ∂ ∂t ψ (11)
The term due to ψ 2 vanishes and we get

dp = ρ ∂ ∂t ∂ψ 1 ∂x dy - ∂ψ 1 ∂y dx (12)

Introduction of polar coordinates

We introduce polar coordinates r, θ in the x,y plane, θ = 0 corresponding to the x-axis, and the radial v r and tangential v θ components of the velocity x = r cos θ and y = r sin θ u = v r cos θ -v θ sin θ and v = v r sin θ + v θ cos θ dψ = udy -vdx becomes dψ = v r rdθ -v θ dr from which we deduce

v r = 1 r ∂ψ ∂θ and v θ = - ∂ψ ∂r (13)
The 2D Laplacian in polar coordinates is

∂ 2 ∂x 2 + ∂ 2 ∂y 2 = ∂ 2 ∂r 2 + 1 r ∂ ∂r + 1 r 2 ∂ 2 ∂θ 2
ψ 1 and ψ 2 are solutions of

∂ 2 ∂r 2 + 1 r ∂ ∂r + 1 r 2 ∂ 2 ∂θ 2 ψ 1 = 0 ( 14
)
∂ 2 ∂r 2 + 1 r ∂ ∂r + 1 r 2 ∂ 2 ∂θ 2 - 1 ν ∂ ∂t ψ 2 = 0 (15)
Equation ( 12) becomes

dp = ρ ∂ ∂t ∂ψ 1 ∂r rdθ - 1 r ∂ψ 1 ∂θ dr (16)

Boundary conditions

We must write boundary conditions on the surfaces of the two cylinders of radii a and b. Using complex notations, the velocity of the inner cylinder is

dx(t) dt = U exp (iωt) with U = iωx 0 (17)
Stokes assumed that the amplitude x 0 is very small with respect to the cylinder radius a i.e. the Keulegan-Carpenter [START_REF] Keulegan | Forces on cylinders and plates in an oscillating fluid[END_REF] number KC = πx 0 /a verifies KC 1. With this assumption, he wrote that the fluid velocity is equal to the cylinder velocity on the surface r = a v r (a, θ) = U cos θ exp (iωt)

v θ (a, θ) = -U sin θ exp (iωt) . (18)
The fluid velocity must vanish on the surface r = b of the outer cylinder

v r (b, θ) = 0 v θ (b, θ) = 0. ( 19
)
Equation (18) proves that ψ 1 and ψ 2 are proportional to sin θ

ψ 1 = U sin θ exp (iωt) F 1 (r) ψ 2 = U sin θ exp (iωt) F 2 (r) (20)
where F 1 (r) and F 2 (r) have the dimension of a length. From eqs. [START_REF] Dolfo | Unsteady Stokes friction force on a cylinder[END_REF]15), we deduce the equations verified by F 1 (r) and F 2 (r)

d 2 F 1 dr 2 + 1 r dF 1 dr - F 1 r 2 = 0 (21) d 2 F 2 dr 2 + 1 r dF 2 dr - F 2 r 2 -κ 2 F 2 = 0, (22)
with κ 2 = iω/ν. We define κ by

κ = 1 + i δ with δ = 2ν ω . (23)
δ is the viscous penetration depth. r n is an obvious solution of eq. ( 21) with n 2 = 1 i.e. n = ±1. We get

F 1 (r) = Aa 2 r + Br. ( 24
)
The introduction of the a 2 factor in the first term makes that A and B are both dimensionless. We multiply eq. ( 22) by r 2 and we introduce z = κr to get

z 2 d 2 F 2 dz 2 + z dF 2 dz -z 2 + 1 F 2 = 0. ( 25
)
Following Watson's book [START_REF] Watson | The relations between modified Bessel functions and their derivatives are given in chapter 3 and their asymptotic expansions in chapter 7[END_REF], the equation

z 2 d 2 F 2 dz 2 + z dF 2 dz -z 2 + ν 2 F 2 = 0 (26)
has two independent solutions which are the modified Bessel functions I ν (z) and K ν (z) so that

F 2 (r) = CaI 1 (κr) + DaK 1 (κr). ( 27
)
The a factor has been introduced so that C and D are also dimensionless. With these results, the stream function ψ is given by

ψ = Aa 2 r + Br + CaI 1 (κr) + DaK 1 (κr) U sin θ exp (iωt) . (28)
Although the modified Bessel functions were not defined in 1851, Stokes was able to calculate many properties of the solutions of eq. ( 22). These results are recalled below.

The unknown constants A, B, C, D are fixed by the boundary conditions in r = a and in r = b. Equation ( 13) gives v r and v θ as a function of F 1 (r) and F 2 (r) and we then use eqs. (18,19) to get

F 1 (a) + F 2 (a) = a F 1 (a) + F 2 (a) = 1 F 1 (b) + F 2 (b) = 0 F 1 (b) + F 2 (b) = 0, (29)
or in explicit form

A + B + CI 1 (α) + DK 1 (α) = 1 -A + B + CαI 1 (α) + DαK 1 (α) = 1 A a b + B b a + CI 1 (β) + DK 1 (β) = 0 -A a 2 b 2 + B + CαI 1 (β) + DαK 1 (β) = 0 ( 30
)
with α = κa and β = κb. For these notations and the use of dimensionless unknowns A, B, C, D, we have followed the paper of Chen et al. [START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF]. Before solving this system, we give the expression of the force as a function of the stream function.

EXPRESSION OF THE FORCE

By symmetry, the only non-vanishing component of the force is along the x-axis and it is proportional to the length l of the cylinder. It is given by

dF x dl = 2π 0 σ • nadθ σ jk = -pδ jk + η ∂v j ∂x k + ∂v k ∂x j . ( 31
)
n is the vector normal to the cylinder surface oriented inward n r = -1 and n θ = 0. We need only σ rr and σ rθ

dF x dl = 2π 0 (σ rr cos θ -σ rθ sin θ) adθ (32)
The components in cylindrical coordinates of the tensor σ jk are given in chapter III of the book Laminar Boundary Layers [START_REF]Unsteady Boundary Layers[END_REF]:

σ rr = -p + η ∂v r ∂r σ rθ = η r ∂ ∂r v θ r + 1 r ∂v r ∂θ = η - v θ r + ∂v θ ∂r + 1 r ∂v r ∂θ (33)

Calculation of σrr

We need the derivative ∂v r /∂r for r = a

∂v r ∂r = 1 r ∂ 2 ψ ∂θ∂r - 1 r 2 ∂ψ ∂θ ∂v r ∂r a = 1 a ∂ 2 ψ ∂θ∂r a - 1 a 2 ∂ψ ∂θ a = 0 (34)
We prove this result thanks to eqs. ( 13) and (18) to get (∂ψ/∂r) a = -v θ = U sin θ exp (iωt). We then derive with respect to θ to get ∂ 2 ψ/∂θ∂r a = U cos θ exp (iωt). The radial velocity v r (a, θ) is related to (∂ψ/∂θ) a and we get (∂ψ/∂θ) a = aU cos θ exp (iωt) so that ∂v r /∂r = 0. As a consequence σ rr = -p.

Calculation of σ rθ

σ rθ = η - v θ r + ∂v θ ∂r + 1 r ∂v r ∂θ a (35)
(∂v θ /∂r) a is given by (∂v θ /∂r) a = -∂ 2 ψ/∂r 2 a . To calculate ∂ 2 ψ/∂r 2 a , we use the equations verified by

ψ, ψ 1 , ψ 2 - ∂ 2 ψ ∂r 2 a = 1 a ∂ψ ∂r a + 1 a 2 ∂ 2 ψ ∂θ 2 a - 1 ν ∂ψ 2 ∂t a ( 36
)
The sum of the two first terms of the r.h.s. of equation (36) vanishes because (∂ψ/∂r) a = U sin θ exp (iωt) and ∂ 2 ψ/∂θ 2 a = -aU sin θ exp (iωt) and we get

- ∂ 2 ψ ∂r 2 a = - 1 ν ∂ψ 2 ∂t a (37)
We then calculate (∂v r /r∂θ) a

∂v r r∂θ a = 1 a 2 ∂ 2 ψ ∂θ 2 a = - U a sin θ exp (iωt) = v θ r a (38)
so that this term cancels the term in -(v θ /r) a in eq. ( 35) and we get

σ rθ = η ∂v θ ∂r = - η ν ∂ψ 2 ∂t a = -ρ ∂ψ 2 ∂t a ( 39
)
Calculation of the force With these results, the force per unit length is given by

dF x dl = a 2π 0 -p a cos θ + ρ ∂ψ 2 ∂t a sin θ dθ (40)
Rather than calculating the pressure p a , Stokes integrates by parts The integrated term obviously vanishes because sin θ vanishes at the bounds. We deduce dp a /dθ from eq. ( 16)

dp a dθ = ρa ∂ ∂t ∂ψ 1 ∂r a (42)
We take ∂/∂t out of the integral to get

dF x dl = ρa ∂ ∂t 2π 0 a ∂ψ 1 ∂r a + (ψ 2 ) a sin θdθ (43)
The values of ∂ψ 1 /∂r and ψ 2 at r = a are given by

∂ψ 1 ∂r a = U sin θ exp (iωt) [-A + B] (ψ 2 ) a = U sin θ exp (iωt) [CaI 1 (κa) + DaK 1 (κa] . (44)
From these results, we get the force per unit length

dF x dl = πa 2 ρU ∂ exp (iωt) ∂t [-A + B + CI 1 (κa) + DK 1 (κa)] = iωπa 2 ρU exp (iωt) [1 -2A] ( 45
)
where we have used the first of the equations (30) to simplify the result.

