P Béjot

B Kibler

E Hertz

B Lavorel

O Faucher

General approach of spatiotemporal modulational instability processes

Keywords: numbers: 42.65.Ky, 42.65.Sf, 42.81.Dp

In this paper, we derive the general exact solution of the modulation instability gain. The solution described here is valid for 1D, 2D, and 3D cases considering any temporal response function of the medium and with possible higher order Kerr nonlinearities. In particular, we show that the gain induced by modulation instability is initial conditions dependent while the usual calculations do not lead to such a dependence. Applications for current and high interest nonlinear propagation problems, such as 1D optical fiber propagation with delayed Raman response and 2D filamentation in gases, are investigated in details. More specifically, we demonstrate that the new 2D model of filamentation based on the balance between higher order Kerr terms leads to a new modulation instability window. The impact of both self-steepening and space-time defocusing effects is also highlighted. Finally, we discuss the influence of the finite time response of the different order electronic Kerr effects on the growth of the expected modulation instability bands.

INTRODUCTION

From the 1960s till today, the modulation instability (MI) process has still remained the subject of numerous experimental and numerical studies in transparent materials [1][2][3][4][5][6]. When a quasi-continuous wave propagates through a nonlinear medium, it can experience, in appropriate phase-matching conditions, spatiotemporal instabilities that manifest themselves by the exponential growth of weak perturbations. More specifically, MI results from the interplay between linear effects, such as group velocity dispersion (GVD) or diffraction, and the Kerr induced nonlinearity. For instance, in optical waveguides, if the focusing (defocusing) nonlinear medium exhibits anomalous (normal) dispersion at the pump frequency ω 0 , because processes such as ω 0 + ω 0 → (ω 0 + ω) + (ω 0 -ω) can be nonlinearly phase matched, spectral bands spontaneously grow symmetrically with respect to the input wave frequency even in absence of any seed. However, both higher order dispersion and birefringence characteristics can strongly impact on the existence of new windows of scalar and vectorial MI effects [7,8]. Similarly, MI can also appear in the spatial domain due to the balance between diffraction and Kerr effect. For instance, it has been shown that multifilamentation [10], i.e., the splitting of an ultrashort ultra high power laser into several distinct structures is a direct manifestation of spatial MI. On the other hand, recent experiments have been able to determine the higher nonlinear refractive indices of gases such as argon, nitrogen, or oxygen [11]. These higher nonlinear indices manifest themselves at high intensity by the cancelation and even a sign reversal of the global nonlinear refractive index of the considered gases. These measurements are of particular importance for the understanding of pulse propagation dynamics since it has been demonstrated that, in some conditions, these indices are responsible for the collapse saturation during the filamentation process [12]. The scope of this paper is then twofold. In the first section, we theoretically study the stability of the nonlinear Schrödinger equation and derive the exact solution for MI. This solution is valid for 1D, 2D, and 3D propagation cases and is established with potential higher order Kerr terms and non-instantaneous temporal responses. The master equation derived here can hence describe any situation of scalar MI. To the best of our knowledge, it is the first time that such an exact solution is derived. Moreover, conversely to usual derivation of MI gain, we show that the gain bands are strongly dependent on the initial noise conditions. In the second section, we apply this new analytical basis to investigate in details the MI process for different configurations and materials. First, according to the recent measurements of higher order Kerr terms of some gases, we demonstrate the existence of a new 2D MI regime directly induced by these new nonlinear indices. Indeed, on-axis MI can occur in the normal dispersion regime when the global nonlinear refractive index becomes negative at very high intensities. Moreover, we analyze how the self-steepening and space-time defocusing terms modify the MI bands. Secondly, we show that the master equation gives also the exact solution of MI in optical fibers when delayed Raman effect is taken into account. In particular, we predict possible complex configurations of MI gain bands using the widely used waveguide category of photonic crystal fibers. Finally, we study the influence of the electronic finite time response on the MI. In particular, we demonstrate that the time response of the Kerr induced nonlinear index has to be relevantly evaluated in order to accurately describe the MI bands growth.

I. MODULATIONAL INSTABILITY ANALYSIS

In this section, we consider a linearly polarized (along a vector u) electric field propagating in a transparent centro-symetric medium. The electric field can be written as

E( -→ r , z, t) = (ε( -→ r , z, t)e ik0z-iω0t + c.c.)u (1)
where c.c. denotes complex conjugation, ε is the slowly varying envelope, -→ r is the transverse direction, and k 0 is the wavevector at the associated pulsation ω 0 .

In the frame propagating at the pump frequency group velocity v g = 1/k (1) and neglecting harmonic generation, ε verifies

∂ z ε = i 2k 0 T -1 △ ⊥ ε + i ∑ n≥2 k (n) n! (i∂ t ) n ε + iT k 0 ∆nε (2)
where

k (n) = ∂ n k ∂ω n | ω0
is the n th -order dispersion coefficient of the medium evaluated at ω 0 and ∆n is the nonlinear refractive index.

△ ⊥ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 in 3D or △ ⊥ = ∂ 2 ∂r 2 + 1 r ∂ ∂r in
2D denotes the transverse Laplacian. Assuming that the medium is transparent, ∆n can be written as ∆n = ∑ j≥1 n 2j R 2j (t) ⋆ |ε| 2j where ⋆ accounts for convolution [13]. Depending on the considered medium, one can consider only n 2 (for instance, for propagation in fiber) or a full development up to n 10 (as shown in [12]). R 2j (t) account for the (2j+1) th order temporal response function of the medium which are all real and verify ∫ ∞ -∞ R 2j (t)dt = 1 and R 2j (t < 0) = 0. R 2 (t) can take into account both electronic induced nonlinear response (generally admitted as instantaneous) and vibrational or rotational response. For instance, for silica fiber propagation problems, one generally takes

R 2 (t) = (1 -f r )δ(t) + f r H(t) with H(t) = f a h a (t) + f b h b (t), where δ is the Dirac function, h a = τ 1 (τ -2 1 + τ -2
2 )e -t/τ2 sin(t/τ 1 ), h b = 2τ b -t τ b e -t/τ b , f r = 0.18, f a = 0.79, f b = 0.21, τ 1 = 12.2 fs, τ 2 = 32 fs, τ b = 96 fs and n 2j (j > 1)=0 [14]. In air propagation problem, one takes R

2 (t) = (1 -f r )δ(t) + f r H(t) where H(t) = τ 1 (τ -2 1 + τ -2
2 )e -t/τ2 sin(t/τ 1 ), f r = 0.5, τ 1 = 62.5 fs, τ 2 = 70 fs with the higher order nonlinear indices n 2j measured recently in [11]. In both cases, the electronic nonlinear response is considered as instantaneous. Finally, T = 1 + ik (1) k0 ∂ t (resp., T -1 = 1 -ik (1) k0 ∂ t ) accounts for self-steepening (resp., for space-time defocusing). Equation 2 allows to describe the propagation of electric field up to the single-cycle limit [15].

The steady state solution (i.e., considering a monochromatic plane wave) of Eq. 2 can be expressed as

ε(z) = √ I 0 e ik0Dnz , ( 3
)
with D n = ∑ j≥1 n 2j I j 0 and I 0 the incident intensity. The stability of the continuous-wave steady state solution of Eq. 2 is examined by studying the evolution of the system in presence of a small complex perturbation a(r, t)

ε(z) = ( √ I 0 + a(r, t) ) e ik0Dnz , (4)
with |a(r, t)| 2 << I 0 . Substituting Eq. 4 into Eq. 2 and linearizing, we find the equation describing the evolution of a(r, t)

∂ z a(r, t) =T -1 i 2k 0 △ ⊥ a(r, t) + i   ∑ n≥2 k (n) n! (i∂ t ) n   a(r, t) + iT k 0 I 0 n 2 eff (t) ⋆ (a(r, t) + a * (r, t)) ,
where

n 2 eff (t) = ∑ j≥1 jR 2j (t)n 2j I j-1 0 . ( 5
)
If one considers a cylindrical symmetry around the propagation axis and considering that a(r, t) =

H ( ∫ ∞ -∞ a(k ⊥ , ω)e -iωt dω
)

, where H is the Hankel transform defined as H(f

)(r) = ∫ ∞ 0 k ⊥ J 0 (k ⊥ r)f (k ⊥ )
dk ⊥ with J 0 the zeroth order Bessel function, one obtains the propagation equation for a(k ⊥ , ω)

∂ z a(k ⊥ , ω) = -i   k 2 ⊥ 2k 0 (1 - k (1) k 0 ω) - ∑ n≥2 k (n) n! ω n   a(k ⊥ , ω) + ik 0 (1 + k (1) k 0 ω)I 0 n 2 eff (ω) ( a(k ⊥ , ω) + a * (-k ⊥ , -ω)) , (6)
where

n 2 eff (ω) = ∫ ∞ -∞ n 2 eff (t)
= a(k ⊥ , ω) + a * (-k ⊥ , -ω) and v = a(k ⊥ , ω) -a * (-k ⊥ , -ω),
u and v then satisfy the following partial differential equations system

∂u ∂z = i(D -u + (D + + S)v) (7) ∂v ∂z = i [(D + + N )u + D -v] ,
where

D + = - k 2 ⊥ 2k 0 + ∑ n≥1 k (2n) (2n)! ω 2n , N = 2k 0 I 0 n 2 eff (ω), D -= k 2 ⊥ k (1) 2k 2 0 ω + ∑ n≥1 k (2n+1) (2n + 1)! ω 2n+1 , S = 2k (1) ωI 0 n 2 eff . Defining K = √ -D + (D + + N ) -S 2
4 and Φ(z) = (D -+ S/2)z, the two solutions of Eq. 7 are

u(z) = e iΦ(z) 2 × ( u 0 cosh(Kz) + i D + v 0 + S 2 u 0 K sinh(Kz) ) v(z) = e iΦ(z) 2 × ( v 0 cosh(Kz) + i (D + + N )u 0 -S 2 v 0 K sinh(Kz)
)

,

where u 0 (k ⊥ , ω) = u(k ⊥ , ω, z = 0) and v 0 (k ⊥ , ω) = v(k ⊥ , ω, z = 0).
Finally, the evolution of the perturbation a(k ⊥ , ω) can be expressed as

a(k ⊥ , ω) = e iΦ(z) ( a 0 cosh(Kz) + i A 0 K sinh(Kz) ) , ( 8
) where A 0 = (D + a 0 + N 2 ( a 0 + a 0 ♯ ) + S 2 a 0 ♯ , with a 0 = a(k ⊥ , ω, z = 0) and a 0 ♯ = a * (-k ⊥ , -ω, z = 0).
In the following numerical simulations, a 0 is the initial noise spectrum calculated as

a 0 = H ( ∫ ∞ -∞ a(r, t, z = 0)e iωt dt
) while a 0 ♯ is calculated as

a 0 ♯ = H ( ∫ ∞ -∞ a * (r, t, z = 0)e iωt dt
) . The evolution of the spatio-temporal noise is then govern by Eq. 8. In particular, MI occurs when R e (K) > 0, i.e., when D + (D + + N ) + S 2 4 < 0, roughly leading to an exponential growth of a. More precisely, MI occurs if

-N - √ N 2 -S 2 2 < D + < -N + √ N 2 -S 2 2
. Equation 8 is a general solution of MI in any nonlinear medium which can present a delayed nonlinear response and higher order nonlinear Kerr terms. However, one has to emphasize that this equation can be extended to a 3D model by replacing k 2 ⊥ by k 2 x +k 2 y and considering a 2D spatial Fourier transform instead of an Hankel transform. Moreover, the 1D case (i.e., for fiber propagation issues) is obtained by setting k ⊥ = 0. To the best of our knowledge, it is the first time that the exact solution of the evolution of the noise is derived. One generally uses an ansatz function for a [8,[19][20][21][22][23]. Using such a method does not allow to retrieve the exact solution for the evolution of a. Instead, one can only evaluate a gain (which does not intrinsically depend on the initial seeding). The fact that the growth is dependent on the initial noise could play a role in rogue wave generation [16]. Moreover, considering the initial conditions dependence of Eq. 8 at a fixed ω, the maximal gain is dependent on the spatial gradient of both a 0 and a 0 ♯ . Consequently, it explains why multifilamentation occurs in the region of strong noise gradient. Since we have neglected the nonlinear terms with respect to a, the only limit of validity of this model is potential cascading phenomena which can occur when |a| 2 becomes non-negligible.

