E Koorambas
email: elias.koor@gmail.com

The interaction between massive and massless gravitons by the perturbing topological field theory

Keywords: number(s): 11.15.-q, 04.60.-m, 95.35. +d, 11.10.Gh Key-Words: gauge field, quantum gravity, dark matter and dark energy, topological field theory Contents 1. Introduction…………………………………………………………………………….. ...... References………………………………………………………………………………………

We test the N.Wu gauge theory of gravity with massive gravitons in the perturbing topological field theory framework. We show that the computation of the correlation function between massive and massless gravitons is reported up to 4-loop and appears to be unaffected by radiative correction. This result ensures the stability of the linking number between massive and massless gravitons with respect to the local perturbation, a result with potential wider applications in cosmology.

Introduction

Various authors have attempted to derive General Relativity from a gauge-like principle, involving invariance of physics under transformations of the locally (i.e., in the tangent space at each point) acting Lorentz or Poincar´e group. ( [START_REF] Utiyama | Invariant theoretical interpretation of interaction[END_REF], [START_REF] Kibble | Lorentz invariance and the gravitational field[END_REF], [START_REF] Wilczek | Riemann-Einstein structure from volume and gauge symmetry[END_REF], [START_REF] Wu | Gauge Model with Massive Gravitons[END_REF]). Gauge theory of gravity is based on the principle of local gauge invariance. Since the model requires strict local gravitational gauge symmetry ( [START_REF] Wu | Gauge Model with Massive Gravitons[END_REF]), gauge theory is a pertubatively renormalizable quantum model. In the original model, all gauge gravitons are massless ( [START_REF] Blagojevic | Gravitation and gauge symmetries I[END_REF]). The massive gravitons were described effectively by Fierz and Pauli ([6]). According to their theory, the existence of massive gravitons would violate the local gauge symmetry of the Lagranian ( [7]).

Building on this earlier work, N.Wu proposed a mechanism which introduces massive gravitons without violating the local gauge symmetry of the Lagranian ( [START_REF] Wu | Gauge Model with Massive Gravitons[END_REF], [START_REF] Wu | Quantum Gauge Theory of Gravity[END_REF], [START_REF] Wu | Gauge Theory of Gravity[END_REF], and [START_REF] Wu | N Renormalizable Quantum Gauge Theory of Gravity[END_REF]).

The third-order gravitational gauge field 3 a C  is massive if mass is very large. Such a field has no contribution to the long-range gravitational force. Long-range gravitational force results exclusively from the contribution of the fouth-order gravitational gauge field 4 a C  and obeys inverse square law.

If the mass term of gravitational gauge field is extremely small, however, the third-order gravitational gauge field 3 the gravitational force range will be about one metre.

Recent results from cosmological observation, especially from Cosmic Microwave Background (CMB) temperature anisotropy, suggest that our Universe is essentially flat and that it consists of mainly dark matter and dark energy [START_REF] Hu | [END_REF]. A natural origin for dark matter and dark energy is to regard it as consisting of massive gravitons. There are indications that those massive gravitons with mass can produce today's acceleration of the Universe ( [12], [13]).

In the traditional gauge treatment of gravity the Lorentz group is localized. The gravitational field is, thus, not represented by gauge potential, but by the metric field g μν .

Here we propose an alternative understanding of gravity resulting from the extension of N.Wu's gauge theory of gravity with massive gravitons into the framework of perturbing topological field theory. Based on this, we calculate the Feynman diagrams for the interaction between massive and massless gravitons in the topological field theory ( [START_REF] Atiyah | The Geometry and Physics of Knots[END_REF], [START_REF] Witten | [END_REF], [START_REF] Schwarz | New topological invariants in the theory of quantized fields[END_REF], [START_REF] Kaul | Complete solution of SU (2) Chern-Simons theory[END_REF], [START_REF] Kaul | [END_REF], [19], [START_REF] Kaul | Chern-Simons theory, knot invariants, vertex models and three-manifold invariants[END_REF], [START_REF] Kaul | Topological Quantum Field Theories -A Meeting Ground for Physicists and Mathematicians[END_REF], and [START_REF] Vitor | A perturbing topological field theories[END_REF]). These calculations provide insights to interactions between gravity and dark energy. They could, therefore, have important repercussions for current cosmological problems.

The interaction between massive and massless gravitons in the topological field theory

Taking Wu's gauge model as our starting point ( [START_REF] Wu | Gauge Model with Massive Gravitons[END_REF], [START_REF] Wu | Quantum Gauge Theory of Gravity[END_REF], [START_REF] Wu | Gauge Theory of Gravity[END_REF], [START_REF] Wu | N Renormalizable Quantum Gauge Theory of Gravity[END_REF], [23], [24], [START_REF] Wu | Fig.1 Linking between 1, 2 [END_REF]), we introduce two gravitational gauge fields  are vectors in Lie algebra ( [START_REF] Wu | Quantum Gauge Theory of Gravity[END_REF], [START_REF] Wu | Gauge Theory of Gravity[END_REF]), they can be expanded as

â a C C P   , 2 2 â a C C P    . ( 1
)
These correspond with two gauge covariant derivatives

() D igC x       , 22 () D iagC x       (2)
and two different field strengths, given by

2 2 2 1 [ , ] 1 [ , ] F D D ig F D D ia          (3)
The Lagrangian of the system is given by 1)

0 22 2 22 2 11 44 ( )( ) 2(
v g F F g F F m g C aC C aC a                                        (4)
where m is the constant mass parameter. The action of the system is given by

4 0 () S d xJ C   (5)
Equation ( 4) gives the mass term in gravitational gauge fields. To obtain the Eigenstates of mass matrix the following rotation is needed 32 42 cos sin cos sin

C C C C C C             (6)
where the angle  is given by

2 1 cos 1 a    , 2 sin 1 a a    (7)
After transformation [START_REF] Fierz | Relativistic wave equation for particles of arbitrary spin in an electromagnetic field[END_REF], the Lagrangian of the system is given by 0 30 30 40 40

11 44 g F F g F F                         (8)
where F a 30μν F a 40μν are given by:

3 0 3 3 4 0 4 4 a a a a a a F C C F C C                 (9)
From the above follows that the three-dimensional gravitational gauge field 3 a C  is massive, with mass m , whereas the four-dimensional gravitational gauge field 4 a C  , is massless.