THE FORCE IN THE ABSENCE OF CONFINEMENT

We first consider the case with b → ∞ and we recall the results obtained by Stokes and by Stuart.

Stokes' results for the force

Stokes expressed the force per unit length of the cylinder in the form

dF x dl = -2πη a δ 2 k dx dt + 1 ω a δ 2 k d 2 x dt 2 (46) = -πa 2 ρ ωk dx dt + k d 2 x dt 2 . ( 47
)
The term proportional to the velocity dx/dt is a friction term and the term proportional to the acceleration is the added mass term, which describes the inertia of the fluid following the cylinder in its motion. The identification of eq. ( 45) with eq. ( 46) relates the quantities k and k to the solution of the system of equations (30) by

k -ik = 2A -1. ( 48
)
The added mass per unit length dm/dl is given by

dm dl = πa 2 ρk. (49)
Stokes calculated asymptotic expansions of k and k in the two limits a δ and a δ. In the low-frequency case, a δ, the quantities (k -1) and k diverge while these quantities multiplied by (a/δ) 2 tend toward 0. We reproduce here the asymptotic behaviors of these last quantities

a δ 2 (k -1) ≈ π/2 L 2 (a/δ) + (π 2 /4) (50) a δ 2 k ≈ - 2L(a/δ) L 2 (a/δ) + (π 2 /4) (51) with L(a/δ) = - ln(2) 2 + γ E + ln a δ , (52)
where γ E is the Euler constant, γ E ≈ 0.577. In the highfrequency case, a δ, we reproduce Stokes's expansions of (a/δ) 2 k and (a/δ) 2 k limited to the 3 dominant terms (there is no constant term in (a/δ) 2 k)

a δ 2 k ≈ a δ 2 + 2 a δ + δ 8a , (53)
a δ 2 k ≈ 2 a δ + 1 - δ 8a . ( 54
)
From these results, one easily deduces the asymptotic behaviors of k and k

k ≈ 1 + 2 δ a + 1 8 δ a 3 , (55)
k ≈ 2 δ a + δ a 2 - 1 8 δ a 3 . (56)
The added mass dm/dl is then given by

dm dl ≈ πa 2 ρ 1 + 2 δ a + 1 8 δ a 3 . ( 57
)
πa 2 ρ is the mass of displaced fluid per unit length of the cylinder and the following terms represent the contribution of the boundary layer of thickness δ. In the sphere case, the expression of the added mass is fully similar [START_REF] Landau | Fluid mechanics[END_REF] but the main term is equal only to half of the mass of the displaced fluid.

Stuart's results

Stuart [START_REF] Stuart | Laminar Boundary Layers[END_REF] has made the calculation using modified Bessel functions. The function I 1 (κr) diverges when r → ∞ (see ref. [START_REF] Watson | The relations between modified Bessel functions and their derivatives are given in chapter 3 and their asymptotic expansions in chapter 7[END_REF]) and this divergence, which is exponential, cannot be compensated by the divergence of r. This proves that B = C = 0 and the system (30) is simplified with two unknowns A and

D A + DK 1 (α) = 1 -A + DαK 1 (α) = 1. ( 58
)
and we get

D = 2 K 1 (α) + αK 1 (α) A = 1 - 2K 1 (α) K 1 (α) + αK 1 (α) = 1 + 2K 1 (α) αK 0 (α) . (59)
A has been simplified thanks to the equality K 1 (α) + αK 1 (α) = -αK 0 (α) (see Watson's book [START_REF] Watson | The relations between modified Bessel functions and their derivatives are given in chapter 3 and their asymptotic expansions in chapter 7[END_REF]). This simplification was introduced by Hussey and Vujacic [START_REF] Hussey | Damping correction for oscillating cylinder and sphere[END_REF]. We thus get

k -ik = 2A -1 = 1 + 4K 1 (α) αK 0 (α) (60)
We have compared numerically the expansions given by Stokes to the exact results given by eq. ( 60). If a/δ < 0.1, the approximate results given by eqs. (50, 51) are accurate with a relative error smaller than 7% for k and 1% for k . If a/δ > 1, the approximate results given by eqs. (55, 56) are also accurate with a relative error smaller than 7% for k and 1% for k .

THE FORCE IN THE GENERAL CASE

If b is finite, we need to solve the system of equations (30). The derivatives of I 1 and K 1 can be replaced by modified Bessel functions either of orders 0 and 1 or of orders 1 and 2. We present these two calculations. The first one was done by Chen, Wambsganss, and Jendrzejczyk [START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF] and, using Mathematica [START_REF][END_REF], we have done the second one which gives slightly simpler expressions.

Calculation of Chen, Wambsganss, and Jendrzejczyk

To solve the system of equations (30), Chen et al. [START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF] have replaced the derivatives I 1 and K 1 using the relations [START_REF] Watson | The relations between modified Bessel functions and their derivatives are given in chapter 3 and their asymptotic expansions in chapter 7[END_REF] zI 1 (z) = zI 0 (z) -I 1 (z)

zK 1 (z) = -zK 0 (z) -K 1 (z). ( 61
)
Their results are expressed by fractions A = A num /∆, B = B num /∆, ... with the same denominator ∆:

A num = -α 2 [I 0 (α)K 0 (β) -I 0 (β)K 0 (α)] +2α [I 1 (α)K 0 (β) + I 0 (β)K 1 (α)] -2αγ [I 0 (α)K 1 (β) + I 1 (β)K 0 (α)] +4γ [I 1 (α)K 1 (β) -I 1 (β)K 1 (α)] B num = 2αγ [I 1 (β)K 0 (β) + I 0 (β)K 1 (β)] +α 2 γ 2 [I 0 (α)K 0 (β) -I 0 (β)K 0 (α)] -2αγ 2 [I 1 (α)K 0 (β) + I 0 (β)K 1 (α)] C num = -2αK 0 (β) -4γK 1 (β) + γ 2 [2αK 0 (α) + 4K 1 (α)] D num = -2αI 0 (β) + 4γI 1 (β) + γ 2 [2αI 0 (α) -4I 1 (α)] , (62)
with the denominator ∆ given by

∆ = α 2 1 -γ 2 [I 0 (α)K 0 (β) -I 0 (β)K 0 (α)] +2αγ [[I 0 (α) -I 0 (β)] K 1 (β) + I 1 (β) [K 0 (α) -K 0 (β)]] +2αγ 2 [(I 0 (β) -I 0 (α)) K 1 (α) + I 1 (α) (K 0 (β) -K 0 (α))] , (63)
where γ = a/b = α/β. We have found some typographical errors in the paper of Chen et al.:

• in the numerator of B, there is a minus sign in front of the term I 0 (β)K 1 (β) whereas there should be a plus sign. We have corrected this error in eq. 62.

• all the quantities A, B, C, D have a sign opposite to the one we have found when we solve the system of equations (30) with the same replacement of the derivatives I 1 and K 1 ;

• their equation 9 gives H = k -ik = -2A -1 so that the final value of the force is exact.

Our results

We have used Mathematica [START_REF][END_REF] to solve the system of equations (30). The results involve the modified Bessel functions I 2 and K 2 which is explained by another possible replacement of I 1 (z) and K 1 (z), namely

I 1 (z) = (I 0 (z) + I 2 (z)) /2 (64) K 1 (z) = -(K 0 (z) + K 2 (z)) /2, (65)
also given by Watson [START_REF] Watson | The relations between modified Bessel functions and their derivatives are given in chapter 3 and their asymptotic expansions in chapter 7[END_REF]. Noting A = A num /∆ , B = B num /∆ ..., we get

A num = αβ 2 [I 2 (α)K 2 (β) -I 2 (β)K 2 (α)] B num = α α 2 (I 0 (β)K 2 (α) -I 2 (α)K 0 (β)) -2 C num = 2 -α 2 K 2 (α) + β 2 K 2 (β) D num = 2 -α 2 I 2 (α) + β 2 I 2 (β) (66)
and the denominator ∆ given by

∆ = α α 2 -β 2 I 0 (β)K 0 (α) + 2αI 0 (β)K 1 (α) +2βI 1 (β)K 0 (α) -α 2 I 2 (α)K 0 (β) +β 2 I 0 (α)K 2 (β) -4 (67)
In our calculation γ = a/b does not appear because we have replaced it by γ = α/β. We have verified that our calculation of H = 2A -1 agrees with value of H given by Chen et al..

APPROXIMATE CALCULATIONS OF THE FORCE ON A CYLINDER IN PRESENCE OF FLUID CONFINEMENT

In this section, we review a series of papers treating this subject. All these papers involve an approximation.