II. RESULTS AND DISCUSSION

A. 2D analysis

In this section, we consider the propagation in Argon of a cylindrical pump beam in order to investigate the impact of its higher order Kerr coefficients which have been recently measured up to n 10 [11]. Moreover, we assume here that all the nonlinear responses are instantaneous, i.e., R 2j (t) = δ(t) ∀j. Figure 1 shows the comparison between the integration of Eq. 2 and the results obtained from Eq. 8 for two different regimes in argon at atmospheric pressure and in a normal dispersion regime. As demonstrated in [12], Eq. 2 can accurately describe the propagation of filaments providing that the plasma contribution stays negligible. The numerical simulations were performed by starting from a CW pump with a superimposed small noise (with a random spectral phase added to each frequency bin) as a seed for the MI. Since in real experiments, the perturbations have their maximum intensity at the laser central frequency, in the frequency domain, and at the center of the pulse, in the spatial domain, we have also checked that using a noise with an amplitude presenting a gaussian distribution centered on the pump frequency in both spectral and spatial domain does not affect qualitatively the MI bands. We clearly note the excellent quantitative agreement between analytical and numerical results, thus confirming the validity of our calculations. It is worth mentioning that calculations using an ansatz would only allow to retrieve the envelope of the gain but not its fine structure.

Figure 1(a) (resp., (b)) displays the analytical (resp., the numerical) MI gain bands for a 20 TW.cm -2 pump propagating trough 1 m of Argon at atmospheric pressure. At low intensities (n 2 eff > 0), the gain displays a X pattern in the (k ⊥ , ω) space. Neglecting higher order dispersion coefficients, four-wave mixing (FWM) processes

ω 0 + ω 0 → (ω 0 + ω) + (ω 0 -ω) are phase matched if the condition N - √ N 2 -S 2 2 ≤ k 2 ⊥ 2k0 -k (2) ω 2 ≤ N + √ N 2 -S 2 2
is respected. At the pump frequency (ω = 0) and considering the temporal response of the medium as instantaneous, instabilities are maximal for k 2 ⊥max = 2k 2 0 I 0 n 2 eff . This instability can then lead to the pulse break-up and consequently to multi-filamentation. Moreover, one can associate to this instability a power P = πI 0 /k 2 ⊥max = λ 2 0 /(8πn 2 eff ) which corresponds to the widely-studied critical power. Neglecting the higher order Kerr terms, if a pulse carries a power above P , then it will lead to a self-similar blowup singularity of Eq. 2 and to the pulse collapse [1].

For other frequencies, instabilities are maximal for

k 2 ⊥max = 2k 0 ( ∑ n≥1 k (2n) (2n)! ω 2n + k 0 I 0 n 2 eff )
. Figure 2 compares the emission angle θ(ω)=k ⊥max /k(ω) obtained with our calculations with those experimentally measured in the visible [24] and in the infrared [25]. Even if MI calculations are intrinsically done with a continuous plane wave with an infinite spatial extension (which does not correspond to filamentation of ultrashort pulses), MI calculations accurately reproduce the conical emission induced during the filamentation. The good agreement between our calculations and the angles measured experimentally consequently confirms that conical emission is indeed generated by off-axis four waves mixing (4WM) during filamentation when n 2 eff >0. The situation significantly changes at higher intensity when n 2 eff < 0. In particular, the X pattern completely disappears but a new mechanism of MI still occurs. As stated beforehand, MI occurs as soon as the condition

D + (D + + N ) + S 2
4 > 0 is respected. This condition is equivalent to

n -< n 2 eff (ω)I 0 < n + , ( 9
)
with

n -= - k 0 D + ( 2 -k (1) 2 ω 2 2k 2 0 ) k (1) 2 ω 2 , n + = - D + 2k 0 .
More particularly, Fig. 3 displays the frequency range where on-axis (k ⊥ =0) MI is possible. When I 0 lies in the range 23.5-31.9 TW.cm -2 , on-axis MI occurs. At higher intensity, ∀ (k ⊥ , ω), n 2 eff I 0 <n -, meaning that no phase matching and consequently, no MI can occur.

As a result, one can distinguish two cases. At moderate intensities (30 TW.cm -2 , Figs. 1(c,d)) such that n -< n 2 eff I 0 < n + , strong on-axis MI is observed. For higher input intensities (40 TW.cm -2 , Figs. 1(e-f)), the gain drastically decreases since the phase matching condition is not fulfilled anymore.

One has to note that Théberge et al. [25] have observed on-axis infrared emission which cannot be explained by off-axis 4WM processes. The MI calculations show that such type of on-axis emission appears as soon as n 2 eff <0.

One should emphasize that the general solution can be used to enhance or reduce the gain at specific frequencies by adjusting the phase and amplitude of the initial seeding. Figure 4 displays the MI gain bands for a pump intensity of 30 TW.cm -2 when a 0 and a 0 ♯ are in phase (red) and out of phase (blue). By adjusting their relative phase, the gain can be enhanced or reduced, the latter being maximal (resp., minimal) when a 0 and a 0 ♯ are in phase (resp., out of phase).

We have shown above that the higher order Kerr terms lead to on-axis MI in the normal dispersion regime. In this paragraph, we study the influence of self-steepening (SS) and space-time defocusing (STD) on the MI band Maioli et al. [24] Present work growth. The effect of these terms has already been discussed in [26]. However, our general expression obtained in section I allows us to better capture the MI dynamics.

To perform this study, we have removed the contributions of these two terms (by setting k (1) = 0) from the analytical and the numerical calculations and we have repeated the same calculations than in the previous section. In Figs. 5(a-b), the intensity of the pump is 20 TW.cm -2 so that n 2 eff > 0. In that condition, neglecting both SS and STD does not lead to significant change of the MI bands. Only a slight deviation of the emission pattern can be noticed at frequencies far from the frequency pump. In the intermediate regime shown in Figs. 5(c-d), both shape and value of the gain change. In particular, neglecting SS and STD leads to a dramatic over-estimation of the MI. This is due to the fact that the coupling between the pump and the MI bands is over-estimated if one neglects SS and STD far from the frequency pump. Hence, without SS and STD, the maximal gain is obtained at ω=0 and ω=2ω 0 while the full calculation gives the frequencies of maximal gain are ω=0.3 ω 0 and ω=1.7 ω 0 (see Fig. 4). At higher intensities, Figs. 5(e,f), the MI gain bands are dramatically over-estimated.

It results from Figs. 5(e,f) that neglecting both SS and STD suppresses the regime where MI is annihilated (high intensity regime). When neglecting both SS and STD, the condition which has to be respected for the MI to occur is reduced to

n 2 eff (ω)I 0 < n + . ( 10
)
When the intensity increases, this condition is always respected which leads to MI as shown in Fig. 6. The lines displayed n2 eff I0 with I0=20 TW.cm -2 (red solid), 30 TW.cm -2 (dashed green), and 40 TW.cm -2 (dotted-dash violet) assuming the Kerr response as instantaneous, i.e., n2 eff (ω)=n2 eff ∀ ω.

B. 1D analysis in presence of Raman induced delayed nonlinear response

In order to underline the generality of the solution derived in section 1, we calculated analytical and numerical solutions of the MI gain induced in a single mode fiber (SMF) in presence of vibrational Raman nonlinearity. We considered the propagation of a 1550 nm 5 GW.cm -2 continuous wave in a 15 cm-long standard SMF (anomalous dispersion k (2) = -2.1 10 -26 s 2 .m -1 and positive nonlinear index n 2 = 2.6 10 -20 m 2 .W -1 ) when vibrational Raman effect is taken into account. Figure 7(a) displays both analytical and numerical MI-Raman gain bands. Since the general solution here takes into account the contributions of both real and imaginary parts of the Raman effect, the global gain is no more symmetric with respect to the pump frequency, with an amplification (resp., absorption) at longer (resp., shorter) wavelengths. We confirm again the excellent quantitative agreement between analytical and numerical results by showing in Fig. 7(b) that the relative error is less than 1%. The slight discrepancies are mainly due to the finite propagation step used during the split-step Fourier numerical simulation. The combined action of parametric and Raman effects was already discussed in the early work of Shen and Bloembergen [17] and later the resulting global (parametric-raman) gain was widely investigated in the context of optical fibers [18,19,21,22]. With our general solution, the results obtained in these works can be retrieved only by calculating the parameter K. However, the gain evaluated with the help of an ansatz function (as shown in Fig. 7(a)) is only a rough estimation of the real gain, in particular around the pump frequency. Moreover, calculating K cannot describe the asymmetry induced by the imaginary part of the Raman response function. When the MI gain bands do not overlap the Raman bands, positive gain is expected in the anti-Stokes band when simply evaluating K, while this part of the spectrum actually undergoes losses as it can be noticed in Fig. 7(a). Frequency detuning (THz) Nowadays, fiber dispersion properties can be engineered to a very high degree, thus allowing phasematching induced by higher order dispersion for ultrabroadband wavelength conversion [27]. For instance, the fourth-order dispersion can induce additional scalar MI bands in the normal dispersion regime [7]. It has been shown that in presence of negative fourth order dispersion near zero dispersion frequency, MI bands can grow up even by pumping in the normal dispersion regime. Consequently, by pumping in the anomalous dispersion regime, it is then possible to observe two pairs of MI sidebands [23]. We consider here this possible complex configuration of the MI process using the waveguide category of photonic crystal fibers (PCF). Figure 8 compares the analytical and numerical gain bands obtained by pumping a photonic crystal fiber which exhibits a low anomalous dispersion between its two zero dispersion wavelengths. Both linear and nonlinear fiber properties can be found in [28]. We used a 2.6 GW.cm -2 pump intensity at 1064 nm and a fiber length of 1 m. It clearly appears that Eq. 8 is also able to perfectly retrieve the two pairs of MI sidebands induced by second and fourth order dispersion terms. Besides the impact of higher order dispersion on the phase matching, our solution describes the asymmetrical shape of MI induced by the Raman gain. Indeed, we have adjusted here the input wave intensity so that the first standard MI sidebands are superposed to the silica Raman gain band (0-30 THz). Consequently, we retrieve again that the global MI-Raman gain is no more symmetric with respect to the pump frequency. As previously, the standard gain evaluation can give a rough estimation of the real gain in specific conditions. Frequency detuning (THz)