To investigate the interaction between the massive gravitational gauge field 3 a C  and the massless gravitational gauge field 4 a C  we use topological field theory.

Our computation of the correlation function between the massive and massless gravitons is analogous with that given in the literature ( [START_REF] Vitor | A perturbing topological field theories[END_REF]).  and 2  (Fig. 1), the computation of the correlation function is reported up to 4-loop. This computation shows that the function is unaffected by radiative correction. This result ensures the stability of the linking number with respect to the local perturbation ( [START_REF] Vitor | A perturbing topological field theories[END_REF]). The local perturbation can be added to the Chern -Simons action given by

3 3 4 3 4 4 3 3 1 2 2 ! a a a eff a S d x C C d xF F F F                     ([22]) (11)
with

4 4 3 3 1 2 FF           
and  being an arbitrary parameter with negative mass dimension, reflecting the power-counting non-renormalizability of the perturbation.

The Feynman diagrams for the interaction between massive and massless gravitons in the topological field theory are similar to those described at [START_REF] Vitor | A perturbing topological field theories[END_REF]. To calculate the correlator function [START_REF] Wu | N Renormalizable Quantum Gauge Theory of Gravity[END_REF] we use the action

,...... 2 4 ! aa eff a S d x C C d xb C d x F F F F                            (12)
Τhe Lagrange multiple b is introduced to this function to implement the Landau gauge. To evaluate the Feynman diagrams we use the following equations: x C y xy

                   (13) 4 3 3 4 3 1 ( ), ( ) ( ) 4 ( ), ( ) ( )
a a C x F y g g x y xy F x F y g x y                              (14)
The Feynman diagrams that contribute to the correlation function [START_REF] Wu | N Renormalizable Quantum Gauge Theory of Gravity[END_REF] are of two-loop order (see Figure 2). Based on [START_REF] Vitor | A perturbing topological field theories[END_REF], it follows that, for the second-, third-and fourth-order diagrams (see Figures 2,[START_REF] Wilczek | Riemann-Einstein structure from volume and gauge symmetry[END_REF][START_REF] Wu | Gauge Model with Massive Gravitons[END_REF], and more generally, for all terms in all possible diagrams between massive and massless gravitons, these Feynman diagrams are proportional to 3 () xy   or its derivatives ( [START_REF] Vitor | A perturbing topological field theories[END_REF]). ( 2 )

3 3 3 3 2 1 2 1 33 1 2 1 2 3 12 4 ( ) ( ) [ ( )][ ( )] [ ( )] I dx dy d z d z x z y z z z z z zz                               (15) 12 (3)
3 3 3 33 3 3 33 [ ( )][ ( )][ ( )] [ ( )][ ( )] I dx dy d z x y x z x z y z y z                                    ( 16
) 12 ( 4 ) 3 3 3 3 4 3 4 3 3 3 3 3 3 3 4 3 4 [ ( )][ ( )][ ( )] [ ( )][ ( )][ ( )][ ( )] I dx dy d z d z x y x z x z y z y z z z z z                                                (17)
Since it is always xy  , the second-, third-and fourth-order diagrams correspond to null correction to the basic diagrams. The correlator function [START_REF] Wu | N Renormalizable Quantum Gauge Theory of Gravity[END_REF], therefore, is the link number to all orders:

4 1 2 3 ( ), ( ) ( , ) eff a S C x C y        (18)

Conclusion

Equation (18) links massive graviton, 3 a C  representing dark energy, with the massless graviton, 4 a C  representing the gravity. This expression of the interaction between gravity and dark energy may have potentially very important implications to cosmology. We find that the interactions between dark energy and gravitons are independent of metric. These interactions are, therefore, within the framework of perturbing topological field theory. The topological properties of these interactions are represented by knots and links. The size, exact shape, location etc of these knots and links are not of immediate concern for the problem at hand (interaction between dark mater/energy and gravitons).

aC

   will contribute to the middle range gravitational force with approximate range hc L m  (where h is the Plank constant and c is the speed of light)

For the case of the two gravitational gauge fields 3 aC  and 4 aC

 34  lying on two smooth, closed, non-intersecting curves 1

12

 12

Fig. 3

 3 Fig.3 Three loop contribution.Fig. 4 Four loop contribution.

Equation (18), that links massive and massless gravitons, is independent of the exact location, size and shape of the two knots. Equation ( 18) depends only on the topological relationship of the knots with each other. This invariant may have a physical interpretation: it may represent the work done to move a massive graviton (dark energy) around one knot in three dimensional space while a massless graviton runs around the other knot.


Clemens Von Sonntag

Sergej Naumov

Keywords: Ozone, Hydroxyl radical, Ozonide radical anion, Oxygen, Thermodynamic cycle

The reaction of *OH with O2, the decay of O3*-and the pKa of HO3* -interrelated questions in aqueous free-radical chemistry

(1), k 1 = 3.5 × 10 9 M -1 s -1 ] [1] while • OH does not, how decays O 3

•-into • OH, and what is the 28 pK a value of the HO 3

• radical? 29

O •-+ O 2 ⇄ O 3 •- (1) 30
In ozone reactions, O 3

•-is a common intermediate. [2][3][4] It may be generated in a direct 31 electron transfer reaction [reaction (2)], [5] but more often an adduct is an intermediate 32 [reactions (3) and ( 4)]. [6][7][8] 33

X + O 3 → X •+ + O 3 •- (2) 34 X + O 3 → + XOOO -→ X •+ + O 3 •- (3)/(4) 35
The formation of O 3