Calculations neglecting the fluid viscosity

This calculation was first done in 1844 by Stokes [START_REF] Stokes | On some cases of fluid motion[END_REF]: only the added mass is not vanishing while the friction force vanishes. The added mass dm/dl and the coefficient k are given by

dm dl = πa 2 ρ 1 + γ 2 1 -γ 2 k = 1 + γ 2 1 -γ 2 . (68)
When γ → 0, the added mass tends toward dm/dl = πa 2 ρ, which is is the limit of eq. ( 57) when δ vanishes, while the added mass dm/dl and k diverge when γ → 1.

In 1965, Hussey and Reynolds [START_REF] Hussey | Effect of a cylindrical boundary on the added mass of two cylinders in liquid helium[END_REF], in order to interpret experiments in superfluid helium, have calculated the effect of a cylindrical boundary on the added mass of one or two cylinders. In the case of a single cylinder, their result agrees with Stokes' result.

The results of Segel [START_REF] Segel | Application of conformal mapping to viscous flow between moving circular cylinders[END_REF] In 1961, Segel [START_REF] Segel | Application of conformal mapping to viscous flow between moving circular cylinders[END_REF] calculated the effect of fluid confinement, using conformal mapping techniques. The two cases of a low or high frequency oscillation corresponding respectively to a 2 /δ 2 1 or 1 were treated separately.

We report here only the results corresponding to the high frequency case a 2 /δ 2 1 because the results in the lowfrequency case have complicated expressions. Equation 5.25 of Segel's paper gives the modification of the coefficients k and k approximately given by eqs. (53, 54) by the presence of the outer cylinder. We have expressed Segel's results with our notations

k ≈ 1 + γ 2 1 -γ 2 + 2 1 -γ 2 × δ a k ≈ 2 1 -γ 2 × δ a . (69)
When γ → 0, these results converge toward the first two terms of Stokes' result for k given by eq. ( 55) and to the first term of Stokes' result for k given by eq. (56). In the limit of vanishing viscosity (i.e. when δ/a → 0), the coefficient k agrees with the result of Stokes [START_REF] Stokes | On some cases of fluid motion[END_REF] given by eq. ( 68).

The results of Siniavskii, Fedotovskii and Kukhtin

In 1980, Siniavskii, Fedotovskii and Kukhtin [START_REF] Siniavskii | Oscillation of a cylinder in a viscous liquid[END_REF] developed an approximate calculation valid if (b -a) δ. They treated separately the boundary layer in which the viscosity is taken into account while the viscosity is neglected outside this layer. When b → ∞, their calculation gives

k ≈ 1 + 2 δ a + 1 4 δ a 2 (70) k ≈ 2 δ a . (71)
The coefficient k agrees with Stokes' result given by eq. (57) for the first two terms but not for the third one and the coefficient k given by eq. ( 71) agrees with the leading term of Stokes' result given by eq. ( 56). If b is finite, their calculation gives the following result for the coefficient

k k ≈ 1 + δ 2a 2 (2b -δ) 2 + (2a + δ) 2 (2b -δ) 2 -(2a + δ) 2 + δ a . (72)
An expansion of this result in powers of δ/a gives

k ≈ 1 + γ 2 1 -γ 2 + 2 1 -γ + γ 2 (1 + γ) (1 -γ) 2 × δ a . (73)
In the limit (δ/a) → 0, this result agrees with the well established result [START_REF] Stokes | On some cases of fluid motion[END_REF][START_REF] Hussey | Effect of a cylindrical boundary on the added mass of two cylinders in liquid helium[END_REF] given by eq. ( 68). The friction force coefficient k is given by

k ≈ 2 1 + γ 3 (1 -γ 2 ) 2 δ a . (74)
We have noticed that 1 + γ 3 / 1 -γ 2 2 = 1 -γ + γ 2 / (1 + γ) so that the term linear in (δ/a) is the same in k and in k .

Expansion of the exact results

Chen, Wambsganss, and Jendrzejczyk [START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF] have given a approximate form of H in the limit where α = (1 + i) a/δ and β = (1 + i) b/δ are both large. Their derivation is based on the asymptotic expansions of the modified Bessel functions I n (z) and K ( z) given by Watson [START_REF] Watson | The relations between modified Bessel functions and their derivatives are given in chapter 3 and their asymptotic expansions in chapter 7[END_REF] :

I n (z) ∼ exp (z) 1 2πz × 1 - 4n 2 -1 1!8z + 4n 2 -1 2 4n 2 -3 2 2! (8z) 2 + ... K n (z) ∼ exp (z) π 2z × 1 + 4n 2 -1 1!8z + 4n 2 -1 2 4n 2 -3 2 2! (8z) 2 + ... (75)
We have limited these expansions to the first terms of the 1/z series and we have neglected terms which are exponentially small if Re (z) is positive and large. If we understand correctly what has been done by Chen et al., the series appearing in eq. ( 75) have been limited to the term equal to 1. Their result involves terms in sinh (β -α) and cosh (β -α) as well as two terms in αγ 1/2 and αγ 3/2 which come from the terms of the type I n (z) K n (z) with the same z (z = α or z = β). When Re (β -α) is positive and large, these terms are negligible with respect to the hyperbolic sine and cosine terms. Moreover, the difference between sinh (β -α) and cosh (β -α) is also negligible in this case. Finally, Chen et al. have not deduced from their calculation of H the values of k and k as a function δ/a and γ.

(γ) = 1+γ 2 1-γ 2 , F1 (γ) = (1-γ+γ 2 ) (1+γ)(1-γ) 2 and F2 (γ) = 1-2γ+6γ 2 -2γ 3 +γ 4 (1+γ)(1-γ) 3
as a function of γ: F0 is represented by the dashed (blue) curve, F1 is represented by the full (red) curve and F2 is represented by the full (green) curve.

Using Mathematica [START_REF][END_REF], we have redone this calculation, including the terms up to 1/z 2 of the series of eq. ( 75) and neglecting the negligible quantities discussed in the previous paragraph. We have expressed β = α/γ and we have expanded the results in powers of 1/α = (1 -i) δ/a up to the second order. We have verified that this procedure gives the same result using the expression of H obtained either by Chen et al. or by ourselves. We thus get the values of k and k up to the second in δ/a, with the expansion coefficients being functions of γ:

k = 1 + γ 2 1 -γ 2 + 1 -γ + γ 2 (1 + γ) (1 -γ) 2 × 2δ a + O δ a 3 k = 1 -γ + γ 2 (1 + γ) (1 -γ) 2 × 2δ a + 1 -2γ + 6γ 2 -2γ 3 + γ 4 (1 + γ) (1 -γ) 3 δ a 2 + O δ a 3 . (76)
Here are some comments:

• in the expansion of k, the term in (δ/a) 2 vanishes;

• we have verified that the expansions of k and k given by eqs. (76) are stable if we increase the number of terms of the 1/z n series of eq. ( 75);

• the limits of k and k when γ → 0 are in perfect agreement with the expansions in (δ/a) given by eqs. (55,56) corresponding to case γ = 0;

• the limit of k when (δ/a) vanishes agrees with Stokes' result given by eq. (68);

• the results of Segel [START_REF] Segel | Application of conformal mapping to viscous flow between moving circular cylinders[END_REF] are in agreement with Stokes' result for the added mass in an inviscid flow but the first order terms in (δ/a) in the expansions of k and k are not in agreement with the results of Siniavski et al. which agree with our results. However our results extend up to a second order term in (δ/a);

• figure 1 presents a plot of the variations of the three non-vanishing coefficients appearing in eqs. (76) as a function of γ. They diverge when γ → 1, the divergence of the coefficient of (δ/a) n being due to the denominator (1 -γ) (n+1) so that the divergence is faster when n increases;

• our procedure can give higher order terms of the expansions of k and k in powers of (δ/a). However, when γ approaches 1, the coefficients of (δ/a) n appear to diverge more rapidly when the order n increases. As a consequence, the use of this expansion is probably limited to lower γ values if the higher order terms of these expansions become important and it is probably better to use the exact results.

2π 0 p

 0 a cos θdθ = p a sin θ| 2π 0 -

FIG. 1 .

 1 FIG. 1. Semi-logarithmic plot of the power expansioncoefficients F0 (γ) = 1+γ 2 1-γ 2 , F1 (γ) =

CONCLUDING REMARKS

In this note, we have first reproduced Stokes' calculation of the force exerted by the surrounding fluid on a cylinder in oscillating motion. We have verified that Stokes asymptotic results are in good agreement with the exact results obtained by Stuart, using modified Bessel functions.

In his 1851 paper, Stokes calculated the effect of confinement for a sphere oscillating inside a larger sphere but, at that time, it was not possible to make the same calculation for a cylinder oscillating inside a larger cylinder because this calculation requires the use of modified Bessel functions. This calculation was first done by Chen, Wambsganss, and Jendrzejczyk [START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF] in 1976 and, using Mathematica [START_REF][END_REF], we have reproduced their calculation and pointed out some misprints in their paper. Moreover, as Mathematica [START_REF][END_REF] uses different relations between modified Bessel functions, we have obtained somewhat simpler expressions of the confinement effect.

We report the results of the approximate calculations of the confinement effect made by Segel [START_REF] Segel | Application of conformal mapping to viscous flow between moving circular cylinders[END_REF] and by Siniavskii, Fedotovskii and Kukhtin [START_REF] Siniavskii | Oscillation of a cylinder in a viscous liquid[END_REF]. We have also completed the calculation of Chen, Wambsganss, and Jendrzejczyk [START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF] and we have obtained a power expansion in δ/a of the two coefficients k and k respectively describing the inertial part (the added mass term) and the friction part of the force. The coefficients of the power expansions of k and k are expressed as functions of the ratio γ = a/b. The results of Siniavskii et al. agree with our results which also involve the next order terms and which could be extended to higher orders.