Frequency detuning (THz)

C. Impact of Kerr electronic time response on the MI bands growth

In this section, we underline the impact of the time response of the different higher order nonlinear Kerr terms on the induced MI. A recent work has highlighted the influence of a finite nonlinear response time [29] on the spatio-temporal MI growth but by using the usual simplified approach. Conversely, our method gives the exact solution even in presence of a delayed nonlinear response without any approximation. Since the systematic study of any situation of MI is out of the scope of the present paper, we have limited our work to 1D MI gain in argon (as for instance relevant in the frame of hollow core fiber propagation [30,31]) with non purely instantaneous nonlinear responses. Since no accurate information about the time response of the higher order nonlinearities are so far available, we used the same expression for all nonlinearities (i.e., the Debye type relaxation model). We write ∀j, R 2j (t) = 1 τ e -t/τ for t > 0 and R 2j (t) = 0 for t < 0, although the different terms exhibit a priori distinct temporal behaviors. We used three distinct values of τ (0, 0.5 fs, and 2 fs) for pump intensities of 20, 30 and, 40 TW.cm -2 which correspond to the three regimes observed when the Kerr effect is supposed to be instantaneous. Figure 9 displays the MI bands calculated after a 1 m propagation through 1 bar of argon with the same initial noise. In the low intensity regime (Fig. 9(a)), while MI is not allowed for τ =0, on-axis MI is allowed when increasing τ . On the contrary, in the intermediate regime (Fig. 9(b)), a reduction by twenty orders of magnitude of the generated MI bands is observed when increasing the time response. Finally, in the high intensity regime (Fig. 9(c)), increasing the time response leads to MI while phase matching cannot be achieved if τ =0. It appears that even for very small values, the Kerr electronic time response is crucial to accurately describe the efficiency of MI bands generation. In particular, we show that increasing the time response from 0 to 2 fs leads to a drastic change of the MI bands in all intensity regimes.

III. CONCLUSION

In this paper, we have derived the general expression of modulation instability in presence of a continuous plane wave pump. The model, valid in 3D, 2D, and 1D, allows to evaluate the MI gain in presence of higher order Kerr terms and any retarded nonlinear response (including the Raman response). We have shown that, if the propagation of a pulse can be described by the equilibrium between the Kerr terms, then on-axis MI bands grow up as soon as the effective nonlinear index becomes negative. We have also highlighted the impact of selfsteepening, space-time defocusing, and finite time non-linear response on the MI band growth. Finally, in the frame of fiber propagation problem, we have shown that our analytical model perfectly describes the MI bands growth in presence of Raman effect and complex dispersive properties. Moreover, we foresee that comparable analytical solutions can be found for vectorial MI, i.e., for propagation through birefringent medium.

IV. ACKNOWLEDGMENT

Figure 1 .

 1 Figure 1. Color Online. Analytical (a) (resp., (c) and (e)) and numerical (b) (resp., (d) and (f)) MI gain (dB) after propagation of a 20 TW.cm -2 (resp., 30 TW.cm -2 and 40 TW.cm -2 ) pump trough 1 m in argon when taking into account both selfsteepening and space-time defocusing.

Figure 2 .

 2 Figure 2. Color Online. Comparison between experimental measurements of conical emission in the visible (black circle) and in the infrared (red square) with MI calculations (blue line).

Figure 3 .Figure 4 .

 34 Figure 3. Color Online. Diagram displaying the spectral region where on-axis MI occurs (grey region) and lying between n+ and n-(black lines).The lines displayed n2 eff I0 with I0=20 TW.cm -2 (red solid), 30 TW.cm -2 (dashed green), and 40 TW.cm -2 (dotted-dash violet) assuming the Kerr response as instantaneous, i.e., n2 eff (ω)=n2 eff ∀ ω.

Figure 5 .

 5 Figure 5. Color Online. Comparison between numerical and analytical calculations of the gain (dB) in the (k ⊥ , ω) space when neglecting both self-steepening and space-time defocusing. (a,b) Gain induced by a 20 TW.cm -2 pump (n2 eff > 0), (c,d) gain induced by a 30 TW.cm -2 pump (n2 eff < 0) and (e,f) gain induced by a 40 TW.cm -2 pump (n2 eff << 0) after 1 m propagation.

2 Figure 6 .

 26 Figure 6. Color Online. Diagram displaying the spectral region where on-axis MI occurs (grey region) when neglecting both SS and STD. This region is below n+ (black line). The lines displayed n2 eff I0 with I0=20 TW.cm -2 (red solid), 30 TW.cm -2 (dashed green), and 40 TW.cm -2 (dotted-dash violet) assuming the Kerr response as instantaneous, i.e., n2 eff (ω)=n2 eff ∀ ω.

Figure 7 .

 7 Figure 7. Color Online. (a) Analytical (red dots) and numerical (solid blue line) MI gain bands calculated after 15 cm propagation in a SMF fiber with a pump of 5 GW.cm -2 in presence of a delayed Raman nonlinearity. The black dashed line represents the gain calculated using an ansatz function. (b) Relative error between numerical and analytical calculations.

Figure 8 .

 8 Figure 8. Color Online. (a) Numerical (blue line) and analytical (red dots) MI gain bands obtained in 1m-long PCF with two zero dispersion frequencies in presence of a delayed Raman nonlinearity. The black dashed line represents the gain calculated using an ansatz function. (b) Relative error between analytical and numerical calculations.

2 Figure 9 .

 29 Figure 9. Color Online. On-axis gain bands after 1 m propagation through Argon with a pump of 20 TW.cm -2 (a), 30 TW.cm -2 (b), and 40 TW.cm -2 (c) with different Kerr electronic time responses (0 fs (blue line), 0.5 fs (green dots), and 2 fs (red triangles)).

  e iωt dt.

	Writing the equation propagation for a * (-k ⊥ , -ω), us-ing the fact that n 2 eff (ω) = n * 2 eff (-ω) (since all R 2j (t)
	are real) and defining u

We gratefully acknowledge K. Hammani, P. Morin, and G. Millot for very fruitful discussions. This work was supported by the ANR COMOC ANR-07-BLAN-0152-01 and FASTQUAST ITN Program of the seventh FP. B. Kibler acknowledges support from the French Agence Nationale de la Recherche project MANUREVA ANR-08-SYSC-019 and from the Conseil Régional de Bourgogne.


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Regulation of lipogenic genes in obesity

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Lipogenesis describes the process of fatty acid and triglycerides synthesis and is obviously of great relevance to obesity. An increase in lipogenesis will lead to fat accumulation, at least if it is not associated with elevated fat utilization. Thus, it appears that in principle the process of lipogenesis is an attractive target for obesity. This paper will review the most recent information on the molecular regulation of lipogenesis, with emphasis on the role of nuclear hormone receptors and other transcription factors.

Transcriptional regulation of lipogenesis in adipose tissue

The effects of various nutrients and hormones on the expression of lipogenic genes in adipose tissue are mediated by a small number of transcription factors, including SREBP-1c. SREBP-1c (also known as Adipocyte Determination and Differentiation-1) belongs to the basic helix-loop-helix-leucine zipper family of transcription factors. It was initially discovered as a key player of adipocyte differentiation [START_REF] Tontonoz | ADD1: a novel helix-loophelix transcription factor associated with adipocyte determination and differentiation[END_REF]. In addition, SREBP-1c plays a critical role in lipogenesis [START_REF] Horton | Sterol regulatory element-binding proteins: activators of cholesterol and fatty acid biosynthesis[END_REF]. SREBP-1c is first synthesized as a precursor form (110 kDa) anchored in the endoplasmic reticulum, which later undergoes a proteolytic cleavage to generate the mature active form of SREBP-1c (50 kDa). In the nucleus, the mature form of SREBP-1c is able to bind to specific sequences (sterol-regulatory elements SRE and E-boxes) located in the promoter gene of SREBP target genes (Figure 1.1). SREBP-1c probably works in tandem with the adipogenic transcription factor PPAR, which is a direct target of SREBP-1c in adipocytes and contains a Sterol Response Element in its promoter [START_REF] Fajas | Regulation of peroxisome proliferator-activated receptor gamma expression by adipocyte differentiation and determination factor 1/sterol regulatory element binding protein 1: implications for adipocyte differentiation and metabolism[END_REF]. It has been proposed that together with an up-regulation of PPAR expression, SREBP-1c is also able to induce PPAR activity by the production of an endogenous ligand, leading to the stimulation of both adipogenesis and lipogenesis (4). Together with PPAR, important lipogenic genes such as Acetyl-CoA Carboxylase-1 and -2 ( ACC), Fatty Acid Synthase (FAS), Stearoyl-CoA Desaturase-1 (SCD-1), Glycerol-3-Phosphate Acyltransferase (GPAT) and Low Density Lipoprotein Receptor (LDL-R) have been identified as direct targets of SREBP-1c in mature 3T3-L1 adipocytes [START_REF] Lay | Insulin and sterol-regulatory element-binding protein-1c (SREBP-1C) regulation of gene expression in 3T3-L1 adipocytes. Identification of CCAAT/enhancer-binding protein beta as an SREBP-1C target[END_REF]. Another gene identified as a SREBP-1c target in adipocytes is INSIG-1, which is one of two recently discovered polytopic membrane proteins of the reticulum endoplasmic [START_REF] Kast-Woelbern | Rosiglitazone induction of Insig-1 in white adipose tissue reveals a novel interplay of peroxisome proliferator-activated receptor gamma and sterol regulatory element-binding protein in the regulation of adipogenesis[END_REF][START_REF] Yang | Crucial step in cholesterol homeostasis: sterols promote binding of SCAP to INSIG-1, a membrane protein that facilitates retention of SREBPs in ER[END_REF][START_REF] Yabe | Insig-2, a second endoplasmic reticulum protein that binds SCAP and blocks export of sterol regulatory element-binding proteins[END_REF]. In the presence of sterols, INSIGs bind to the SREBP cleavage-activating protein (SCAP), a critical escort protein required for the cleavage and activation of the SREBP family of membrane-bound transcription factors. It appears that INSIG tethers the SCAP-SREBP complex to the endoplasmic reticulum. Recent in vitro studies have shown that stable over-expression of INSIG-1 in 3T3-L1 preadipocytes results in defective adipocyte differentiation, which is likely due to impaired fatty acid and triglyceride synthesis [START_REF] Li | Insig-1 "brakes" lipogenesis in adipocytes and inhibits differentiation of preadipocytes[END_REF]. Although the relevance of the SREBP-1c mediated upregulation of INSIG-1 in adipose tissue remains to be demonstrated, it can be speculated that it provides a feedback mechanism by which nuclear SREBP-1c can modulate its maturation and, by limiting the processing of SREBPs precursors, maintain a check on lipogenesis.