•-in ozone reactions is of considerable importance in water treatment. 36

There is a general agreement that O 3

•-must be the precursor of • OH that is made responsible 37 for the degradation of ozone-refractory micropollutants in the treatment of drinking water and 38 wastewater with ozone. In this context it may be noted that upon reacting with ozone, the 39 organic matter contained in wastewater produces as much as 13% • OH with respect to ozone 40 consumed. [9] Beyond this general statement that O 3

•-must be the precursor of • OH, there is, 41 however, a continuing discussion as to how O (-6)]. [10] 44

HO 3 • ⇄ O 3 •-+ H + (5) 45 • OH + O 2 ⇄ HO 3 • (6) 46
A pK a (HO 3 • ) = 6.15 (revised to 8.2, Bühler, private communication) has been estimated 47 from these experiments, [10] and this mechanism is also found in the most recent review on the 48 chemistry of ozone in aqueous solution. [4] 49

F o r P e e r R e v i e w 3
A pK a as high as this is doubtful as HO 3

• should be more acidic than HO 2

• whose pK a = 4.8 50 is well established. [11] In fact, Bielski and Schwarz [12] developed an experimental relationship 51 for the pK a values of O x H y acids [equation ( 7)], wherefrom one arrives at pK a (HO 3

• ) = -2.0. 52 pK a (O x H y ) = 19 -7 × (O/H) (7) 53
We have, therefore, suggested that O 3

•-decays according to equilibria (1) and ( 8) 54

[pK a ( • OH) = 11.8]. [7;13] 55

• OH ⇄ O •-+ H + (8) 56
From the above, it can be seen that the reaction of • OH with O 2 and the pK a of HO 3

• radical 57 are interconnected by a thermodynamic cycle (Fig. 1). 58 59 The Gibbs energy of reaction (6) has been calculated by various approaches (Table 1 Among the above methods, the most reliable one proved to be G1 (for a discussion see 73

-H + OH + O 2 HO 3 O + O 2 O 3 -H + 60
Ref. [8] ) followed by B3LYP/6-311+G(d,p)/PBF. Calculations of charged species as required 74 for the Gibbs energy of equilibrium (1) are inherently poorer. G1 does not allow calculating 75 charged species. Here B3LYP/6-311+G(d,p)/PBF has been tested to be the best. Indeed, the 76 calculated value using this method for comes closest to the experimental value. For 77 equilibrium (1), a reliable experimental value is available, and for further discussions one can 78 neglect the calculated ones. 79

The Gibbs energy of reaction (6) relates to the equilibrium constant of reaction (6) 80 according to equation (9). 81 lnK = -∆G 0 /RT (9) 82

From the above value of +47 kJ mol -1 for the Gibbs energy of equilibrium (6), an 83 equilibrium constant of K 6 = 10 -8.2 is calculated. The anion of the hydroxyl radical, O •-, reacts 84 fast with O 2 (k 1 = 3.5 × 10 9 M -1 s -1 ) giving rise to O 3

•-, and details of equilibrium (1) are well 85 established (K 1 = 10 5.7 M, k -1 = 1.94 × 10 3 s -1 ). [1] Thus, the Gibbs energy for reaction (1) is 86 ∆G 0 = -32 kJ mol -1 . A Gibbs energy of ∆G 0 = +67 kJ mol -1 for reaction (8) is calculated from 87 pK a ( • OH) = 11.8. 88

Based on the thermodynamic cycle (Fig. 1), a Gibbs energy of ∆G 0 = -12 kJ mol -1 is 89 arrived at for reaction (5). This relates to a pK a value of the HO 3

• radical of -2.1 in good 90 agreement with Bielski and Schwarz relationship [12] that yields pK a (HO 3 This good agreement supports our earlier statement [7] that the reported [10] pK a (HO 3 • ) = 6.15 92

[revised to 8.2 (Bühler, private communication)] must have been an experimental artifact (so-93 called "reactivity pK"). 94

As the rate constant for reaction (-5) must be in the order of 10 10 M -1 s -1 , [14] the rate of 95 deprotonation of HO 3 • , reaction (5), must be in the order of 10 12 s -1 . This reaction is in 96 competition with its decay into • OH and O 2 [reaction (-6)]. Upon calculating the Gibbs energy 97 of the reaction of H • with O 3 , the adduct HO 3

• proved to be sufficiently stable to qualify as an 98 intermediate [reactions (10) and (-6); ∆G 0 overall = -411 kJ mol -1 ]. 99

H • + O 3 → HO 3 • (10) 100
In other reactions of ozone with free radicals; such adducts are often only transition states 101 (Naumov and von Sonntag, submitted). This points to a lifetime of HO 3

• beyond several 102 vibrations, that is, k -6 < 10 12 s -1 . With K 6 in the order of 10 -8 , k 6 must be < 10 4 M -1 s -1 . Form 103 the pulse radiolysis study of the reaction of H • with O 3 , carried out at pH 2, [15] there was no 104 indication that O 3

•-is present at about 1 µs. Thus the lifetime of HO 3

• must be < 10 7 s. This 105 narrows the range to 0.1 > k 6 < 10 4 M -1 s -1 . In any case, the reaction of • OH with O 2 must be 106 very slow, as generally assumed, and can thus be neglected for most purposes. 107

CONCLUSION 109

The equilibrium constants for the reactions of O 2 with • OH and its anion, O •-, differ by as 110 much as fourteen orders of magnitude. The generally accepted statement that the • OH radical 111 does not react with O 2 is substantiated here by bracketing the rate constant: 0.1 > k 6 < 10 4 M -1 112 s -1 . It is now established that HO 3