Experimental tests of the confinement effect on the added mass and on the friction force have been done by Chen et al. [START_REF] Chen | Added mass and damping of a vibrating rod in confined viscous fluids[END_REF] and also by Siniavskii et al. [START_REF] Siniavskii | Oscillation of a cylinder in a viscous liquid[END_REF]. In both cases, the experimental results have been found in good agreement with their calculations. We have also used our calculation of the confinement effect to analyze our measurements of the friction force on a cylinder oscillating inside another cylinder [START_REF] Dolfo | Unsteady Stokes friction force on a cylinder[END_REF]: the confinement correction appears to be quite necessary for a correct interpretation of the experimental results.


Dennis A Infante-Rodríguez

Juan L Monribot-Villanueva

Klaus Mehltreter

Gloria L Carrión

Jean-Paul Lachaud

A Carlos Velázquez-Narváez

Víctor M Vásquez-Reyes

Jorge E Valenzuela-González

José A Guerrero-Analco

Infante -Rodriguez

; Valenzuela-González

Phytochemical characteristics of leaves determine foraging rate of the leaf-cutting ant Atta mexicana (Smith) (Hymenoptera: Formicidae)

Keywords: Plant-insect interactions, leaf-cutting ants, ant fungal symbiont, plant metabolomics

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

INTRODUCTION

Leaf-cutting ants are distributed along a wide range of Neotropical habitats, including native and anthropic ecosystems [START_REF] Cherrett | A review of the status of leaf-cutting ants and their control[END_REF][START_REF] Vasconcelos | Changes in leaf-cutting ant populations (Formicidae: Attini) after the clearing of mature forest in Brazilian Amazonia[END_REF][START_REF] Wirth | Annual foraging of the leaf-cutting ant Atta colombica in a semideciduous rain forest in Panama[END_REF]. They perform an important role in plant defoliation, and are pests in some agricultural and silvicultural plantations [START_REF] Cherrett | History of the leaf-cutting ant problem[END_REF][START_REF] Hölldobler | The ants[END_REF]Montoya-Lerma et al. 2012). Workers of leaf-cutting ants forage on leaves, flowers, and fruits of different plant species in order to cultivate their symbiotic fungus Leucoagaricus gongylophorus (Möller) Singer (Weber 1966;[START_REF] Littledyke | Direct ingestion of plant sap from cut leaves by the leafcutting ants Atta cephalotes (L.) and Acromyrmex octospinosus (Reich) (Formicidae, Attini)[END_REF][START_REF] Quinlan | The role of fungus in the diet of the leaf-cutting ant Atta cephalotes (L.)[END_REF].

Leaf-cutting ants are considered as generalists due to their adaptability to exploit a wide range of plant species [START_REF] Littledyke | Direct ingestion of plant sap from cut leaves by the leafcutting ants Atta cephalotes (L.) and Acromyrmex octospinosus (Reich) (Formicidae, Attini)[END_REF][START_REF] Quinlan | The role of fungus in the diet of the leaf-cutting ant Atta cephalotes (L.)[END_REF][START_REF] Fine Licht | Forage collection, substrate preparation, and diet composition in fungus-growing ants[END_REF]. Nevertheless, diverse plant species are foraged at different intensities, and some plants are completely avoided [START_REF] Howard | Leafcutting ant diet selection: the role of nutrients, water, and secondary chemistry[END_REF][START_REF] Howard | Toxicity of terpenoid deterrents to the leaf-cutting ant Atta cephalotes and its mutualistic fungus[END_REF].

Previous studies suggested that each species of leaf-cutting ants exhibits preferences for particular plants [START_REF] Cherrett | Some factors involved in the selection of vegetable substrate by Atta cephalotes (L.) (Hymenoptera: Formicidae) in tropical rain forest[END_REF][START_REF] Rockwood | Plant selection and foraging patterns in two species of leaf-cutting ants (Atta)[END_REF][START_REF] Hubbell | Chemical leaf repellency to an attine ant: seasonal distribution among potential host plant species[END_REF]Hölldober and Wilson 1990), which might be influenced by certain mechanical and chemical characteristics of the foliage [START_REF] Cherrett | Some factors involved in the selection of vegetable substrate by Atta cephalotes (L.) (Hymenoptera: Formicidae) in tropical rain forest[END_REF][START_REF] Waller | Leaf-cutting ants and live oak: The role of leaf toughness in seasonal and intraspecific host choice[END_REF][START_REF] Berish | Leaf-cutting ants (Atta cephalotes) select nitrogen rich forage[END_REF][START_REF] Hubbell | Chemical leaf repellency to an attine ant: seasonal distribution among potential host plant species[END_REF].

Plant chemical defense strategies include the production of specific secondary/specialized metabolites [START_REF] Fürstenberg-Hägg | Plant defense against insect herbivores[END_REF]. Several groups of secondary metabolites, such as alkaloids [START_REF] Miyashira | Influence of caffeine on the survival of leaf-cutting ants Atta sexdens rubropilosa and in vitro growth of their mutualistic fungus[END_REF], terpenes [START_REF] Littledyke | Olfactory responses of the leaf-cutting ants Atta cephalotes (L.) and Acromyrmex octospinosus (Reich) (Hymenoptera: Formicidae) in the laboratory[END_REF][START_REF] Pagnocca | Toxicity of sesame extracts to the symbiont fungus of leaf-cutting ants[END_REF][START_REF] Verza | Preliminary studies on the effects of d-limonene to workers of the leaf-cutting ant Atta sexdens rubropilosa and its implications for control[END_REF]phenols (Nichols-Orians 1991) have been reported to be toxic for leaf-cutter ants [START_REF] Cherrett | Some factors involved in the selection of vegetable substrate by Atta cephalotes (L.) (Hymenoptera: Formicidae) in tropical rain forest[END_REF][START_REF] Boulogne | Insecticidal and antifungal chemicals produced by plants: a review[END_REF][START_REF] Van Bael | Fungal-fungal interactions in leaf-cutting ant agriculture[END_REF], or to inhibit the growth and development of their symbiotic fungus [START_REF] Miyashira | Influence of caffeine on the survival of leaf-cutting ants Atta sexdens rubropilosa and in vitro growth of their mutualistic fungus[END_REF][START_REF] Lobo-Echeverri | Effects of Capsicum baccatum and C. frutescens against Atta cephalotes (Hymenoptera: Formicidae) and the symbiotic fungus Leucoagaricus gongylophorus[END_REF]. Indeed, the symbiont fungus of some Atta species have been reported to be able to detoxify certain phenolic compounds present in the leaves of foraged plants (Cherret 1980;[START_REF] Powell | Factors influencing the growth of Attamyces bromatificus, a symbiont of attine ants[END_REF][START_REF] Cm | Condensed tannins, attine ants, and the performance of a symbiotic fungus[END_REF]. In addition, the ant workers obtain certain enzymes (e.g. laccases or polyphenol oxidases) from the fungus; these fungal enzymes pass through the ant gut without being denatured [START_REF] Fine Licht | Symbiotic adaptations in the fungal cultivar of leaf-cutting ants[END_REF], are added by the ants through fecal drops placed on the processed plant material [START_REF] Rockwood | Plant selection and foraging patterns in two species of leaf-cutting ants (Atta)[END_REF]). These enzymes help detoxify some compounds present in leaves [START_REF] Aylward | Leucoagaricus gongylophorus produces diverse enzymes for the degradation of recalcitrant plant polymers in leaf-cutter ant fungus gardens[END_REF], an ability which ants have lost from their genome [START_REF] Nygaard | Reciprocal genomic evolution in the ant-fungus agricultural symbiosis[END_REF]. Therefore, the range of plants that leaf-cutting ants can harvest appears to have become broader through fungal-derived enzymatic detoxification techniques.

Few studies have described the foraging activity of the Mexican leaf-cutting ant Atta mexicana [START_REF] Mintzer | Foraging activity of the Mexican leafcutting ant Atta mexicana (F. Smith), in a Sonoran desert habitat (Hymenoptera, Formicidae)[END_REF][START_REF] Zavala-Hurtado | Influence of leaf-cutting ants (Atta mexicana) on performance and dispersion patterns of perennial desert shrubs in an inter-tropical region of Central Mexico[END_REF][START_REF] Mehltreter | Leafcutter ants as test organisms for leaf quality of ferns[END_REF], and none of them has associated host plant preferences with the chemical composition of their food plants. Although little studied, the fungal symbiont (hereafter FS) of A. mexicana has been identified as L. gongylophorus based on morphological features and DNA sequencing [START_REF] Espinoza | In vitro isolation and identification of Leucoagaricus gongylophorus from Atta mexicana (Hymenoptera: Formicidae) fungal garden[END_REF][START_REF] Vigueras | Growth and enzymatic activity of Leucoagaricus gongylophorus, a mutualistic fungus isolated from the leaf-cutting ant Atta mexicana, on cellulose and lignocellulosic biomass[END_REF]. The foraging behavior of A. mexicana modulated by the occurrence of secondary metabolites (e.g. alkaloids and phenols) in leaves could be expected to be similar to that of other leaf-cutting ant species (as above mentioned).