Apart from SREBP, several other nuclear factors play a critical role in the transcriptional control of adipose lipogenesis. Numerous loss and gain of function experiments have shown that the nuclear hormone receptor PPAR plays a pivotal role in adipocyte differentiation (reviewed in [START_REF] Rosen | PPARgamma : a nuclear regulator of metabolism, differentiation, and cell growth[END_REF]), and is also an important activator of lipogenesis. Several lipo-and/or adipogenic genes have already been identified as direct PPAR targets, including Lipoprotein Lipase, CD36, Adipocyte Lipid Binding Protein and cytosolic Glycerol 3-phosphate dehydrogenase. Recently, INSIG-1 was reported as a novel PPAR target gene (Fig. 1) [START_REF] Kast-Woelbern | Rosiglitazone induction of Insig-1 in white adipose tissue reveals a novel interplay of peroxisome proliferator-activated receptor gamma and sterol regulatory element-binding protein in the regulation of adipogenesis[END_REF]. Considering that INSIG-1 appears to inhibit lipo-and adipogenesis, it is hard to reconcile this observation with the pro-adipogenic role of PPAR [START_REF] Li | Insig-1 "brakes" lipogenesis in adipocytes and inhibits differentiation of preadipocytes[END_REF].

Likely, fine-tuning of adipogenesis is achieved by balanced expression of several opposing factors that include PPAR, SREBP-1c, and INSIG.

Another recently identified direct target gene of PPAR in adipocytes is Acyl-CoA Binding Protein (ACBP) [START_REF] Helledie | The gene encoding the Acyl-CoA-binding protein is activated by peroxisome proliferator-activated receptor gamma through an intronic response element functionally conserved between humans and rodents[END_REF]. It was proposed that ACBP would repress PPAR-mediated transactivation induced by exogenous fatty acids, thereby inhibiting 3T3-L1 adipogenesis. However, additional research is necessary to establish the potential effects of ACBP on PPAR transactivation.

With respect to the role of PPAR in lipid storage in mature fat cells, it has been observed that both heterozygous PPAR mutant mice and adipose specific PPAR-deficient mice exhibit smaller fat stores on a high fat diet [START_REF] Kubota | PPAR gamma mediates high-fat diet-induced adipocyte hypertrophy and insulin resistance[END_REF][START_REF] Miles | Improved insulinsensitivity in mice heterozygous for PPAR-gamma deficiency[END_REF][START_REF] He | Adipose-specific peroxisome proliferator-activated receptor gamma knockout causes insulin resistance in fat and liver but not in muscle[END_REF]. In adipose specific PPAR-deficient mice mRNA expression of genes involved in both lipogenesis and adipogenesis were strongly down-regulated, illustrating an absolute requirement for PPAR [START_REF] He | Adipose-specific peroxisome proliferator-activated receptor gamma knockout causes insulin resistance in fat and liver but not in muscle[END_REF][START_REF] Zhang | Selective disruption of PPARgamma 2 impairs the development of adipose tissue and insulin sensitivity[END_REF]. Fibrosis, macrophage infiltration and hypertrophy, implying loss of more than 80% of adipocytes, were evident in mutant fat. Thus, PPAR is not only essential for the early steps of the adipogenesis program but also for post-differentiation and survival of mature white adipocytes in vivo [START_REF] He | Adipose-specific peroxisome proliferator-activated receptor gamma knockout causes insulin resistance in fat and liver but not in muscle[END_REF][START_REF] Imai | Peroxisome proliferatoractivated receptor gamma is required in mature white and brown adipocytes for their survival in the mouse[END_REF].

While it is clear that SREBP-1c and PPAR are extremely important regulators of adipo-and lipogenesis, recent studies have drawn attention to other nuclear hormone receptors. One of these receptors is the Estrogen-Related Receptor alpha ERR Deletion of ERR in mice reduces body weight and peripheral fat deposits, while food consumption and energy expenditure are unaffected [START_REF] Luo | Reduced fat mass in mice lacking orphan nuclear receptor estrogen-related receptor alpha[END_REF]. At the gene level, expression of several enzymes involved in lipogenesis was down-regulated in white adipose tissue. In line with this, de novo lipogenesis was shown to be reduced in the mutant animals, suggesting a stimulatory role of ERR on lipogenesis. These data suggest that ERR functions as a metabolic regulator with an important effect on fat synthesis.

Transcriptional regulation of hepatic lipogenesis

Consumption of large amounts of carbohydrates stimulates the conversion of glucose to fatty acids by upregulating glycolytic and lipogenic enzymes. The effects of carbohydrate feeding are mediated by insulin and glucose, which activate distinct signalling pathways. Upregulation of lipogenic genes by insulin partially occurs via the Upstream Stimulatory Factor (USF). USFs are ubiquitous bHLHleucine zipper transcription factors that are able to form homo and/or heterodimers. They modulate expression of genes such as FAS and ACC by direct binding to promoter gene sequences called E-Boxes. Studies with mice lacking USF1 and/or USF2 have provided compelling evidence for their role in the stimulatory effect of insulin and glucose on lipogenesis. In USF1-/-or USF2-/-mice, FAS expression in liver was strongly impaired after a fasting/refeeding cycle, demonstrating that USFs are critical factors for the transcriptional activation of FAS by diet [START_REF] Casado | Essential role in vivo of upstream stimulatory factors for a normal dietary response of the fatty acid synthase gene in the liver[END_REF]. At the molecular level, USFs exerted their stimulatory effect on FAS transcription via an E-box motif located in the promoter of the FAS gene. This motif is likely shared with SREBP-1, which also is a potent activator of FAS gene expression. Studies with mice lacking or over-expressing SREBP-1c have indicated that this transcription factor is responsible for the coordinate induction of numerous lipogenic genes in liver, including ACC, FAS, SCD-1, GPAT, ACL, Malic Enzyme, Glucose-6 Phosphate Dehydrogenase, Spot 14, and Long Fatty Acid Elongase. Accordingly, over-expression SREBP-1c was associated with a dramatic build-up of hepatic triglycerides [START_REF] Shimomura | Increased levels of nuclear SREBP-1c associated with fatty livers in two mouse models of diabetes mellitus[END_REF]. At the present time, SRE have been identified in the promoter gene of FAS [START_REF] Bennett | Sterol regulation of fatty acid synthase promoter. Coordinate feedback regulation of two major lipid pathways[END_REF], ACC [START_REF] Magana | Sterol regulation of acetyl coenzyme A carboxylase promoter requires two interdependent binding sites for sterol regulatory element binding proteins[END_REF], ACL [START_REF] Sato | Transcriptional regulation of the ATP citrate-lyase gene by sterol regulatory element-binding proteins[END_REF], GPAT [START_REF] Ericsson | Identification of glycerol-3-phosphate acyltransferase as an adipocyte determination and differentiation factor 1-and sterol regulatory element-binding protein-responsive gene[END_REF], Spot 14 [START_REF] Koo | Glucose and insulin function through two distinct transcription factors to stimulate expression of lipogenic enzyme genes in liver[END_REF], SCD-1 [START_REF] Tabor | Identification of conserved cis-elements and transcription factors required for sterol-regulated transcription of stearoyl-CoA desaturase 1 and 2[END_REF], and ACBP [START_REF] Sandberg | The gene encoding acyl-CoA binding protein is subject to metabolic regulation by both SREBP and PPARalpha in hepatocytes[END_REF]. The last gene encodes a small intracellular protein that is able to bind long chain fatty acyl-CoAs.

Although the role of ACBP in vivo is not very clear, it can be speculated that ACBP participates in de novo fatty acid biosynthesis. Indeed, high expression levels of ACBP have been observed in both hepatocytes and adipocytes and in liver ACBP expression is stimulated by insulin and repressed by fasting.

While the expression of many lipogenic enzymes is enhanced by insulin, optimal transcription of most lipogenic genes requires high carbohydrate levels as well. In concordance with this observation, glucose has been identified as a potent activator of lipogenesis not only by acting as substrate, but also as an important regulatory molecule. Recent data indicate that increased glucose metabolism activates an intracellular signaling pathway, probably involving xylulose 5-phosphate, that transcriptionally regulates genes encoding lipogenic enzymes via the Carbohydrate Response Element Binding Protein "ChREBP", a basic helix-loop-helix/leucine zipper transcription factor. ChREBP was identified and purified by taking advantage of its binding to the carbohydrate response element "ChRE" within the promoter of the L-type pyruvate kinase gene. ChREBP is stimulated by high concentrations of glucose that promote both translocation of ChREBP from the cytosol to the nucleus coupled with binding of ChREBP to a ChRE [START_REF] Yamashita | A glucose-responsive transcription factor that regulates carbohydrate metabolism in the liver[END_REF]. Importantly, ChREBP becomes active in response to high glucose concentrations in liver independently of the insulin level. Mice with a targeted disruption of the ChREBP gene were recently generated, showing reduced expression of several glycolytic genes [START_REF] Iizuka | Deficiency of carbohydrate response element-binding protein (ChREBP) reduces lipogenesis as well as glycolysis[END_REF].

In addition, the mRNA levels of ACC1, ACC2, LCE, ME, SCD1 and FAS were significantly lower in the ChREBP-/-mice compared to the wild-type littermates. Follow-up studies indicated that FAS, ACC and Spot 14 are direct targets of ChREBP with a ChRE present in their promoters [START_REF] Ishii | Carbohydrate response element binding protein directly promotes lipogenic enzyme gene transcription[END_REF][START_REF] Shih | Definition of the carbohydrate response element of the rat S14 gene. Evidence for a common factor required for carbohydrate regulation of hepatic genes[END_REF].

These results suggest that ChREBP, apart from its predictable role in glucose utilization, is also of importance for fatty acid biosynthesis.

Knock-out mice have also turned out to be an invaluable tools to demonstrate a critical role for C/EBP in hepatic lipogenesis. Deletion of C/EBP in combination with leptin deficiency caused diminished lipogenic gene expression and was associated with a significant decrease in hepatic triglyceride content [START_REF] Matsusue | Hepatic CCAAT/enhancer binding protein alpha mediates induction of lipogenesis and regulation of glucose homeostasis in leptin-deficient mice[END_REF]. Development of the fatty liver was exacerbated by high fat feeding but much less so in liver-specific C/EBP null mice. Thus, in addition to its functional role in adipogenesis, C/EBP stimulates hepatic lipogenesis (reviewed in [START_REF] Rosen | Transcriptional regulation of adipogenesis[END_REF].