• is a very strong acid with a pK a near -2. With this revised 113 pK a value at hand, modeling of ozone reactions in aqueous solution can now be carried out 114 with much higher accuracy. The Gibbs energy of reaction (6) was calculated using a number of approaches. Density 118 Functional Theory (DFT) calculations were carried out using B3LYP correlation 119 functional [16;17] (Jaguar 7.6 program) [18] . The molecular geometries of all calculated molecules 120 were optimized in water at the B3LYP/6-311+G(d,p)/PBF level of theory. The interactions 121 between the molecule and the solvent were evaluated by Jaguar's Poisson-Boltzmann solver 122 (PBF). [19] Gibbs free energies (G 0 ) were calculated at the same level of theory at a standard 123 temperature of 298.15 K and a pressure of 1 atm using unscaled frequencies. Reaction 124 parameters were calculated as the difference of the Gibbs free energies ∆G 0 between reactants 125 and products. Calculations were also made in water (self-consistent reaction field 126 SCRF=COSMO models) [20;21] at Gaussian-1 (G1) [22;23] , Gaussian-3 (G3MP2B3) [24] , and DFT 127 B3LYP/6-311+G(3df,3pd) levels (as implemented in Gaussian 03) [25] . 128 (1), k 1 = 3.5 × 10 9 M -1 s -1 ] [1] while • OH does not, (ii) how decays O 3

The reactions O

• • • •--- -+ O 2 ⇄ ⇄ ⇄ ⇄ O 3 • • • •--- -(1), H + + O • • • • - -- -⇄ ⇄ ⇄ ⇄ • • • • OH (8), • • • • OH + O 2 ⇄ ⇄ ⇄ ⇄ HO 3 • • • • (6) and H + + 15 O 3 • • • •-⇄ ⇄ ⇄ ⇄ HO 3 • • • • (5)
•-into • OH, and (iii) what 28 is the pK a value of the HO 3

• radical? 29

O •-+ O 2 ⇄ O 3 •- (1) 30
In ozone reactions of organic compounds, O 3

•-is a common intermediate. [2][3][4] It may be 31 generated in a direct electron transfer reaction [reaction (2)], [5] but more often an adduct is an 32 intermediate [reactions (3) and ( 4)]. [6][7][8] 33

X + O 3 → X •+ + O 3 •- (2) 34 X + O 3 → + XOOO -→ X •+ + O 3 •- (3)/(4) 35
The formation of O 3

•-in ozone reactions is of considerable importance in water treatment. 36

There is a general agreement that O 3

•-must be the precursor of • OH that is made responsible 37 for the degradation of ozone-refractory micropollutants in the treatment of drinking water and 38 wastewater with ozone. In this context it may be noted that upon reacting with ozone, the 39 organic matter contained in wastewater produces as much as 13% • OH with respect to ozone 40 consumed. [9] Beyond this general statement that O 3

•-must be the precursor of • OH, there is, 41 however, a continuing discussion as to how O 3

•-is converted into • OH. 42

According to a pulse radiolysis study, O 3

•-is protonated [reaction (-5)], and HO 3

• thus 43 generated looses O 2 [reaction (-6)]. [10] 44

HO 3 • ⇄ O 3 •-+ H + (5) 45 • OH + O 2 ⇄ HO 3 • (6) 46
A pK a (HO 3 • ) = 6.15 (revised to 8.2, Bühler, private communication) has been estimated 47 from these experiments, [10] and this mechanism is also found in the most recent review on the 48 chemistry of ozone in aqueous solution. [4] 49

F o r P e e r R e v i e w 3
A pK a as high as this is doubtful as HO 3

• should be more acidic than HO 2

• whose pK a = 4.8 50 is well established. [11] In fact, Bielski and Schwarz [12] developed an experimental relationship 51 for the pK a values of O x H y acids [equation ( 7)], wherefrom one arrives at pK a (HO 3

• ) = -2.0. 52 pK a (O x H y ) = 19 -7 × (O/H) (7) 53
We have, therefore, suggested that O 3

•-decays according to equilibria (1) and ( 8) 54

[pK a ( • OH) = 11.8]. [7;13] 55

• OH ⇄ O •-+ H + (8) 56
From the above, it can be seen that the reaction of • OH with O 2 and the pK a of HO 3

• radical 57 are interconnected by a thermodynamic cycle (Fig. 1). 58 The Gibbs energy of reaction (6) has been calculated by various approaches (Table 1 Among the above methods, the most reliable one proved to be G1 (for a discussion see 73

Ref. [8] ) followed by B3LYP/6-311+G(d,p)/PBF. Calculations of charged species as required 74 for the Gibbs energy of equilibrium (1) are inherently poorer. G1 does not allow calculating 75 charged species. Here B3LYP/6-311+G(d,p)/PBF has been tested to be the best. Indeed, the 76 calculated value using this method for comes closest to the experimental value. For 77 equilibrium (1), a reliable experimental value is available, and for further discussions one can 78 neglect the calculated ones. 79

The Gibbs energy of reaction (6) relates to the equilibrium constant of reaction ( 6) 80 according to equation (9). 81 lnK = -∆G 0 /RT (9) 82

From the above value of +47 kJ mol -1 for the Gibbs energy of equilibrium (6), an 83 equilibrium constant of K 6 = 10 -8.2 is calculated. The anion of the hydroxyl radical, O •-, reacts 84 fast with O 2 (k 1 = 3.5 × 10 9 M -1 s -1 ) giving rise to O 3

•-, and details of equilibrium (1) are well 85 established (K 1 = 10 5.7 M, k -1 = 1.94 × 10 3 s -1 ). [1] Thus, the Gibbs energy for reaction (1) is 86 ∆G 0 = -32 kJ mol -1 . A Gibbs energy of ∆G 0 = +67 kJ mol -1 for reaction (8) is calculated from 87 pK a ( • OH) = 11.8. 88

Based on the thermodynamic cycle (Fig. 1), a Gibbs energy of ∆G 0 = -12 kJ mol -1 is 89 arrived at for reaction (5). This relates to a pK a value of the HO 3

• radical of -2.1 in good 90 agreement with Bielski and Schwarz relationship [12] that yields pK a (HO 3 This good agreement supports our earlier statement [7] that the reported [10] pK a (HO 3 • ) = 6.15 92

[revised to 8.2 (Bühler, private communication)] must have been an experimental artifact (so-93 called "reactivity pK"). 94