In this study, we evaluated the effect of secondary metabolites in leaves of different plant species on the foraging behavior of A. mexicana. For this, we selected three plant species usually well foraged and three species usually avoided (even when easily accessible) by this ant. Specifically, the study had the following aims: I) to evaluate the foraging rate of ants on each of the six plant species, II) to determine the total content of alkaloids, phenols, terpenes flavonoids, and, tannins, present in the leaves of these plants, III) to evaluate the effect of leaf extracts on the ant's foraging rate, IV) to test the effect of selected compounds, identified by metabolomic analysis in the leaves of tested plants, on the foraging rate of ants and the growth of symbiotic fungus.

METHODS

Plant Material. In order to verify the foraging preferences of ants we selected young leaves of three species that, according to preliminary observations, were well foraged and three species that were avoided by ants. The first group comprised Trema micrantha (Rosales, Cannabaceae), Styrax glabrescens (Ericales, Styracaceae) and Rosa x alba (Rosales, Rosaceae) [locally named in Spanish as "ixpepe", "azar de monte" and "rosa"], whereas the second group comprised Citrus reticulata (Sapindales, Rutaceae), Coffea arabica (Rubiales, Rubiaceae) and Psidium guajava (Myrtales, Mytaceae) [locally named in Spanish as "mandarina", "café" and "guayaba"]. These observations were performed using laboratory-raised colonies of A. mexicana. Young fresh leaves of three different individuals of each plant species were collected in the Botanical Garden of the Instituto de Ecología,A.C.,Xalapa,Veracruz,Mexico (19° 30' 51.5" N,96° 56' 32.31" W), and from an adjacent fragment of cloud forest.

Foraging Rate of Leaf-Cutting Ants. Six experimental sub-colonies without queens that were obtained from six different maternal laboratory-reared colonies were used for this experiment. Each of these sub-colonies (hereafter named experimental units, EU) was built using a plastic jar of 250 mL containing 10 g of FS mycelium and included ant larvae, pupae and workers. The plastic jar was connected by a plastic tube to a foraging arena consisting of a rectangular plastic container (1.5 L capacity). A water supply was provided by using two glass vials filled with water and plugged with cotton. Before the experiments, sub-colonies were left undisturbed for five days under controlled conditions (25 ± 1 °C and 90% relative humidity). For each plant species, 30 leaf discs of 0.5 cm in diameter were obtained from three different plants (ten discs from each plant) were offered to each EU in a random order. For each test, the 30 discs of one plant species were placed in each foraging arena. We measured the foraging rate of ants after 24 h by counting the number of discs that had been taken into the nest (cf. [START_REF] Bruce | Allometric scaling of foraging rate with trail dimensions in leafcutting ants[END_REF]. The remaining discs were removed and discarded. After a 2 h period without access to any leaf discs, a new set of 30 leaf discs from a different plant species were placed in the foraging arena. These experiments were concluded once each EU had been offered leaf material of each plant species. In order to determine the effect of plant species on the foraging rate of leaf-cutting ants, we fitted a Generalized Linear Mixed Model. The number of foraged leaf-discs was considered as the response variable and as predictors the fixed additive effects of offered plant species (treatment), and sub-colony as a random factor. The model was fitted using the linear mixed effect (lme) of the nlme library of R software (R Development Core Team 2008), and the predictor variables were ranked based on the p-value obtained from likelihood ratio tests. Ranked variables were tested using a P value of <0.05, as described by [START_REF] Zuur | Mixed effects models and extensions in ecology with R[END_REF].

Crude Leaf Extracts. Young leaves from three different individuals of each plant species were freeze-dried (LABCONCO FreeZone 1) for 7 days under vacuum at -56 °C. Dry leaves were pulverized with a porcelain mortar and a pestle, and stored in sealed plastic bags until further use. Crude extracts of leaves were obtained in triplicate samples using an accelerated solvent extraction system (ASE 350, Thermo Scientific, USA), according to a protocol for targeting the extraction of polar and medium polar compounds [START_REF] Juárez-Trujillo | Caracterización del aceite y harina obtenido de la semilla de uva silvestre (Vitis tiliifolia)[END_REF]. A mixture of 3 g of lyophilized leaves from tested individuals (n=3) of each plant species and 1 g of diatomaceous earth was poured into a 34 mL glass cell. Each cell was filled with methanol (HPLC grade; Sigma-Aldrich, St. Louis, USA) under a pressure of 1,500 psi and heated at 60 °C during 5 min. Then, the cells were washed with 30% of the cell volume. Aliquots of 1 mL of each sample were placed into a 1.5 mL centrifuge tube, and formic acid (MS grade; Sigma-Aldrich, St. Louis, USA) was added to obtain a final concentration of 0.1%. Samples were filtered and placed into 1.5 mL UPLC vials for further analysis using a liquid chromatograph coupled to a mass spectrometer (LC-MS).

The remaining extract of each sample was dried under a fume hood, and this crude extract of leaves was used for the preliminary analyses of total secondary metabolites.

Ant Foraging Rate of Paper-discs Impregnated with Plant Extracts. Six new EU were used for this experiment. Thirty paper discs (0.5 cm diameter) impregnated with a solution of 1 mg/mL of total crude leaf extract dissolved in methanol were allowed to dry and were then placed in the foraging arena of each sub-colony following previously published procedures (Roses 1990;[START_REF] Bigi | Activity of Ricinus communis (Euphorbiaceae) and ricinine against the leaf-cutting ant Atta sexdens rubropilosa (Hymenoptera: Formicidae) and the symbiotic fungus Leucoagaricus gongylophorus[END_REF]). The removal rate of impregnated paper discs by ants after 24 h, was measured by counting the number of discs that had been taken into the nest. The remaining discs were removed and discarded and a new set of 30 paper discs impregnated with extract from a different plant species were offered. This experiment was concluded once each EU had been offered discs impregnated with extracts of each of the plant species.

In order to determine the effect of plant species on the foraging rate of leaf-cutting ants, we fitted a General Linear Mixed Model with the number of foraged leaf-discs as the response variable and as predictors the fixed additive effects of plant species (treatment), and subcolony as a random factor. The model was fitted using R software (R Development Core

Team 2008).

Total Content Analyses of Alkaloids,Terpenes Phenols Flavonoids, and, Tannins, . Spectrophotometric methods were applied to the crude extracts of leaves obtained from three individuals per plant species, in order to determine the total content of alkaloids, flavonoids and tannins by using the method of [START_REF] Tambe | Phytochemical screening and anthelmintic activity of wood and leaves of Hibiscus tiliaceus Linn[END_REF], and the total phenols following the method of [START_REF] Singleton | Analysis of total phenols and other oxidation substrates and antioxidants by means of folin-ciocalteu reagent[END_REF]. The method described by [START_REF] Ferguson | A textbook of pharmacognosy[END_REF] was used as a proxy for the quantification of total terpenes. The resulting values were compared among plant species by one-way ANOVA followed by Tukey's test using R software (R Development Core Team 2008).

Chemical Biomarkers Identification by Untargeted Metabolomics Approach. The chromatographic system used for untargeted metabolomics analysis was an Ultra-High Performance Liquid Chromatograph (UPLC) Class I of Waters coupled to a Synapt HDMi mass spectrometer. Chromatography was carried out in an Acquity BEH column (1.7 µm, 2.1 x 50 mm) with a column and sample temperatures of 40 °C and 15 °C, respectively.

The mobile phase consisted of (A) water and (B) acetonitrile, both with 0.1% formic acid (SIGMA). The gradient conditions of the mobile phases were 0-13 min linear gradient 1-80% of B, 13-14 min isocratic at 80% of B, 14-15 min linear gradient 80-1% of B, 15-20 min isocratic at 1% of B. We injected 1 µL of sample extracts at a flow rate of 0.3 mL/min.

Mass spectrometric analysis was performed with an electrospray ionization source in positive mode with a capillary, sampling cone and source offset voltages of 3,000 V, 40 V and 80 V, respectively. The source temperature was 100 °C and the desolvation temperature was 20 °C. The desolvation gas flow was 600 L/h and the nebulizer pressure was 6.5 Bar. Leucine-enkephalin was used as the lock mass (556.2771, [M+H] + ). The conditions used for MS analysis were: mass range 50-1200 Da, Function 1 CE, 6 V, function 2 CER 10-30 V, scan time 0.5 sec. Data were acquired and processed with MassLynx (version 4.1) and MarkerLynx (version 4.1) software of Waters™ Corp., in order to identify chemical biomarkers through Principal component analysis (PCA) and Orthogonal Projections to Latent Structures discriminant analysis (OPLS-DA). Retention times and protonated masses were considered at a noise threshold of 10,000 counts, and smoothing was applied. Pareto scaling was applied to generate the score plots. For the Splots, the variables that contributed to discrimination between two plant species were considered as potential biomarkers. Ant foraging rate on Leaf-Discs Impregnated with Selected Compounds. Three compounds, identified in leaves of experimental plants by targeted and untargeted metabolomics approaches, were used in their pure form to assess their effect on the foraging rate of ants. Caffeine and chlorogenic acid, both found at high concentrations in C. arabica leaves (a plant rejected by ants), were selected for this test, as well as quercetin because several quercetin derivatives were detected in high concentrations in P. guajava (another rejected plant) and Rosa x alba (a well foraged plant). For this purpose, five EU from five laboratory ant colonies were set up. Ten leaf-discs of 0.5 cm in diameter were cut from young leaves of Rosa x alba (the best foraged plant) and impregnated with a 3 mM solution of each selected compound, using distilled water as a solvent. This concentration was chosen based on preliminary experiments performed with the FS in which 3 mM solutions of these compounds showed inhibition of fungal growth (Suppl. Fig. 2). Solutions were mixed by vortex for 2 min prior to application to leaf discs. Leaf-discs treated only with water alone were used as a solvent control, and dry leaf-discs without water as a foliar control. Leaf-discs were offered to ant sub-colonies in a randomized order. After 2 h of exposure to the ants, the number of removed discs by ants was recorded, and the remaining discs were removed and discarded. To detect treatment effects on foraging rates, we used the non-parametric Kruskal-Wallis test followed by Dunn's post hoc multiple comparison between groups using R software (R Development Core Team 2008).