The liver X receptor is a nuclear hormone receptor that is highly expressed in liver and that is activated by oxysterols. It was first shown to play an important role in the feed-forward control of bile acid synthesis from cholesterol by upregulating cholesterol 7-alpha hydroxylase expression. More recent studies indicate that LXRs are also potent stimulators of lipogenesis in liver in mice. Functional LXR Response Elements were identified in the promoters of the PPAR, SREBP-1c and FAS gene [START_REF] Repa | Regulation of mouse sterol regulatory element-binding protein-1c gene (SREBP-1c) by oxysterol receptors, LXRalpha and LXRbeta[END_REF][START_REF] Schultz | Role of LXRs in control of lipogenesis[END_REF]. Accordingly, activation of LXRs by synthetic agonists leads to a robust induction of lipogenic genes that is translated into an increase of both plasma triglyceride and phospholipid levels [START_REF] Schultz | Role of LXRs in control of lipogenesis[END_REF]. Studies with LXRs mutant null mice further established the key role of LXRs as factors that mediate the effect of insulin on lipogenesis. [START_REF] Tobin | Liver X receptors as insulin-mediating factors in fatty acid and cholesterol biosynthesis[END_REF]. In cultured primary hepatocytes insulin increases the half-life of LXR mRNA, which in turn increases LXRprotein.

Although both insulin and LXR stimulate SREBP-1c gene expression, they do so by different pathways. According to Hegarty et al. LXR activation stimulates the production of the precursor SREBP-1c protein, yet this effect is not translated into the mature nuclear form [START_REF] Hegarty | From The Cover: Distinct roles of insulin and liver X receptor in the induction and cleavage of sterol regulatory elementbinding protein-1c[END_REF]. In contrast, insulin was shown to efficiently and rapidly stimulate the cleavage and maturation of the precursor SREBP-1c form. The molecular explanation for this differential effect may lie with INSIG-2, which is upregulated by LXR. According to this scenario, INSIG-2 retains the mature form of SREBP-1c in the endoplasmic reticulum, which would thus escape the maturation process.

The nuclear hormone receptor PPAR plays a pivotal role in the adaptive response to fasting by upregulating many genes involved in hepatic fatty acid oxidation, ketogenesis and gluconeogenesis. In this context, it is remarkable that PPAR upregulates several genes involved in fatty acid elongation and desaturation, including SCD-1, SCD-2, delta 5 desaturase, delta 6 desaturase, LCE, TER and KAR (reviewed in [START_REF] Mandard | Peroxisome proliferator-activated receptor alpha target genes[END_REF][START_REF] Kersten | Peroxisome proliferator-activated receptor alpha mediates the adaptive response to fasting[END_REF] and unpublished data). The enzymes SCD-1 and SCD-2 catalyze the conversion of stearic acid into oleic acid and are thus essential for the synthesis of mono-and poly-unsaturated fatty acids. An important question that arises is why a single factor such as PPAR would stimulate both fatty acid oxidation and fatty acid elongation/desaturation. It has been proposed that upregulation of desaturases expression by PPAR may generate unsaturated fatty acids, which are agonists for PPAR. Alternatively, it is possible that the high turn-over of plasmatic phospholipids membrane requires a constant level of (unsaturated) free fatty acids to prevent cell death. The story is even more complex, since delta-5, -6 and -9 desaturase expressions are also under the transcriptional control of SREBP-1c, which is generally considered to have a function completely opposite to that of PPAR [START_REF] Matsuzaka | Dual regulation of mouse Delta(5)-and Delta(6)-desaturase gene expression by SREBP-1 and PPARalpha[END_REF].

PPAR mediates the stimulatory effect of poly-unsaturated fatty acids (PUFAs) on hepatic fatty acid oxidation. However, PUFAs also inhibit hepatic lipogenesis by a process that is PPAR-independent.

Studies by a variety of groups subsequently showed that PUFA suppress lipogenesis by inhibiting expression of SREBP-1c [START_REF] Sekiya | Polyunsaturated fatty acids ameliorate hepatic steatosis in obese mice by SREBP-1 suppression[END_REF][START_REF] Xu | Sterol regulatory element binding protein-1 expression is suppressed by dietary polyunsaturated fatty acids. A mechanism for the coordinate suppression of lipogenic genes by polyunsaturated fats[END_REF][START_REF] Yahagi | A crucial role of sterol regulatory element-binding protein-1 in the regulation of lipogenic gene expression by polyunsaturated fatty acids[END_REF][START_REF] Clarke | Dietary polyunsaturated fatty acid regulation of gene transcription[END_REF]. At the molecular level, it has been reported that PUFA are able to act as competitive LXR antagonists which would result in the downregulation of SREBP-1c and other lipogenic genes (44) [START_REF] Yoshikawa | Polyunsaturated fatty acids suppress sterol regulatory element-binding protein 1c promoter activity by inhibition of liver X receptor (LXR) binding to LXR response elements[END_REF]. However, others have argued that down-regulation of SREBP-1 by PUFAs is independent of LXR [START_REF] Pawar | The role of liver X receptoralpha in the fatty acid regulation of hepatic gene expression[END_REF]. Further studies are required to determine the precise molecular mechanisms responsible for this discrepancy.

Hepatic lipogenesis in steatosis

A major metabolic consequence of obesity is insulin resistance, which is usually accompanied by storage of triglycerides in the liver. It is believed that the excess release of free fatty acids from adipose tissue lipolysis accounts for the triglycerides accumulation in liver. While PPAR is barely expressed in liver under basal conditions, its expression is markedly increased in animals models of insulin resistance and fatty liver, suggesting a role for PPAR in hepatic steatosis [START_REF] Chao | Adipose tissue is required for the antidiabetic, but not for the hypolipidemic, effect of thiazolidinediones[END_REF]. Indeed, specific deletion of hepatic PPAR improves fatty liver in two mouse models of hepatic steatosis (ob/ob and AZIP), yet worsens hyperglycemia and insulin resistance [START_REF] Matsusue | Liver-specific disruption of PPARgamma in leptindeficient mice improves fatty liver but aggravates diabetic phenotypes[END_REF][START_REF] Gavrilova | Liver peroxisome proliferator-activated receptor gamma contributes to hepatic steatosis, triglyceride clearance, and regulation of body fat mass[END_REF]. In addition, forced expression of PPAR1 in liver of PPAR-deficient mice using adenovirus leads to the strong induction of both lipogenic and adipogenic genes (including aP2, adipsin, LPL, CD36, FAS, SCD-1), indicating transformation of hepatocytes towards adipocyte-like cells [START_REF] Yu | Adipocyte-specific gene expression and adipogenic steatosis in the mouse liver due to peroxisome proliferator-activated receptor gamma1 (PPARgamma1) overexpression[END_REF]. Similarly, forced expression of PPAR2 in hepatocytes was shown to lead to both hepatic lipid accumulation and induction of lipogenic and adipogenic gene expression [START_REF] Schadinger | PPAR{gamma}2 regulates lipogenesis and lipid accumulation in steatotic hepatocytes[END_REF]. Thus, over-expression of PPAR appears to be both necessary and sufficient for hepatic steatosis. Accordingly, in certain pathological conditions, over-expression of PPAR in liver may contribute to steatosis.

Nuclear hormone receptors and the control adipocytokine gene expression

In the past few years it has become clear that fat tissue is not merely a storage depot for excess fat but also is an active endocrine tissue, which secretes a number of biologically active proteins with putative role in metabolic syndrome. Several proteins secreted by adipocytes that affect satiety and/or energy homeostasis have been identified. These so-called adipocytokines include resistin, adiponectin, acylation-stimulating protein, FIAF, visfatin, leptin, adipsin, plasminogen activator inhibitor-1, renin angiotensin system, metallothioneins and the inflammatory cytokines interleukin-6, tumor necrosis factor- "TNF-"and tumor growth factor-. These molecules have been linked to a wide range of clinical abnormalities such as obesity, atherosclerosis, insulin resistance and type 2 diabetes mellitus (reviewed in (52)). For example, adipose TNF- mRNA levels are increased in several models of genetically induce obesity and the same is true for the plasma resistin level. Both proteins seem to play a role in obesity-induced insulin resistance, at least in mice. While resistin and TNF- diminish insulin sensitivity, a large body of evidence in mice and human indicates that adiponectin improves insulin sensitivity. Indeed, plasma levels of adiponectin were found to be decreased in obese patients and were positively correlated with insulin resistance [START_REF] Lihn | Adiponectin: action, regulation and association to insulin sensitivity[END_REF] [START_REF] Trujillo | Adiponectin--journey from an adipocyte secretory protein to biomarker of the metabolic syndrome[END_REF].

An adipocytokine that has major effects on triglyceride synthesis in adipocytes is acylation stimulating protein "ASP" [START_REF] Cianflone | Critical review of acylation-stimulating protein physiology in humans and rodents[END_REF]. The uptake and re-esterification of nonesterified fatty acids by adipose tissue is enhanced by ASP, probably by stimulating the activity of diacylglycerol acyltransferase "DGAT", a key enzyme in triglyceride synthesis. Moreover, ASP stimulates glucose uptake and transport in human adipocytes, thereby providing the substrates for further triglyceride synthesis.

An interesting newly (re)discovered adipocytokine is visfatin [START_REF] Fukuhara | Visfatin: A Protein Secreted by Visceral Fat that Mimics the Effects of Insulin[END_REF]. Injection of recombinant visfatin was shown to decrease blood glucose in insulin resistant or deficient mice. Furthermore, like insulin, visfatin stimulated glucose uptake by cultured adipocyte and muscle cells and decreased glucose output by hepatocytes. Interestingly, visfatin was found to stimulate triglyceride synthesis and accumulation in adipocytes to a similar extent as insulin. The precise molecular mechanism behind this observation is unclear but is likely linked to the insulin pathway, since visfatin was shown to bind the insulin receptor.

Thiazolidinediones influence the expression of many of these adipocytokines both positively and negatively. Leptin, TNF and resistin are negative targets of TZDs (57) Resistin might mediate the effect of TZD on insulin sensitivity, at least in mice. Adiponectin, in contrast, is a direct positive target gene of PPAR [START_REF] Iwaki | Induction of adiponectin, a fat-derived antidiabetic and antiatherogenic factor, by nuclear receptors[END_REF], which provides a molecular explanation for the induction of adiponectin gene expression by TZDs. The fasting-induced adipose factor FIAF/ANGPTL4 is also a direct positive target of PPAR, expression of which is upregulated by TZDs in both mouse and human adipocytes (59 and unpublished data). Visfatin might be regulated by PPAR ligands as well, although this remains to be demonstrated. The possible stimulatory role of PPAR on visfatin gene expression would be of interest in the context of type 2 diabetes, since recombinant visfatin protein was shown to promote glucose uptake by adipocytes and to suppress glucose release by hepatocytes [START_REF] Fukuhara | Visfatin: A Protein Secreted by Visceral Fat that Mimics the Effects of Insulin[END_REF]. In summary, PPAR not only has a role in adipose lipogenesis and adipogenesis but also controls different endocrine pathways that are critical for whole body energy homeostasis.

Targeting lipogenesis for obesity ?