As the rate constant for reaction (-5) must be in the order of 10 10 M -1 s -1 , [14] the rate of 95 deprotonation of HO 3 • , reaction (5), must be in the order of 10 12 s -1 . This reaction is in 96 competition with its decay into • OH and O 2 [reaction (-6)]. Upon calculating the Gibbs energy 97 of the reaction of H • with O 3 , the adduct HO 3

• proved to be sufficiently stable to qualify as an 98 intermediate [reactions (10) and (-6); ∆G 0 overall = -411 kJ mol -1 ]. 99

H • + O 3 → HO 3 • (10) 100
In other reactions of ozone with free radicals; such adducts are often only transition states 101 (Naumov and von Sonntag, submitted). This points to a lifetime of HO 3

• beyond several 102 vibrations, that is, k -6 < 10 12 s -1 . With K 6 in the order of 10 -8 , k 6 must be < 10 4 M -1 s -1 . Form 103 the pulse radiolysis study of the reaction of H • with O 3 , carried out at pH 2, [15] there was no 104 indication that O 3

•-is present at about 1 µs. Thus the lifetime of HO 3

• must be < 10 7 s. This 105 narrows the range to 0.1 < k 6 < 10 4 M -1 s -1 . In any case, the reaction of • OH with O 2 must be 106 very slow, as generally assumed, and can thus be neglected for most purposes. 107

CONCLUSION 109

The equilibrium constants for the reactions of O 2 with • OH and its anion, O •-, differ by as 110 much as fourteen orders of magnitude. The generally accepted statement that the The Gibbs energy of reaction ( 6) was calculated using a number of approaches. Density 118 Functional Theory (DFT) calculations were carried out using B3LYP correlation 119 functional [16;17] (Jaguar 7.6 program) [18] . The molecular geometries of all calculated molecules 120 were optimized in water at the B3LYP/6-311+G(d,p)/PBF level of theory. The interactions 121 between the molecule and the solvent were evaluated by Jaguar's Poisson-Boltzmann solver 122 (PBF). [19] Gibbs free energies (G 0 ) were calculated at the same level of theory at a standard 123 temperature of 298.15 K and a pressure of 1 atm using unscaled frequencies. Reaction 124 parameters were calculated as the difference of the Gibbs free energies ∆G 0 between reactants 125 and products. Calculations were also made in water (self-consistent reaction field 126 SCRF=COSMO models) [20;21] at Gaussian-1 (G1) [22;23] , Gaussian-3 (G3MP2B3) [24] , and DFT 127 B3LYP/6-311+G(3df,3pd) levels (as implemented in Gaussian 03) [25] . 128 (1), k 1 = 3.5 × 10 9 M -1 s -1 ] [1] while • OH does not, (ii) how decays O 3

reactions O • • • •--- -+ O 2 ⇄ ⇄ ⇄ ⇄ O 3 • • • •--- - (1)
, H + + O • • • • - -- -⇄ ⇄ ⇄ ⇄ • • • • OH (8), • • • • OH + O 2 ⇄ ⇄ ⇄ ⇄ HO 3 • • • • (6) and H + + O 3 • • • •-⇄ ⇄ ⇄ ⇄ 16 HO 3 • • • • (5)
•-into • OH, and (iii) what 29 is the pK a value of the HO 3

• radical? 30

O •-+ O 2 ⇄ O 3 •- (1) 31
In reactions of ozone with organic compounds in aqueous solution, O 3

•-is a common 32 intermediate. [2][3][4] It may be generated in a direct electron transfer reaction [reaction (2)], [5] but 33 more often an adduct is an intermediate [reactions (3) and ( 4)]. [6][7][8] 34

X + O 3 → X •+ + O 3 •- (2) 35 X + O 3 → + XOOO -→ X •+ + O 3 •- (3)/(4) 36
This is depicted for the reaction of ozone with 1,3,5-trimethoxybenzene in Fig. 1, where 37 From the above, it can be seen that the reaction of • OH with O 2 and the pK a of HO 3

• radical 67 are interconnected by a thermodynamic cycle (Fig. 2). 68 The Gibbs energy of reaction (6) has been calculated by various approaches (Table 1 Among the above methods, the most reliable one proved to be G1 (for a discussion see 83

Ref. [8] ) followed by B3LYP/6-311+G(d,p)/PBF. Calculations of charged species as required 84 for the Gibbs energy of equilibrium (1) are inherently poorer. G1 does not allow calculating 85 charged species. Here B3LYP/6-311+G(d,p)/PBF has been tested to be the best. Indeed, the 86 calculated value using this method for comes closest to the experimental value. For 87 equilibrium (1), a reliable experimental value is available, and for further discussions one can 88 neglect the calculated ones. 89

The Gibbs energy of reaction (6) relates to the equilibrium constant of reaction ( 6) 90 according to equation (9). 91 lnK = -∆G 0 /RT (9) 92

From the above value of +47 kJ mol -1 for the Gibbs energy of equilibrium ( 6), an 93 equilibrium constant of K 6 = 10 -8.2 is calculated. The anion of the hydroxyl radical, O •-, reacts 94 fast with O 2 (k 1 = 3.5 × 10 9 M -1 s -1 ) giving rise to O 3

•-, and details of equilibrium (1) are well 95 established (K 1 = 10 5.7 M, k -1 = 1.94 × 10 3 s -1 ). [1] Thus, the Gibbs energy for reaction (1) is 96 ∆G 0 = -32 kJ mol -1 . A Gibbs energy of ∆G 0 = +67 kJ mol -1 for reaction (8) is calculated from 97 pK a ( • OH) = 11.8. 98

Based on the thermodynamic cycle (Fig. 1), a Gibbs energy of ∆G 0 = -12 kJ mol -1 is 99 arrived at for reaction (5). This relates to a pK a value of the HO 3

• radical of -2.1 in good 100 agreement with Bielski and Schwarz relationship [12] that yields pK a (HO 3