Assays on the Effect of Selected Compounds on Growth of the Ant-Symbiotic Fungus.

Identification of the FS strain was corroborated as Leucoagaricus gongylophorus by using morphological criteria previously described by [START_REF] Espinoza | In vitro isolation and identification of Leucoagaricus gongylophorus from Atta mexicana (Hymenoptera: Formicidae) fungal garden[END_REF] and [START_REF] Vigueras | Growth and enzymatic activity of Leucoagaricus gongylophorus, a mutualistic fungus isolated from the leaf-cutting ant Atta mexicana, on cellulose and lignocellulosic biomass[END_REF]. Caffeine, chlorogenic acid, and quercetin were tested for this assay of the FS growth. In order to obtain a pure strain of the fungus, serial cultures of 1 mg samples of mycelium obtained from a laboratory-established ant nest were grown on 90 mm Petri dishes containing 20 mL of potato-dextrose-agar medium (PDA) and incubated at 25 °C. This fungal strain was used for comparison of the growth rate by using Petri dishes containing five different treatments: PDA (as culture media growth control), PDA mixed with one of each of the three tested compounds (caffeine, chlorogenic acid and quercetin), and PDA plus thiophanate-methyl, a commercial antifungal compound as a growth inhibition control [START_REF] Holley | Distribution of thiabendazole and thiophanate-methyl resistant strains of Helminthosporum solani and Fusarium sambucinum in Alberta potato storages[END_REF]. Compounds were tested at 0.03, 0.3 and 3 mM. Each treatment was replicated three times (Suppl. Fig. 2). These concentrations were chosen based on the chlorogenic acid and quercetin concentrations found in the leaves of C. arabica (1692.2 µg/g of fresh tissue) and P. guajava (17.9 µg/g of fresh tissue).

Concentrations of 0.03, 0.3 and 3 mM corresponded to 10.2, 102.3 and 1023.0 µg/g of chlorogenic acid in PDA medium, respectively, 8.7, 87.3 and 872.7 µg/g of quercetin in fresh PDA medium, respectively, and 5.6, 56.0 and 560.7 µg/g of caffeine in PDA medium, respectively. Caffeine was not quantified but it has been reported at a concentration of up to 7100 µg/g of fresh tissue [START_REF] Ashihara | Biosynthesis of caffeine in leaves of coffee[END_REF]. For chlorogenic acid and caffeine, the maximum concentration tested (3 mM) was less than the concentration determined and reported in leaves. For quercetin, the maximum concentration tested (3 mM) was higher than the concentration found in P. guajava leaves.

Each treatment was replicated three times. A circle of 1.5 cm in diameter and 0.5 cm in height of mycelium previously cultured on PDA was placed in the center of each Petri dish.

Mycelium growth, measured as the diameter (in mm), was recorded weekly during one month. The percentage of fungal growth inhibition (GI) was calculated using the equation: GI = [(DC-DT)/DC] × 100; where DC is the average mycelial diameter of the control treatment, and DT is the individual mycelial diameter. The mean percentage was calculated for each treatment and data were then analyzed by one-way ANOVA and Tukey's test using R software (R Development Core Team 2008).

RESULTS

Ant Foraging Rate of Leaf-discs. The percentage of leaf-discs foraged by the ants (Fig. 1) was significantly influenced by the plant species (F 5, 25 = 13.449; P < 0.001). Variation among EU was not significant (F 1, 4 = 0.245; P= 0.6462). The most preferred species were Rosa x alba, S. glabrescens, and T. micrantha that had 98.8%, 89.4% and 54.4% removal after 24 h, respectively. In contrast, C. arabica, C. reticulata and P. guajava were mostly avoided by the ants with 20.6%, 14.4% and 14.4% removal respectively, confirming our preliminary field observations. Ant Foraging of Paper-discs Impregnated with Plant Extracts. Ant foraging of paper discs was significantly influenced by the plant extract offered (F 6, 30 = 6.733; P < 0.01).

Variation among EU was not significant (F 1, 4 = 0.219; P = 0.664). Leaf-cutting ants had a preference for paper-discs impregnated with extracts of Rosa x alba (35.6% of disks removed), C. reticulata (32.2%), T. micrantha (12.8%) and S. glabrescens (4.4%). In addition, in line with the results of the foliar behavior experiment, ant workers rejected paper discs impregnated with extract of C. arabica and P. guajava with foraging percentages of 1.7% and 0%, respectively. This suggests that rejection of these species could be due to their chemical composition. Paper discs impregnated with the solvent alone were also avoided (1.1%) (Fig. 2).

Total Content of Alkaloids, Terpenes, Phenols, Flavonoids and Tannins. The plant species differed significantly in all concentrations of secondary metabolites with the exception of terpenes (F5, 12 = 0.675, P > 0.05; Table 1). For C. arabica (F5,12 = 13.93, P < 0.001) and P. guajava (F5, 12 = 23.34, P < 0.001), both species rejected by ants, displayed the highest alkaloid and phenol contents, respectively. C. arabica also had the second highest content of phenols, whereas C. reticulata, the other rejected plant species, had an intermediated concentration of phenols and alkaloids, and the lowest content of flavonoids and tannins. Rosa x alba, the most preferred plant species, exhibited the lowest concentration of alkaloids and phenols. T. micrantha, presented the second lowest concentration of alkaloids and phenols, whereas S. glabrescens, showed the second highest concentration of phenols and the highest concentration of flavonoids and tannins.

Untargeted and Targeted Metabolomics. One of the main aims of this work was to identify key chemical compounds that could explain the ant foraging preferences. To achieve this, we used two different metabolomics approaches: untargeted and targeted analyses. The first one was performed to identify the possible presence of groups based on chemical similarities among the plant extracts tested and to correlate them with ant foraging preferences. The ordination of plant species according to their chemical profiles classified them into three groups. S. glabrescens and C. reticulata each represented an individual group, whereas the remaining four species comprised a third group (Fig. 3). The third group contained two rejected and two accepted plant species. In order to identify key compounds we performed two OPLS-DA, first between one accepted (Rosa x alba) and one rejected (P. guajava) plant species (Fig. 4), and the second between two rejected plant species, P. guajava and C. arabica (Fig. 5). When the chemical profiles of Rosa x alba and P. guajava were compared, we observed peaks with retention times of 3.8 and 4.1 min for P. guajava and 4.3 min for Rosa x alba as markers, and these peaks were tentatively identified as quercetin derivatives because they share the ion 303.0505 (+/-5 ppm). In a search for this selective ion in the Rosa x alba and P. guajava chromatograms, we detected more quercetin derivatives in P. guajava (3.9, 4.0 and 4.2 min). A similar scenario was observed when C. arabica and P. guajava chemical profiles were compared (Fig. 5). For P. guajava, we tentatively identified the same chemical markers as in Rosa x alba. In C. arabica, we tentatively identified the alkaloids caffeine and trigonelline, and the phenolic compound mangiferine (Fig. 5).

Most of the chemical markers that were tentatively identified based on the accurate mass spectrometric fingerprints by the untargeted metabolomics approach were phenolic compounds. Consequently, we performed a phenolic targeted metabolomics study in order to confirm the identity and quantity of the putatively phenolic compounds previously identified. The targeted metabolomics study was performed with a dynamic multiple reaction monitoring method with 60 phenolic authentic standards. The phenolic compounds identified and quantified by this strategy are shown in Table 2 and the full list of the compounds searched is shown in Supplementary Table 1. A total of 30 compounds were identified and quantified in the leaf methanolic extracts of the six plant species (Table 2), which represented phenolic acids ( 16), flavonoids (12) and coumarins (2). The most abundant phenolic compounds of the three preferred species were (+)-catechin, vanillic acid and quercetin 3-glucoside for S. glabrescens; (+)-catechin for T. micrantha and quercetin 3-D-galactoside, (+)-catechin and quercetin for Rosa x alba. Procyanidin B2 and 3-coumaric acid were only detected in Rosa x alba. The most abundant phenolic compounds of the three rejected plants were chlorogenic acid, (-)-epicatechin, (+)-catechin and some quercetin derivatives in C. arabica, compared to shikimic and ellagic acids, (+)catechin, quercetin and its derivates in P. guajava. C. reticulata contained mainly rutin, scopolin, ferulic and chlorogenic acids.