In principle, the process of lipogenesis appears as an attractive target for the treatment of obesity. However, two major issues likely impose severe restrictions on the applicability of lipogenesis inhibitors in obesity management. The first issue is that conversion of carbohydrates and amino acids to fatty acids appears to be of minor significance in human individuals consuming a regular diet that is high in fat. Whereas in mice conversion of carbohydrates to fatty acids is very active, detailed metabolic studies have shown that this situation is not analogous in humans. Thus, inhibiting an enzyme such as FAS in liver and adipose tissue is unlikely to have major consequences in humans. This is not true for enzymes involved in synthesis of triglycerides from fatty acids, which are highly active in humans in a variety of organs. The second issue that questions the usefulness of targeting lipogenesis is that inhibition of lipogenesis in fat tissue, without concomitantly activating energy expenditure, will merely cause a redistribution of fat storage from adipose tissue to other tissues such as liver and pancreas, a process which is now considered highly undesirable. Indeed, it seems that the effectiveness of thiazolidinediones toward ameliorating insulin resistance is positively correlated with their effect on subcutaneous fat gain, suggesting that at least for diabetes management strategies should promote preferential storage of fat in subcutaneous stores.

Conclusions and perspectives

In conclusion, lipogenesis is tighly controlled by nutritional, hormonal and transcription factors. In the last few years, major advances have been made to identify the complex regulatory networks involved in the regulation of lipogenesis. One of the most pressing challenges that researchers will have to deal with is to extend their findings from rodents to humans. Finally, although lipogenesis may appear as an attractive target for the pharmacological treatment of obesity, it has become increasingly clear that inhibiting triglyceride storage in subcutaneous fat tissue is more likely to do harm than any good.

  fax: (+31) 317 483342 e-mail: stephane.mandard@wur.nl

Fig. 1 . 1 :

 11 Fig. 1.1: Regulation of lipogenic gene expression in adipocyte.

Fig. 2 . 1 :

 21 Fig. 2.1: Regulation of lipogenic gene expression in hepatocyte.

Acknowledgements:

The work of the authors is supported by the Netherlands Organization for Scientific Research (NWO), the Dutch Diabetes Foundation, the Wageningen Centre for Food Sciences (WCFS), and the Royal Netherlands Academy of Arts and Sciences. The authors would like to thank Rinke Stienstra for critical reading of the manuscript.

Bibliography


Iván Iglesias-García

Isabel Brandariz

Emilia Iglesias

Fluorescence study of tetracaine-cyclodextrin inclusion complexes

Keywords: Cyclodextrins, Tetracaine, Fluorescence, Inclusion complex Cyclodextrins, tetracaine, fluorescence, stability constants

The steady state fluorescence emission from of the local anesthetic tetracaine (TCA) in watersolvent mixtures and in the presence of α-, β-, and γ-cyclodextrin (CD) was investigated at various pH values. Emission was observed from the locally (LE) and the intramolecular charge transfer (ICT) excited states. The TCA:CD system found to be characterized by 1:1 associate in every case. The association constants of each complex were determined.

Graphical Abstract

Tetracaine shows dual fluorescence in watersolvent mixtures and in aqueous cyclodextrin solutions; the formation of 1:1 inclusion complexes has been studied in α-, β-, and γ-CD.

Introduction

It is generally believed that interaction of the anesthetic molecules with membrane lipids or membrane proteins leads to the inactivation of neuronal ion channel activity. 1 Tetracaine [TCA, 2-dimethylaminoethyl-4(N-butylamine) benzoate] is one of the most widely studied tertiary amine local anesthetic, 2,3 The absorption and the ability of this drug to cross biological membranes are pH-dependent, since TCA can exist in neutral, singly or doubly positively charged form. The use of natural cyclodextrins as drug carriers provides the solution to many difficulties in drug formulation that includes poor bioavailability, limited shelf life, or restricted utility when the drug is irritating to skin, membranes, tissues, and so on.

Natural cyclodextrins (CDs) are water soluble cyclic-oligosaccharides that behave in aqueous solutions as pre-organized hosts that form hydrophobic cavities with hydrophilic external walls. [START_REF] Szejtli | Cyclodextrin Technology. Topics in inclusion science[END_REF][START_REF][END_REF] The formation of inclusion complex means that compatibility is reached between the host and the guest in terms of the polarity and stereochemistry; in this process, the guest molecule experiences a non-polar environment and possesses a decreased freedom for bulk and intramolecular rotations, in the rigid nonpolar CD cavity. This ability of CDs to accommodate guest molecules of the appropriate size in their cavities has been utilized to control photophysical and photochemical properties 6,7 or chemical reactivity 8,9 of molecules. The restricted shape and size and the hydrophobic nature of the cavities of CDs offer the opportunity to carry out selective phototransformations and to investigate specific aspects of photochemical mechanisms. In addition, in chromophoric species that can be entirely enclosed into de CD cavity, direct protection from quencher can take place as long as the probe remains inside the cavity. The analysis of the changes in physical and chemical properties of the complexed guest provides information about the structure of the complex, the stoichiometry, and stability.

The intramolecular charge transfer (ICT) phenomenon, firstly introduced by Grabowski and coworkers 10 to explain the dual fluorescence emission observed by Lippert et al. 11 subject of several studies from the early 1970s. [12][13][14][15][16][17][18][19][20][21][22][23][24] On the basis of the fluorescence features, such as maxima of emission bands, fluorescence lifetimes and relative intensities of the short-wavelength (SW) due to locally excited state (LE) and long-wavelength (LW) due to the ICT emissions of DMABN complexes, different geometries were proposed for the inclusion in the αor β-CD cavities.

Recently, the study of fluorescence emission and of chemical reactivity of novocaine (2diethylamino-ethyl,4-aminobenzoate) has been carried out in aqueous β-CD solutions. 25 In the present study we report on the influence of complexation with α-, β-, and γ-CD on the TCA excited state properties; the obtained results provide information on the effect of environment on LE and ICT emissions and on the stability and geometry of the inclusion complexes.

Experimental

High purity tetracaine•HCl and cyclodextrins were purchased from Sigma-Aldrich. Dioxane, DMSO, MeOH, and acetonitrile (uvasol grade) were from Merck. Twice distilled water obtained under potassium permanganate solution, were used for preparing solutions and buffers. Concentrated stock solution of the hydrochloric anesthetic salt was prepared in water and frequently renewed. The concentration of TCA was determined by weighing the appropriate quantity of salt and the consequent dilution of the stock solution.

Potentiometric measurements were carried out in a jacket cell that was kept at constant temperature (25 °C) by circulating water from a thermostat. Purified nitrogen was bubbled through the solutions to ensure thorough homogenization and CO 2 removal. A Crison microBu 2030 automatic burette furnished with a 2.5 mL syringe for dispensing the titrant was used. The burette was controlled via a computer that was used to read the emf values from a Crison micropH 2000 pHmeter, connected to two electrodes: a glass electrode Radiometer pHG211 and a reference electrode Radiometer REF201.

Absorption spectra were recorded on a Uvikon-Kontron 942 double beam spectrophotometer fitted with thermostated cell holders at 25 ºC. The concentration of tetracaine was about 6.5×10 -5 M.

Data acquisition and analysis of UV-vis spectra were performed with software supported by the manufacturer and converted to ASCII format for their analysis with common packet programs.

Steady-state fluorescence spectra were recorded on an SLM Aminco-Bowman series 2 spectrofluorimeter fitted with a 150WXenon lamp at 25 o C. Data acquisition and analysis of fluorescence spectra were performed with the Fluorescence Data Manager Software supported by Aminco. Excitation and emission slits were fixed at 4 and 2 nm, respectively, the excitation wavelength was set at 300 nm, and the emission intensity of fluorescence was recorded at 346 nm. In all measurements the emission spectra, collected from 315 to 400 nm, was obtained from solutions of constant tetracaine concentration of ~9 µM, choose in the way that the absorbance of the sample at 300 nm (the excitation wavelength) was lower than 0.2 absorbance units, and variable CD concentration.

Results

Tetracaine is a tertiary amine that also contains a secondary amine group bonded to the phenyl ring (Scheme 1).

Scheme 1

The absorption spectrum of TCA (62 µM) in water at 25 ºC is characterized by two structureless bands with λ max =226 nm and λ max =309 nm. Both the longer (LW, assigned to the So→S 1 transition) and the shorter (SW, assigned to the So→S 2 transition) wavelength bands exhibit a slight blue shift in low dielectric media, Table 1. In strong aqueous acid media (pH<2) the LW absorption markedly decreases, whereas, in mild acid (or neutral) aqueous media, the maximum was observed at 310 nm, but shifts to 306 nm in alkaline medium. The pK a values of tertiary and secondary amine groups of TCA are largely different. The corresponding values have been determined by potentiometry at 0.10 M ionic strength NaCl. The results are reported in Table 2 along with the values determined from the spectroscopic titration method 26 and from literature data. [START_REF] Perrin | pKa Prediction for organic acids and bases[END_REF] Therefore, three different absorbing species of tetracaine can be considered on varying the pH of the medium, which correspond to TcH 2 +2 , TcH + and Tc (Scheme 1). At very low pH, the strongest absorption was observed at 228 nm (SW, due to π→π* transition) whereas, either mild acid or alkaline medium favour the low energetic n→π* transition (the auxochrome secondary amine group is deprotonated) and a strong absorption at 310/306 nm (LW) was respectively observed for the acid and alkaline medium. A progressively blue shift of LW and SW bands on increasing pH was observed.

The fluorescence spectrum of tetracaine in water is strongly affected by the medium acidity showing dual fluorescence emission and the intensity is strongly enhanced with the pH of the medium.

Figure 1 shows the corresponding spectra recorded in strong acid (pH<1.5), mild acid (pH~5), basic (pH~10), and in alkaline medium. The different investigated pH regions reflect the appearance of also three different fluorescent species of TCA showing distinct fluorescence quantum yield. As the pH regions in fluorescence are similar to those observed in the ground state, it suggest no significant changes on the pK a value of TCA in the excited state in accordance with the Fırster approximation. 26,[START_REF] Förster | [END_REF] In addition, at pH>4 (existence of TcH + or Tc) dual fluorescence is observed. The emitting chromophore in tetracaine is the benzene ring, and emission occurs from both the lowest (π, π*) singlet state (LE) and the intramolecular charge transfer (ICT) state, which is originated from the N-lone pair of the secondary amine group (acting as the electron donor group) to π-orbitals of the benzene ring or to the carbonyl group that is characterized by a high electron accepting character. An alternative view could attribute the dual emission to a π→π* transition involving a considerable charge transfer character. The scan [2] in Fig. 1 corresponds to the emission of TcH + species, that is, when the N-atom of the secondary amine group is deprotonated. Scans [3] and [4] were obtained when the tertiary amine group is also deprotonated. The emission through ICT state decreases faster than the LE emission as the non-polar solvent concentration increases. Plotting the fluorescence emission intensity, I F , for both maximum emission bands, as a function of the solvent concentration, result the Figs. 2a and2b, respectively to the emission at 357 (LE) and 377 nm (ICT). Two different behaviours are clearly observed; for both dioxane and DMSO solvents, high non-linear fluorescence emission was observed; whereas, in the presence of acetonitrile good straight lines were drawn; methanol can be seen as an intermediate situation. In S 1 state, the polarity of the secondary amine group is further increased due to the increased charge transfer interactions from this group to the carbonyl moiety, and the nitrogen-lone pair of the secondary amine group acts as a better proton donor. Table 1 reports the fluorescence intensity ratio of LE to ICT states measured for the same solvent concentration, along with solvent properties [START_REF] Reichardt | Solvent and Solvent Effects in Organic Chemistry[END_REF] and characteristics of the absorption spectra. It can be noted the decisive role of H-bonding with the solvent -the case of water and MeOH-to stabilized the optimal configuration required to observe emission from the ICT state (low ratio of I F (LE)/I F (ICT) determined at 10 M solvent concentration).