• ) = -2.0. 101
This good agreement supports our earlier statement [7] that the reported [10] pK a (HO 3 As the rate constant for reaction (-5) must be in the order of 10 10 M -1 s -1 , [14] the rate of 105 deprotonation of HO 3 • , reaction (5), must be in the order of 10 12 s -1 . This reaction is in 106 competition with its decay into • proved to be sufficiently stable to qualify as an 108 intermediate [reactions (10) and (-6); ∆G 0 overall = -411 kJ mol -1 ]. 109

H • + O 3 → HO 3 • (10) 110
In other reactions of ozone with free radicals; such adducts are often only transition states 111 (Naumov and von Sonntag, submitted). This points to a lifetime of HO 3

• beyond several 112 vibrations, that is, k -6 < 10 12 s -1 . With K 6 in the order of 10 -8 , k 6 must be < 10 4 M -1 s -1 . Form 113 the pulse radiolysis study of the reaction of H • with O 3 , carried out at pH 2, [15] there was no 114 indication that O 3

•-is present at about 1 µs. Thus the lifetime of HO 3

• must be < 10 7 s. This 115 narrows the range to 0.1 < k 6 < 10 4 M -1 s -1 . In any case, the reaction of • OH with O 2 must be 116 very slow, as generally assumed, and can thus be neglected for most purposes. 117

CONCLUSION 119

The equilibrium constants for the reactions of O 2 with • OH and its anion, O •-, differ by as 120 much as fourteen orders of magnitude. The generally accepted statement that the • OH radical 121 does not react with O 2 is substantiated here by bracketing the rate constant: 0.1 > k 6 < 10 4 M -1 122 s -1 . It is now established that HO 3

• is a very strong acid with a pK a near -2. With this revised 123 pK a value at hand, modeling of ozone reactions in aqueous solution can now be carried out 124 with much higher accuracy. 125

EXPERIMENTAL 127

The Gibbs energy of reaction (6) was calculated using a number of approaches. Density 128 Functional Theory (DFT) calculations were carried out using B3LYP correlation 129 functional [16;17] (Jaguar 7.6 program) [18] . The molecular geometries of all calculated molecules 130 were optimized in water at the B3LYP/6-311+G(d,p)/PBF level of theory. The interactions 131 between the molecule and the solvent were evaluated by Jaguar's Poisson-Boltzmann solver 132 (PBF). [19] Gibbs free energies (G 0 ) were calculated at the same level of theory at a standard 133 parameters were calculated as the difference of the Gibbs free energies ∆G 0 between reactants 135 and products. Calculations were also made in water (self-consistent reaction field 136 SCRF=COSMO models) [20;21] at Gaussian-1 (G1) [22;23] , Gaussian-3 (G3MP2B3) [24] , and DFT 137 B3LYP/6-311+G(3df,3pd) levels (as implemented in Gaussian 03) [25] . 138

6 ( 1 ) 11 clemens 23 INTRODUCTION 25 F

 61112325 Leibniz-Institut für Oberflächenmodifizierung (IOM), Permoserstrasse 15, D-04318 7 Leipzig, Germany, email: sergej.naumov@iom-leipzig.de. (2) Max-Planck-Institut für 8 Bioanorganische Chemie, Stiftstrasse 34-36, P.O. Box 101365, D-45470 Mülheim an der 9 Ruhr, Germany. (3) Universität Dortmund, Fachbereich Bio-und Chemieingenieurwesen, 10 Lehrstuhl Umwelttechnik, Emil-Figge-Strasse 70, 44227 Dortmund, Germany, e-mail: are interconnected by a thermodynamic cycle. For equilibria (1) and (8) 16 reliable equilibrium constants, and hence Gibbs energies are available (∆ ∆ ∆ ∆G 0 strong acid. This value agrees with a value of -2.0 obtained from the 21 Bielski and Schwarz relationship for pK a values of O x H y compounds. Reaction (6) must 22 be very slow, 0.1 < k 6 < 10 4 M -1 s -1 . free-radical chemistry in aqueous solution has been troubling free-radical 26 chemist for a long time. Why reacts the anion of the • OH radical, O • -, fast with O 2 [reaction 27

Figure 1 .

 1 Figure 1. Thermodynamic cycle involving O 2 , • OH/O •-and HO 3

  Leibniz-Institut für Oberflächenmodifizierung (IOM), Permoserstrasse 15, D-04318 7 Leipzig, Germany, email: sergej.naumov@iom-leipzig.de. (2) Max-Planck-Institut für 8 Bioanorganische Chemie, Stiftstrasse 34-36, P.O. Box 101365, D-45470 Mülheim an der 9 Ruhr, Germany. (3) Universität Dortmund, Fachbereich Bio-und Chemieingenieurwesen, 10 Lehrstuhl Umwelttechnik, Emil-Figge-Strasse 70, 44227 Dortmund, Germany, e-mail: 11 clemens@vonsonntag.de 12 13 14

  are interconnected by a thermodynamic cycle. For equilibria (1) and (8) 16 reliable equilibrium constants, and hence Gibbs energies are available (∆ ∆ ∆ ∆G 0 (1) = -32 kJ 17 mol -1 , ∆ ∆ ∆ ∆G 0 (8) = 67 kJ mol -1 ). For reaction (6), a Gibbs energy of ∆ ∆ ∆ ∆G 0 (6) = 47 kJ mol -1 (K 6 18 = 10 -8.2 M) has now been calculated by G1. From the thermodynamic cycle one hence 19 arrives at ∆ ∆ ∆ ∆G 0 (5) = -12 kJ mol -1 ). This relates to pK a (HO 3 very strong acid. This value agrees with a value of -2.0 obtained from the 21 Bielski and Schwarz relationship for pK a values of O x H y compounds. Reaction (6) must 22 be very slow, 0.1 < k 6 < 10 4 M -1 s -1 . free-radical chemistry in aqueous solution have been troubling free-radical 26 chemist for a long time: (i) why reacts the anion of the • OH radical, O • -, fast with O 2 [reaction 27