Foraging Rate of Leaf-Cutting Ants of Leaf-Discs Impregnated with Pure Compounds. The foraging of leaf-discs of Rosa x alba was negative influenced by impregnation of discs with caffeine, quercetine and chlorogenic acid [H (4) = 15.417, P < 0.01] (Fig. 6).

Growth Inhibition on Ant-Symbiotic Fungus. Fungal growth was significantly affected by the three tested compounds at 28 days (F 4, 5 = 310.4, P < 0.001; Fig. 7). Caffeine had an inhibitory effect on fungal growth across all the tested concentrations (0.03-3 mM), whereas quercetin and chlorogenic acid had a moderated inhibitory effect at 3 mM (Suppl.

Fig. 1 and Suppl. Fig. 3).

DISCUSSION

Leaf-cutting ants are considered to select or to avoid plants [START_REF] Roces | Olfactory conditioning during the recruitment process in a leaf-cutting ant[END_REF][START_REF] Roces | Odour learning and decision-making during food collection in the leafcutting ant Acromyrmex lundi[END_REF][START_REF] Herz | Delayed rejection in a leaf-cutting ant after foraging on plants unsuitable for the symbiotic fungus[END_REF]Saverscheck et al. 2010;[START_REF] Arenas | Avoidance of plants unsuitable for the symbiotic fungus in leafcutting ants: Learning can take place entirely at the colony dump[END_REF]. Some plant secondary metabolites fulfill specific ecological defense roles against insects [START_REF] Rehman | Role of phenolics in plant defense against insect herbivory[END_REF][START_REF] War | Mechanisms of plant defense against insect herbivores[END_REF][START_REF] Pino | Plant secondary metabolites as an alternative in pest management. I: Background, research approaches and trends[END_REF] or fungi [START_REF] Bennett | Transley Review No.72. Secundary metabolities in plant defence mechanisms[END_REF]. High concentrations of certain alkaloids and phenols have plant defensive attributes against herbivores and microorganisms [START_REF] Villard | Defence mechanisms of Ficus: pyramiding strategies to cope with pests and pathogens[END_REF], and could be involved in the avoidance behavior of A. mexicana on two less preferred plants in our foraging tests, because these plants exhibited the highest concentration of alkaloids (C. arabica) and phenols (P. guava), and coffee leaves also showed the second highest concentration of phenols.

The main alkaloid produced by C. arabica leaves is caffeine, a methylxantine with insecticidal and fungicidal properties [START_REF] Rizvi | Antifungal properties of 1,3,7trimethylxanthine, isolated from Coffea arabica[END_REF][START_REF] Nathanson | Caffeine and related methylxanthines: possible naturally occurring pesticides[END_REF][START_REF] Albuquerque | Mammalian nicotinic acetylcholine receptors: from structure to function[END_REF], which has been reported to cause inhibition to the growth of the FS of Atta sexdens [START_REF] Miyashira | Influence of caffeine on the survival of leaf-cutting ants Atta sexdens rubropilosa and in vitro growth of their mutualistic fungus[END_REF]. Our results confirmed caffeine as a chemical marker for coffee leaves, and showed that it promotes an avoidance behavior by A. mexicana on the offered rose leaves treated with this compound. We also observed a strong inhibitory effect of caffeine on the growth of the FS of A. mexicana. Some phenolic compounds produced by plants have also be implicated in defense against herbivores and pathogens [START_REF] Rehman | Role of phenolics in plant defense against insect herbivory[END_REF][START_REF] War | Mechanisms of plant defense against insect herbivores[END_REF]. One of the most relevant physiological effects of phenols on herbivores is astringency (Ozawa et al. 1987), and some of them possess insecticidal or fungicidal properties [START_REF] Boulogne | Insecticidal and antifungal chemicals produced by plants: a review[END_REF]. Coffee leaves contained the highest concentrations of three phenolic compounds: chlorogenic acid, (-)epicatechin, and quercetin 3,4-di-O-glucoside. Chlorogenic acid and (-)-epicatechin exhibit both fungistatic and fungicidal properties [START_REF] Martínez | Chlorogenic acid is a fungicide active against phytopathogenic fungi[END_REF][START_REF] Yamaji | The role of catechin and epichatechin in chemical defense against damping-off fungi of current-year Fagus crenata seedlings in natural forest[END_REF]Ichihara 2011, Hirasawa and[START_REF] Hirasawa | Multiple effects of green tea catechin on the antifungal activity of antymicotics agains Candida albicans[END_REF]. In the present study, ants avoided foraging leaf discs of rose impregnated with chlorogenic acid, and this phenolic compound caused a weak inhibition of the mycelial growth of the FS.

According to ethnobotanical studies, P. guajava is a traditional medicinal plant with a great versatility of uses, including the treatment of fungal infections [START_REF] Pérez-Gutiérrez | Psidium guajava: A review of its traditional uses, phytochemistry and pharmacology[END_REF], and the antifungal effects have been attributed to the presence of phenolic compounds, especially to quercetin and its glycoside derivatives [START_REF] Metwally | Monograph of Psidium guajava L. leaves[END_REF][START_REF] Dakappa | A review on the medicinal plant Psidium guajava Linn. (Myrtaceae)[END_REF][START_REF] Morais-Braga | Phenolic composition and medicinal usage of Psidium guajava Linn.: Antifungal activity or inhibition of virulence?[END_REF]. [START_REF] Chang | Unique bioactive polyphenolic profile of guava (Psidium guajava) budding leaf tea is related to plant biochemistry of budding leaves in early dawn[END_REF] noted that the main components of infusions of P. guajava leaves were quercetin, myrcetin, catechin, and gallic and ellagic acids. In addition to these compounds (except myrcetin), we detected 15 other compounds, mainly phenolic acids and flavonoids, and some of them (shikimic, 4coumaric, and ellagic acids, kaempferol, and kaempferol 3-O-glucoside) were present at high concentrations. Our results revealed quercetin derivatives as chemical markers of guava. Certain glucoside derivatives of quercetin such as quercetin-3-O-α-L-rhamnoside have been reported to cause a weak in vitro inhibition in the growth of the FS of A. sexdens [START_REF] Bicalho | Evaluation of the toxicity of Virola sebifera crude extracts, fractions and isolated compounds on the nest of leaf-cutting ants[END_REF]. Although, quercetin only caused a weak inhibitory effect on the growth of the FS in our assays, in comparison to caffeine, quercetin significantly decreased the removal of rose leaf-discs by ants. Foragers of A. mexicana rejected mandarin leaves, while paper discs impregnated with crude extract of this plant were foraged. This suggests that the rejection of C. reticulata leaves was not related to the phytochemical profile obtained from methanol extracts. Mandarin leaves did not contain high concentrations of alkaloids or phenols in comparison to the remaining tested plants. However, scopoletin was detected in the mandarin extracts, which is a coumarin with antifungal activity [START_REF] Gnonlonfin | Review scopoletin -A coumarin phytoalexin with medicinal properties[END_REF][START_REF] Rehman | Role of phenolics in plant defense against insect herbivory[END_REF].

In fact, other coumarins inhibit the growth of the FS of Atta sexdens [START_REF] Godoy | Inhibition of the symbiotic fungus of leaf-cutting ants by coumarins[END_REF]. Some other compounds not analyzed in the present study could be responsible for their antrepellent effects. For instance, the lipid non-polar fraction from mandarin leaf extracts was reported to repel Acromyrmex octospinosus [START_REF] Jones | Chemical deterrency of Citrus reticulata (Blanco) to the leaf-cutting ant, Acromyrmex octospinosus (Reich)[END_REF], and some of mandarin essential oils showed antifungal properties, likely due to terpenes [START_REF] Eldahshan | Comparison of chemical and antimicrobial studies of Egyptian mandarin leaves and green branches volatile oil[END_REF]. This result suggests that a mixture of attractive and repellent compounds could occur in leaves, and the contrasting effect of this mixture could provoke conflicts to the ant behavior that would explain the sudden changes of foraging preferences reported for Atta species, in which leaves are initially well-harvested and latterly avoided. This behavior, known as delayed rejection, may continue for several months [START_REF] Knapp | Factors controlling foraging patterns in the leafcutting ant Acromyrmex octospinosus (Reich)[END_REF][START_REF] Herz | Delayed rejection in a leaf-cutting ant after foraging on plants unsuitable for the symbiotic fungus[END_REF].