Another parameter that conditions the ICT state emission is the restricted motion imposed to the fluorophore. The nature of the CD microvessel constraints nuclear motions and may stabilize conformations which are less favoured in free solution. On the other hand, the hydrophobic nature of the cavity can affect photoprocesses which are sensitive to solvent polarity or dielectric properties. For that, we investigated the effect of rigid hosts such as cyclodextrins (CDs) in the emission of fluorescence of tetracaine. As mentioned elsewhere, CDs are very soluble in water but their cavity is hydrophobic. Therefore, hydrophobic species can be "extracted" from water and enclosed in the CDs, with a consequent change in the luminescence properties.

Figure 3a shows the steady-state fluorescence spectra of tetracaine as a function of the acidity of the medium at constant [β-CD]; whereas, Fig. 3b shows the effect of β-CD concentration on the fluorescence emission intensity recorded in aqueous solutions of 0.067 M of sodium acetate. Under all experimental conditions, the addition of β-CD at increasing concentrations (up to 9 mM) induces normal spectral variations characterized by a progressive blue-shift of the maximum to ca 357 nm and large enhancement of emission intensity with respect to water (compare Fig. 1 and3a). In addition, the emission through the LE state is favoured over that of ICT state -the band at 357 nm increases with both the β-CD concentration (Fig. 3b) and with the stability of the complex -. The former statement is, in fact, the consequence of the latter, since as more CD is present in the medium, the quantity of the TCA forming complexes increases in accordance with eq. ( 1) for the case of 1:1 stoichiometry of the inclusion complexes. This behaviour is indicative of inclusion complexes formation of 1:1 stoichiometry. It can also be seen, the strong effect of pH on the stability of the complex: the highest increase in I F provoked by the lowest cyclodextrin concentration was observed in aqueous carbonate/bicarbonate buffer.

To understand the concept of stability of the inclusion complexes, one need to know which tetracaine species is included into the CD-cavity. Then, by considering the acid-base equilibrium of tetracaine stated in Scheme 1, and that of cyclodextrin, which includes the ionization of a secondary HO-group in alkaline medium (the pK a is 12.2) [START_REF]The pK a values determined at 25 ºC by potentiometry are 12.33, 12.20, and 12.08, respectively to α-, β-, and γ-cyclodextrin; see Comprehensive Supramolecular Chemistry[END_REF] , four possible different complexes of 1:1 stoichiometry can be postulated that, a priori, could be TcH 2 +2

•CD, TcH + •CD, Tc•CD, or Tc•CD -, i.e. neutral CD complexed with dication-, monocation, and neutral forms of tetracaine and ionized CD complexed with neutral tetracaine. The presence of two acid-base equilibrium in TCA molecule, along with the own acid-base equilibrium in cyclodextrin, and four possible complexation equilibrium processes, makes the CD-TCA system a very complex one. Nevertheless, the system results notably simplified when the experiments are performed under determined working conditions; in addition, the main driving forces for complex formation is the hydrophobic effect that considers the exclusion of the hydrophobic guest from the bulk water solvent when intermolecular interactions are not favourable, accompanied by the release of high energy water molecules from the CD cavity. This fact explains why the double protonated TCA molecule (TcH 2

+2

) does not form inclusion complexes; nevertheless, even in strong acid medium one observes spectral variations over the whole CD concentration range. The Scheme 2 accounts for the experimental facts, in which the TcH + species undergoes complexation with CD by the equibrium process of constant K 1 c or K 2 c expressed by eqs. ( 2) and (3), respectively. The "extraction" of this species from water and enclosed into de CD cavity, shifts the acid-base equilibrium, K

, towards the basic form of TCA, i.e., TCA is a weaker base in the presence of CD.

Scheme 2 [ ][ ] [ ][ ] CD TcH H CD • TcH K 2 2 c 1 + + + = (2) ; [ ] [ ][ ] CD TcH CD • TcH K c 2 + + = (3)
Under the experimental conditions of the work, i.e., [TCA] t <<[CD], we must consider firstly, that, in strong acid, HCl, the stoichiometric tetracaine concentration is given by:

c = K 2 c K 1 H . ( ) ( ) [ ] ( )[ ] ( ) [ ] + + + + + + = + + = H K K H f with CD H f K 1 CD H f K I H f I I H 1 H 1 c 2 c 2 c F w F F (4)
Solid curves corresponding to the down two set points displayed in Fig. 4a were drawn in the fit of eq. ( 4) to the experimental data read at both 357 (filled symbols) and 377 (open symbols). The non linear regression analysis of this equation to the experimental data yield the values for the optimized parameters, I F c and K 2 c

, listed in Table 3, first arrow.

In mild acid medium, i.e., in aqueous buffered solutions of acetic acid-acetate of pH>4, as the pK 1 H has been measured equal to 2.27, then the [H + ]<<K 1 H and eq. ( 4) simplifies to eq. ( 5). Solid curves for the upper two set-points in Fig. 4a correspond to the fit of eq. ( 5) to the experimental data read also at both 357 (filled) and 377 (open) symbols, when the optimized parameters take the values reported in Table 3.

[ ] [ ]

CD K 1 CD K I I I c 2 c 2 c F w F F + + = (5)
In mild basic medium, i. e., in aqueous buffered solutions of carbonate-bicarbonate buffer of pH~10, the tertiary amine group of tetracaine is deprotonated (pK 2 H =8.39). As tetracaine is now a neutral molecule, the hydrophobic effect is increased, since the strong ion-dipole interactions are now not possible, and one may expect a higher value for the association constant of TCA with CD-cavity.

The equilibrium reaction of eq. ( 6), lead us to an equation of type (5) where K 2 c must be replaced by K 3 c that corresponds to the equilibrium constant of the inclusion complexes formed between both neutral forms of TCA and CD. The experimental data are displayed in Fig. 4b and the fit to the corresponding eq. (of type 5) results in the values of the optimized parameters listed in Table 3. As expected, the value of K 3 c is nearly 3-fold that of K 2 c

; the difference measures the relative importance of the hydrophobic effect against the ion-dipole interactions (the energy of water release from the cavity is in both cases the same).

(

) 6
Finally, in alkaline medium, tetracaine is a neutral molecule but the host, cyclodextrin, is an anion; that is, a secondary -OH group in ionized (pK a ~12.2). This group is located at the main entrance of CD cavity, therefore, when it is ionized, the hydration level is more important and difficult the inclusion process of a hydrophobic guest. Thus, the equilibrium reaction of eq. ( 7) can be proposed. In order to study the effect of the size of the cyclodextrin cavity, we analyzed the effect of both αand γ-cyclodextrin. Figure 5 shows the variation of the fluorescence emission intensity as a function of the cyclodextrin concentration. In every case, the plot of I F against [CD] describes saturation curves, due to the inclusion complex formation of 1:1 stoichiometry. The non-linear regression analysis of the experimental data to eq. ( 5) -valid for experiments performed in either mild acid or mild basic medium-afford the values for the optimized parameters listed in Table 4, along with the results obtained in the presence of β-CD for the sake of comparison. It can be concluded that β-CD cavity combines optimal geometry and polarity conditions to complex TCA molecules, whereas γ-CD yields the less stable complex. It is worthy to mention the restricted motion imposed by the narrow α-CD cavity, in which case the I F c due to LE emission is more than 3-fold the value measured in γ-CD cavity, where both the polarity (dielectric constant equal to 70 against 47 for α-CD) [START_REF]The pK a values determined at 25 ºC by potentiometry are 12.33, 12.20, and 12.08, respectively to α-, β-, and γ-cyclodextrin; see Comprehensive Supramolecular Chemistry[END_REF] and the rotation freedom are high.

The experimental data were also fitted to the linearized form of eqs. ( 4) or (5), that is, the

Discussion

The UV-vis spectra of TCA indicate no significant changes in the pK a of the ground state with respect to that of the excited state. Excitation at the lowest energy absorption band (λ max ~300 nm) yields dual fluorescence, which emission intensity increases drastically with both the pH and the apolar character of the medium, represented here by aqueous-solvent mixtures or aqueous cyclodextrins solutions.

Tetracaine forms 1:1 inclusion complexes with α-, β-, or γ-cyclodextrin; only the protonated (at the tertiary amine group) or neutral TCA molecules are able to included into the CD-cavity; however, the highest complex stability was obtained in conditions of neutral host and guest, i. e. at pH ~10 in bicarbonate-carbonate buffered solutions. The size of the CD cavity affects both the complex stability and dual fluorescence emission.

In aqueous solvent mixtures up to 80% v/v solvent/water, TCA shows dual non-linear fluorescence, except for the case of acetonitrile, in which medium nearly straight lines of I F vs [solvent] was obtained. By contrast, in aqueous-cyclodextrin solutions, the I F vs [CD] increase at low CDconcentrations and levels off at high concentration of the host, when almost all TCA molecules are forming complexes.

In aqueous buffered solutions of acetic acid-acetate, or even in aqueous sodium acetate solutions, the quotient between the fluorescence intensities of LE (~357 nm) and of ICT (~377 nm) is nearly 1, scan 2 in Fig. 1; that is, the ICT emission state is as important as the LE emission. This fact is contrary to the behaviour observed in those fluorophores bearing a dialkylamine group bonded in the pposition of the phenyl ring to the acceptor group like, for instance, methyl p-dimethylamine benzoate, p-(N,N-dimethyl-amine) benzoic acid, 22 which in pure water emit only LE fluorescence at short wavelength. We could attribute this difference to the ability of TCA to form H-bonding with water through the secondary amine group, a fact that would enhances the stability of ICT. In this sense, the ratio of I LE /I ICT increases proportional to the solvent concentration in the aqueous mixtures, showing the greatest effect in aqueous dioxane mixtures (Fig. 6a), i.e. in the less polar and aprotic solvent (see Table 2 for solvent properties), in where, the ICT state is not stabilized and TCA emits nearly LE fluorescence.

In aqueous CD-solutions, the plot of I LE /I ICT against [CD] increases also with [CD], but drawing saturation curves that level off at high CD; in addition, the I LE /I ICT vs.

[CD] profiles show similar shape for α-, β-, or γ-cyclodextrin; Fig. 6b shows representative data for the case of β-CD. Again, this behaviour is contrary to that observed with e.g. DEABA in which case, firstly, the ICT emission intensity is higher than the LE emission (i.e. I ICT /I LE increases with [β-CD]), and secondly, the observed behaviour in β-CD is opposite to that seen with α-CD. 22,23 Our results indicate that, the CD-cavity provides a less polar and aprotic environment that favours LE emission over the ICT state, because of the destabilization of the latter due to the absence of H-bonding interactions and the constrained microenvironment that difficult the necessary rotation to singly positively charged on the tertiary amine N-atom or. Geometrical optimization of TCA structure with AM1 method [START_REF] Dewar | Program HyperChem 5.0, Hypercube[END_REF] yields the form and dimensions of TCA displayed in Scheme 3 and compared to that of cyclodextrin.