Figure 1 . 3 •

 13 Figure 1. Thermodynamic cycle involving O 2 , • OH/O •-and HO 3 • /O 3 •-

6 ( 1 ) 12 13

 6112 Leibniz-Institut für Oberflächenmodifizierung (IOM), Permoserstrasse 15, D-04318 7 Leipzig, Germany, email: sergej.naumov@iom-leipzig.de. (2) Max-Planck-Institut für 8 Bioanorganische Chemie, Stiftstrasse 34-36, P.O. Box 101365, D-45470 Mülheim an der 9 Ruhr, Germany. (3) Instrumentelle Analytische Chemie, Universität Duisburg-Essen, 10Universitätsstrasse 5, 45117 Essen, Germany, e-mail: clemens@vonsonntag.de 11 In the reactions of ozone with organic compounds in aqueous solution,

  are interconnected by a thermodynamic cycle. For equilibria (1) and (8) 17 reliable equilibrium constants, and hence Gibbs energies are available (∆ ∆ ∆ ∆G 0 strong acid. This value agrees with a value of -2.0 obtained from the 22 Bielski and Schwarz relationship for pK a values of O x H y compounds. Reaction (6) must 23 be very slow, 0.1 < k 6 < 10 4 M -1 s -1 . free-radical chemistry in aqueous solution have been troubling free-radical 27 chemist for a long time: (i) why reacts the anion of the • OH radical, O • -, fast with O 2 [reaction 28

39

  electron transfer competes with the formation of an adduct [reaction (b)From this zwitterionic adduct, three competing reactions are evident from the products that 40 are formed. Release of singlet dioxygen [reaction (c)] gives rise to the phenol [reaction (d)].

Figure 2 . 3 •

 23 Figure 2. Thermodynamic cycle involving O 2 , • OH/O •-and HO 3 • /O 3 •-

  8.2 (Bühler, private communication)] must have been an experimental artifact (so-103 called "reactivity pK"). 104

  .15 K and a pressure of 1 atm using unscaled frequencies. Reaction 134

  3

  3

Table 1 .

 1 Compilation of experimental und calculated Gibbs energies (kJ mol -1 ) 71

	).

Table 1 .

 1 Compilation of experimental und calculated Gibbs energies (kJ mol -1 ) 71

	).

Table 1 .

 1 Compilation of experimental und calculated Gibbs energies (kJ mol -1 ) 81

	).

  • OH and O 2 [reaction (-6)]. Upon calculating the Gibbs energy 107 • with O 3 , the adduct HO 3

	of the reaction of H
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	6

http://mc.manuscriptcentral.com/poc Journal of Physical Organic Chemistry

[reactions (e) and (f)]. The quinone is thought to arise from the hydrotrioxide formed upon 43 proton transfer [reactions (g) and (h)]. Details are discussed elsewhere [5] . 44

The formation of O 3

•-in ozone reactions is of considerable importance in water treatment. 45

There is a general agreement that O 3

•-must be the precursor of • OH that is made responsible 46 for the degradation of ozone-refractory micropollutants in the treatment of drinking water and 47 wastewater with ozone. In this context it may be noted that upon reacting with ozone, the 48 organic matter contained in wastewater produces as much as 13% • OH with respect to ozone 49 consumed. [9] Beyond this general statement that O 3

•-must be the precursor of • OH, there is, 50 however, a continuing discussion as to how O 3

•-is converted into • OH. Without having solved 51 this question, it is not possible to simulate such ozone reactions with some confidence. 52

According to a pulse radiolysis study, O 3

•-is protonated [reaction (-5)], and HO 3

• thus 53 generated looses O 2 [reaction (-6)]. [10] 54

A pK a (HO 3 • ) = 6.15 (revised to 8.2, Bühler, private communication) has been estimated 57 from these experiments, [10] and this mechanism is also found in the most recent review on the 58 chemistry of ozone in aqueous solution. [4] 59 A pK a as high as this is doubtful as HO 3

• should be more acidic than HO 2

• whose pK a = 4.8 60 is well established. [11] In fact, Bielski and Schwarz [12] developed an experimental relationship 61 for the pK a values of O x H y acids [equation ( 7)], wherefrom one arrives at pK a (HO 3

) 63

We have, therefore, suggested that O 3

•-decays according to equilibria (1) and ( 8) 64

[pK a ( • OH) = 11.8]. [7;13] 65


Francois Vermeulen
email: francois.vermeulen@uzleuven.be

Nathalie Feyaerts

Marijke Proesmans

Kris De Boeck

MD Vermeulen François

MD PhD Proesmans Marijke

M Sc Feyaerts Nathalie

MD PhD De Boeck Kris

François Vermeulen

Nasal potential measurements on the nasal floor and under the inferior turbinate : does it

Keywords: Cystic fibrosis, diagnosis, Nasal potential difference Nasal potential measurements : nasal floor vs turbinate Cystic fibrosis, diagnosis

4 .7H20 0.30 g/l) + amiloride 100 µM (solution 3), chloridefree solution + amiloride 100 µM + 106 isoprenalin 10 µM (solution 4), each at 5 ml/minute during 3 minutes and until a stable value is reached.

107

The final chloridefree solution containing amiloride, isoprenaline and ATP 100 µM (solution 5) is 108 perfused during 1 to 2 minutes and until a peak value is reached. The perfusion line passes through a 109 thermostatic bath set up at a temperature of 43°C, to heat the solutions to 35-37°C, measured at the tip 110 of the exploring catheters. The amiloride response is the potential difference between measurement at 111 the end of perfusion with solution 1 and the end of solution 2. Total chloride response is the difference 112 between potential at the end of perfusion with solution 2 and at the end of solution 4. ATP response is 113 the potential difference at the end of perfusion of solution 4 and the peak potential during solution 5,

114

and is used as positive control. Ratio between amiloride response and total chloride response is 115 calculated as e (total chloride response/amiloride response) 10 . TransPD is the difference between potential at the end of 116 perfusion with solution 1 and at the end of perfusion with solution 4 (figure 1 CF and control subjects. Using the classical cut-off of -5 mV, sensitivity and specificity were 94.6 % and 175 97.1% under the nasal turbinate, and 94.1 % and 98.2 % on the nasal floor. To reach a sensitivity of 176 100%, the cut-off for total chloride response would be -8 mV for the floor (with a specificity of 82.1 %) 177 and -10 mV for the turbinate (with a specificity of 78.5 %).