Despite the relatively high concentrations of some secondary metabolites, the leaf-discs of Rosa x alba and S. glabrescens were well foraged in our foraging experiments. Leaves of Rosa spp. have been previously reported to be well-accepted by leaf-cutting ants [START_REF] Arenas | Avoidance of plants unsuitable for the symbiotic fungus in leafcutting ants: Learning can take place entirely at the colony dump[END_REF][START_REF] Herz | Delayed rejection in a leaf-cutting ant after foraging on plants unsuitable for the symbiotic fungus[END_REF]. Although Rosa x alba leaves had the lowest concentrations of alkaloids and phenols, they possessed a remarkable diversity of phenolic acids and flavonoids, with the highest concentrations of protocatechuic, caffeic and ferulic acids. [START_REF] Liu | In vitro and in vivo antioxidant activity of ethanolic extract of white button mushroom (Agaricus bisporus)[END_REF] detected these substances in high concentrations in extracts of the fungus Agaricus bisporus, a genus of the same family as Leucoagaricus; these compounds could have no harmful effects on the FS of A. mexicana. However, some other compounds of Rosa spp., such as gallic and salicylic acid, can have antifungal properties (Dixit and Uphadyay 1976). The ant preference for rose leaves and paper discs impregnated with rose extracts suggest a positive response by workers of A. mexicana to specific chemical compounds present in the leaves of this plant. Ant workers are exposed to plant juices from cut surfaces of leaves during foraging and preparation of the substrate for the fungus garden [START_REF] Littledyke | Direct ingestion of plant sap from cut leaves by the leafcutting ants Atta cephalotes (L.) and Acromyrmex octospinosus (Reich) (Formicidae, Attini)[END_REF]. A positive response to this plant could therefore arise though direct ingestion of plant sap and cellular fluids containing these compounds.

Although S. glabrescens had the highest concentrations of total flavonoids and tannins, it was well foraged by the ants. The most abundant phenolic compounds identified in the leaves of this plant were vanillic acid, (+)-catechin, and quercetin 3-glucoside. These results could be related to the ability of FS to detoxify phenolic compounds, as it has been reported to the FS of other species of leaf-cutting ants (e.g. Cherret 1980; [START_REF] Nygaard | Reciprocal genomic evolution in the ant-fungus agricultural symbiosis[END_REF]. Additionally, some plant phenols play a beneficial role in herbivore insects, especially flavonoids. This includes protection against pathogens and predators [START_REF] Simmonds | Flavonoid-insect interactions: recent advances in our knowledge[END_REF], stimulation to either food acceptance or rejection [START_REF] Schoonhoven | An inventory of taste in caterpillars: each species its own key[END_REF], phagostimulation [START_REF] Blaney | Electrophysiological activity in insects in response to antifeedants[END_REF][START_REF] Van Drongelen | Contact chemoreception of host plant specific chemicals in larvae of various Yponomenta species (Lepidoptera)[END_REF], and increased reproduction [START_REF] Burghardt | Flavonoid wing pigments increase attractiveness of female common blue (Polyommatus icarus) butterflies to mate-searching males[END_REF].

Interestingly, grouping based on the comprehensive chemical profiles detected by PCA analyses (that summarized the complete dataset, comprising 698 Exact Mass/Retention Time pairs [EMRTs]), suggests a relative similarity among some accepted and rejected plants, remarkably between Rosa x alba (preferred) and P. guajava (rejected). Nevertheless, differences more detailed in leaf chemical composition detected by the targeted and untargeted metabolomics analyses, the concentration of compounds, and the presence of specific chemical markers (EMRTs with higher levels to P. guajava) could explain these differences in the foraging preferences of A. mexicana.

Food plant selection of leaf-cutting ants is a complex behavior involving many plant traits, including the composition and concentrations of secondary metabolites. In our study, leaves of two of the less-preferred plants (C. arabica and P. guajava) had higher concentrations of alkaloids and phenolic compounds. In the case of C. reticulata leaves (the other not foraged plant), our results suggest that this rejection was due to some additional foliar features.

Foragers avoided collecting leaf discs of the most preferred plant (Rosa x alba) after treatment with the three pure tested compounds: caffeine, quercetin and chlorogenic acid.

These results suggest that ants are able to perceive the presence of these compounds, which also had a strong or moderate inhibitory effect on the growth of the FS. Several other phenolic compounds identified in leaves of tested plants have been reported to exhibit antifungal properties. In the future, it would be necessary to explore the existence of synergistic or antagonist relationships among these compounds and their effects on the foraging behavior of ants and on the growth of its FS.

Overall, our results showed that secondary metabolites play an important role in the process of plant selection by A. mexicana. This study represents the first contribution that has investigated simultaneously the possible interrelation between foraging preferences of A. mexicana, the phytochemical profiles of their food plants, and the effect of these phytochemical compounds on the FS of A. mexicana. In addition, we present the first phytochemical screening for T. micrantha and S. glabrescens, two valuable tropical and subtropical tree species [START_REF] Quintanar-Isaías | La madera de Trema micrantha (L.) Blume de Veracruz, México[END_REF][START_REF] Gonsoulin | A revision of Styrax (Styracaceae) in North America, Central America, and the Caribbean[END_REF].

  Identification and Quantification of Individual Phenolic Compounds by Targeted Metabolomics Approach. The identification and quantification of individual phenolic compounds was performed with an UPLC Agilent 1290 series coupled to a tandem mass spectrometer (Agilent 6460, triple quadrupole, QqQ) using dynamic multiple reaction monitoring (dMRM) as the acquisition method, following the protocol described by Juárez-Trujillo et al. (2017). Chromatographic separations were carried out on a ZORBAX SBC18 column (1.8 µm, 2.1 × 50 mm) (Agilent Technologies) with a column temperature of 40 °C. The mobile phase consisted of (A) water and (B) acetonitrile, both containing 0.1% formic acid. The gradient conditions for the mobile phase were: 0 min 1% B, 0.1-40 min linear gradient 1-40% B, 40.1-42 min linear gradient 40-90% B, 42.1-44 min isocratic 90% B isocratic, 44.1-46 min linear gradient 90-1% B, 46.1-47 min 1% B isocratic. The flow rate was 0.1 mL/min, and the sample injection volume was 5 µL. The ESI source was operated in positive and negative ionization modes at a desolvation temperature of 300 °C, cone gas (N2) flow of 5 L/min, nebulizer pressure at 45 psi, sheath gas temperature of 250 °C, sheath gas flow of 11 L/min, capillary voltage (positive and negative) of 3,500 V, and nozzle voltage (positive and negative) of 500 V. For the quantification of each phenolic compound, a calibration curve in a concentration range of 0.3 to 30 µM was constructed (R 2 values = 0.99 were considered for the linearity range) and quantities were established by using Mass Hunter Workstation Software, version B.06.00 (Agilent Technologies).Values are expressed as µg/g of sample (dry weight) [see TableS1for supplementary information].

Table 1 .

 1 Total content of secondary metabolites [mg/g] in plant species rejected or accepted by the leaf-cutter ant A. mexicana. Total content of SM [average (n=3) ± SD]. Different letters indicate significant differences among species (Tukey's test, P < 0.05). Numeric superindexes refer to chemical equivalents of: berberine (1), gallic acid (2), and quercetin (3).

	Plant species	Alkaloids 1	Terpenes	Phenols 2		Flavonoids 3		Tannins 2
	Rejected
	C. arabica	5.4 ± 1.76 a	10.2 ± 2.05 a	81.1 ± 0.02 b	24.6 ± 6.12 c	2.6 ± 0.05 e
	C. reticulata	0.8 ± 0.41 b	8.5 ± 2.53 a	42.8 ±19.90	bc	2.5 ± 0.32	d	0.3 ± 0.01	f
	P. guajava	0.8 ± 0.78 b	9.2 ± 1.03 a	145.1 ± 16 a	32.3± 4.44	bc	17.6 ± 1.7	b
	Accepted
	S. glabrescens	1.9 ± 0.82 b	11.6 ± 4.51 a	46.8 ± 2.36	bc	48.7 ± 7.50	a 25.7 ± 0.51	a
	T. micrantha	0.8 ± 0.22 b	11.3 ± 1.71 a	32.5± 14.72 c	23.4 ± 2.80 c	4.5 ± 0.05	d
	Rosa x alba	0.4 ± 0.06 b	10.2 ± 1.63 a	17.7 ± 4.93 c	40.7± 8.41	ab	13.7 ± 0.40

c

Table 2 .

 2 Phenolic compounds (μg/g) of methanolic extracts of three rejected and three preferred food plant species of A. mexicana. Different letters indicate significant differences among metabolites for each species (Tukey's test, P < 0.05) [average (n=3) ± SD].

ACKNOWLEDGEMENTS

We thank Diana Sánchez Rangel and Randy Ortiz Castro for the access to their laboratory facilities; Dora Luz Martínez Tlapa, Javier Tolome Romero, Ariadna Martínez Virués, Daniela Cela Cadena, Ofelia Ferrera Rodríguez, and José Benjamín Rodríguez Haas for technical assistance at laboratory. We also thank Trevor Williams, Luis Javier Fuentes Jacques and Renato Portela Salomaõ for helpful suggestions and critical comments on the manuscript. Dennis Adrian Infante-Rodríguez acknowledges the fellowship granted by the Consejo Nacional de Ciencia y Tecnología (CVU 409930) and the Doctoral Program of the Instituto de Ecología, A.C. Finally, we thank to the "Secretaría de Agricultura, Ganadería, Desarrollo Rural, Pesca y Alimentación" (SAGARPA) for the financial support to the Laboratory of Natural Products Chemistry, Department of Advanced Molecular Studies of the Instituto de Ecología A.C. (INECOL) for the development of the analytical method by LC-ESI-MS-MS for plant phenolic identification (Project SAGARPA-INECOL 80056).

Supplementary material