Scheme 3

Conclusions

Tetracaine shows dual fluorescence in either water solvent mixtures or aqueous cyclodextrin solutions. The ratio of normal emission intensity to the ICT emission increases with CD concentration and level off at high concentrations, whereas in water-solvent mixtures straigth lines were observed.

The behaviour is different to the most studied fluorophores of type dimethylaminebenzonitrile. On the other hand, the stability of inclusion complex formed between tetracaine and cyclodextrins depends on the pH of the medium and on the cyclodextrin nature; the most stable complex was observed with β-CD in mild basic medium of pH~10. While, with αand β-CD the results suggest the existence of two different inclusion complexes, with γ-CD one conformation for the inclusion complex suffices to explain the experimental observations. µ/D (b) ILE / IICT (c) λmax (log ∈) (a) Relative solvent dielectric constant; (b) dipole moment in Debyes (1D= 3.34×10 -30 C•m); (c) HAc/Ac -(1:1) 0.048 M (b) 3.6 0.258 2.09±0.02 922±45 (c) 0.212 1.37±0.01 924±47 (c) HAc/Ac -(1:1) 0.167 M (b) 3.6 0.255 2.04±0.02 991±50 (c) 0.225 1.34±0.02 989±50 (c) HAc/Ac -(1:1) 0.067 M (b) 10.4 0.451 4.50±0.02 1090±20 (c) 0.490 3.03±0.01 1113±18 (c) NaAc 0.067 M (b) 10.4 0.563 5.26±0.04 1412±40 (c) 0.535 3.40±0.03 1380±43 (c) HCO3 -/CO3 (a) eq. 5 value of f(H + )=0.17; (b) buffer solutions of acetic acid (HAc) acetate (Ac -); (c) eq. 6, K2 c ; (d) buffer solutions of carbonatebicarbonate, pH 10.2; (e) eq. 6, value of K3 c ; (f) eq. 6, value of K4 (a) buffer solutions of acetic acid-acetate; (b) eq. 6, value of K2 c

  in 4-(dimethyl)amino benzonitrile, DMABN), is strongly dependent on the medium polarity and viscosity and has been the

( 1 )

 1 In this sense, the fluorescence emission intensities, read under different experimental conditions and plotted in Fig.4, increase with β-CD concentration and level off at high cyclodextrin concentration.

[

  TCA] o =[TcH 2 +2 ]+[TcH + ]+[TcH•CD + ]. Secondly, taking into account that the quantum emission yield of TcH 2 +2 species (tetracaine dication) is negligible, then the measured fluorescence intnsity is due to both the protonated tetracaine in the bulk water phase (TcH + ) and included into the CD-cavity (TcH•CD + ). Taking into account the different quantum yield (Φ) for free and included species, the total fluorescence emission is given by I F =ε w ℓI o Φ w [TcH + ] w +ε c ℓI o Φ c [TcH•CD + ], where ε is the molar absorption coefficient, ℓ=1 cm the path light, and I o is the absorbed intensity. This equation can be into eq. (4) by considering together eqs. (2) and (3). In this equation, I F w and I F c represent the fluorescence intensities corresponding to the TCA free and complexed with CD, respectively, and K 1

3 c 3 c, because under these conditions of pH~pK 2 H,,

 332 Taking into account that the tetracaine concentration is much lower than that of CD and proceeding as in the previous cases, we can easily arrive to an equation of type(5) to relate the changes in fluorescence emission intensity with the cyclodextrin concentration in alkaline medium, where K 2 c must be replaced by K4 c that corresponds to the stability constant of the inclusion complexes formed between the neutral form of TCA and the ionized cyclodextrin, CD -. The experimental data are displayed also in Fig.4band the fit to the corresponding eq. (of type 5) results in the values of the optimized parameters listed in Table3. These data indicate that K of sodium acetate, is, in fact, a combined result of both K 2 c and K TCA exists as both TcH + and Tc. In the same sense, at [OH -]=0.048 M where the percentage of ionized β-CD is not higher than 80%, the value of K i.e. the true K 4 c should be lower.As a summary, the preceding mechanistic treatment implies that the variation of I F as a function of [CD] should follow the eq. (4), or (5) depending on the experimental conditions of the experiments, stability equilibrium constants of the inclusion complexes formed between protonated tetracaine and CD (TcH•CD + ), neutral tetracaine and CD (Tc•CD), and neutral tetracaine and ionized CD (Tc•CD -), respectively, and I F w and I F c refer to the fluorescence intensity readings obtained, under each experimental conditions, in water and at a CD concentration corresponding to the complete guest inclusion, respectively.

Benesi-

  Hildebrand relation used for these types of complexation processes and expressed in terms of fluorescence intensity with the assumption that [CD]>> [TCA•CD], i.e., the reciprocal plot of (I F -I F w ) against 1/[CD]. In every case, perfect straight lines were obtained (plots not shown) and rendering similar values of K i c and I F c to those obtained from the non linear regression analysis.

  orthogonal position between donor and acceptor groups. The saturation value obtained in I LE /I ICT vs. [CD] plots depends on the nature of TCA molecule forming complexes, i.e.

1 .Scheme 2 .Scheme 3 .Figure 1 .

 1231 Scheme 1. Acid-base ionization equilibrium of tetracaine in water

Figure 2 .

 2 Figure 2. Plot of fluorescence intensities read at (a) 357 nm and at (b)377 nm as a function of the solvent concentration in the water-solvent mixtures; symbols correspond to ( )dioxane, ( )dimethylsulphoxide, DMSO; ( )acetonitrile, AN, and ( )methanol, MeOH.

Figure 3 .

 3 Figure 3. Emission spectra of TCA 10.4 µM in (a)aqueous-β-cyclodextrin (1.29 mM) solutions of [1] hydrochloric acid 0.033 M; [2]acetic acid-acetate buffer 0.066 M pH 4.78; [3]alkaline medium of [OH -]=0.048 M, and [4]carbonate-bicarbonate buffer 0.033 M pH 10.2, and (b)in aqueous sodium acetate 0.067M as a function of β-cyclodextrin concentration equal to [1] 0; [2]0.026, [3]0.052; [4]0.10, [5]0.16 mM. λ ex =315 nm.

Figure 4 .

 4 Figure 4. Variation of the fluorescence emission intensities of 10.4 µM tetracaine dissolved in (a) ( , )aqueous HCl 0.033M, and in ( , )aqueous 0.067 M acetic acid-acetate (1:1) and (b) ( , )aqueous 0.048 M of NaOH, and in ( , )aqueous 0.033 M carbonate-bicarbonate (1:1) buffer, as a function of β-cyclodextrin concentration: (open) symbols correspond to 377 nm or ICT emission state, (solid) symbols correspond to 357 nm or LE emission state.

Figure 5 .

 5 Figure 5. Variation of the fluorescence emission intensities of 10.4 µM TCA dissolved in water (a) ( , )at 0.033 M carbonate-bicarbonate (1:1) buffer and ( , )at 0.067 M acetic acid-acetate (1:1) as a function of α-CD concentration; the inset shows the indicated expanded plot, and (b) ( , )at 0.033 M carbonate-bicarbonate (1:1) buffer, as a function of γ-CD concentration; (open) symbols correspond to 377 nm or ICT emission state, (solid) symbols correspond to 357 nm or LE emission state.

Figure 6 .

 6 Figure 6. Plot of the ratio I LE /I ICT as a function of (a)solvent concentration in the aqueous mixture ( )dioxane, ( )acetonitrile, ( )methanol; (b)β-CD concentration in ( )strong acid, HCl; ( )mild acid, acetic acid-acetate; ( )mild base, carbonate-bicarbonate, and ( )alkaline medium.

Figure 1 .Figure 2 .Figure 2 .Figure 3 .Figure 3 .Figure 4 .Figure 4 .Figure 5 .Figure 5 .Figure 5 .Figure 6 .Figure 6 .

 122334455566 Figure 1. Emission spectra of TCA 10.4 µM in aqueous solutions of [1] hydrochloric acid 0.033 M; [2]acetic acid-acetate buffer 0.066 M pH 4.78; [3]carbonate-bicarbonate buffer 0.033 M pH 10.2, and [OH<->]=0.048 M; λ<ex>=315 nm. 148x114mm (300 x 300 DPI)

Table 1 .

 1 Solvent properties, values of the ratio of fluorescence emission intensities of LE to ICT states, and UV-vis absorption data.

	TABLES
	solvent	εr (a)

Table 2 .

 2 Concentration and mixed dissociation constants of tetracaine determined by potentiometry and spectrophotometric titrations at 25 ºC

	Method	Potentiometry	Spectrophotometry
	pK1 H	2.241±0.005 (a)	2.27±0.02 (b) ref. 26
	pK2 H	8.390±0.001 (a)	8.24±0.28 ref . 27
	(a) This work, I=0.1 M, NaCl; (b) I =0.12 M, buffers, ref. 26

Table 3 .

 3 -Experimental conditions and optimized parameters obtained in the fitting process of eqs.(5), or (6) to the experimental data, which result in the study of the influence of β-cyclodextrin concentration on the steady-state fluorescence emission of tetracaine.

			λ = 357 nm, LE emission state	λ = 377 nm, ICT emission state
	Medium	[TCA]/µM	IF w	IF c	Ki c /M -1	IF w	IF c	Ki c /M -1
	HCl 0.035 M	10.4	0.706 (a)	4.08±0.06	1174±16 (a)	0.708 (a)	2.80±0.05	1178±50 (a)

Table 4 .

 4 -Experimental conditions and parameters obtained in the study of the influence of cyclodextrins on the fluorescence emission of 10.4 µM tetracaine aqueous solutions.

			λ = 357 nm, LE emission state	λ = 377 nm, ICT emission state
	Medium	cyclodextrin
			IF w	IF c	K3 c /M -1	IF w	IF c	K3 c /M -1
	HAc/Ac -(1:1) 0.067 M (a)	α-CD	0.451	4.80±0.20	46±4 (b)	0.495	2.7±0.1	49±6 (b)
	HCO3 -/CO3 -2 buffer (1:1) 0.033 M	α-CD	0.65	15.35±0.22	125±4	0.61	6.5±0.1	141.5±5.0
	HCO3 -/CO3 -2 buffer (1:1) 0.033 M	β-CD	0.656	9.02±0.07	3464±145	0.63	4.96±0.03	3530±130
	HCO3 -/CO3 -2 buffer (1:1) 0.033 M	γ-CD	0.65	4.7±0.1	65±4	0.62	2.60±0.04	76±5

Acknowledgment

Financial support from the Dirección General de Investigación (Ministerio de Educación y Ciencia) of Spain and FEDER (Project CTQ2005-07428/BQU) and from Dirección General de Programas y Transferencia de Conocimiento (Ministerio de Ciencia e Innovación) of Spain (Project CTQ2008-04429/BQU) is gratefully acknowledged. I. I.G. also thanks the Ministerio de Educación y Cicncia of Spain for a research training grant.