179

To assess repeatability, the measurement was repeated in 41 controls and 16 CF patients, resulting in 180 two acceptable tracings in 34 controls and 14 CF patients for the nasal floor and 38 controls and 16 CF 181 patients under the turbinate. Median and IQR of differences between first and second measurement are 182 given in Table 1. Correlation between first and second measurement was adequate to excellent, with a 183 Spearman r coefficient for total chloride response of 0.71 (p<0.001) for the nasal floor and 0.78 184 (p<0.001) for the turbinate. Assessment of repeatability for total chloride response between the first and 185 second measurement by the Bland and Altman method is shown in figure 3. There was no correlation

205

and the power to discriminate between controls and CF subjects was equivalent.

206

Measurements on the floor of the nose resulted in a small but significant increase in non-interpretable 207 tracings, mainly because of more frequent tracings with low potential changes (amplitude of both 208 response to amiloride and total chloride response less than 5 mV). Low total chloride response is 209 suggestive of CF, and a cut-off of 5 mV has been chosen to differentiate control patients from CF, based 210 on earlier publications 17 . However, if both the amiloride response and the total chloride response are 211 weak, the latter is possibly the result of an overall poor mucosal ion transport or a poor contact of the 212 electrode with the mucosa, and not of a defective CFTR protein. Therefore, we chose to consider such 213 tracings with weak potential changes as non-interpretable. Further supporting this approach, 214 hyperpolarization during perfusion with ATP was also low (less than 5 mV) in all of these 'flat' tracings.

215

The reasons for more frequent flat tracings from the nasal floor could be diverse. The nasal floor is more 216 exposed to temperature changes, dry air or chemical irritants. The proportion of ciliated cells is lower in 217 mucosae from the nasal floor (around 50%) than at the inferior surface of the turbinate (75%) 6 .

218

Therefore, reduction of the number of 'functional' ciliated cells below a critical number leading to 219 changes in NPD is likely to occur earlier on the nasal floor.

221

If nasal potential difference is to be used in clinical studies of CFTR modulators, efforts should be made

225

patients harbouring a nonsense mutation 15 . This suggests correction in treated CF subjects to 226 approximately 40 % of the total chloride response in control subjects (-12 +/-7 mV) 15 . The number of 227 patients to be included in a placebo controlled study of a CFTR modulator would be slightly lower if 228 measurements were done on the nasal floor, even after correction for the higher number of non-229 acceptable tracings (Table 2).

245

Measurements from the inferior turbinate benefit from a higher rate of interpretable tracings, which

251

The authors have no conflict of interests to disclose. We are grateful to Bram Verstockt for technical 252 assistance and data collection, and to all the trial participants.

25

However, a small number of patients have an inconclusive sweat test and only one or even no mutation 26 found after genetic analysis. For these patients, further assessment of CFTR function is indicated 2,3 .

28

The initial pathophysiological step leading to the multi-organ clinical manifestations of cystic fibrosis (CF)

84

following the methods previously published 18,19 . To minimize movement, the subject sits upright with his 85 chin on a headrest. The head is in flexion to allow the perfused solutions to drip out of the nose.

168

Ability to discriminate between CF and control subjects was examined with ROC curves and optimal cut-169 offs were calculated separately for each measurement technique. e (total chloride response/amiloride response) and 170 transPD discriminate best. Cut-offs were similar for both methods, with a cut-off of 0.75 for e (total chloride 171 response/amiloride response) , and 7.5 mV for transPD resulting in a sensitivity and a specificity of 100% . Area and the power to discriminate between controls and CF subjects was equivalent.

206

Measurements on the floor of the nose resulted in a small but significant increase in non-interpretable 207 tracings, mainly because of more frequent tracings with low potential changes (amplitude of both 208 response to amiloride and total chloride response less than 5 mV). Low total chloride response is 209 suggestive of CF, and a cut-off of 5 mV has been chosen to differentiate control patients from CF, based 210 on earlier publications 17 . However, if both the amiloride response and the total chloride response are 211 weak, the latter is possibly the result of an overall poor mucosal ion transport or a poor contact of the 212 electrode with the mucosa, and not of a defective CFTR protein. Therefore, we chose to consider such 213 tracings with weak potential changes as non-interpretable. Further supporting this approach, 214 hyperpolarization during perfusion with ATP was also low (less than 5 mV) in all of these 'flat' tracings.

215

The reasons for more frequent flat tracings from the nasal floor could be diverse. The nasal floor is more 216 exposed to temperature changes, dry air or chemical irritants. The proportion of ciliated cells is lower in 217 mucosae from the nasal floor (around 50%) than at the inferior surface of the turbinate (75%) 6 .

218

Therefore, reduction of the number of 'functional' ciliated cells below a critical number leading to 219 changes in NPD is likely to occur earlier on the nasal floor.

225

patients harbouring a nonsense mutation 15 . This suggests correction in treated CF subjects to 226 approximately 40 % of the total chloride response in control subjects (-12 +/-7 mV) 15 . The number of 227 patients to be included in a placebo controlled study of a CFTR modulator would be slightly lower if 228 measurements were done on the nasal floor, even after correction for the higher number of non-229 acceptable tracings (Table 2).

MgS0

  , panel a and b) 117 At the end of the measurement, position of the catheters is checked visually. For quality control, skin 118 potential and offset of the electrodes and of the closed circuit are measured at the start and the end of 119 the procedure. Only skin potentials more negative than -30 mV and electrode or circuit offsets below 5 120 mV are allowed. Tracings were discarded if the response to amiloride and total chloride response were floor tracings (first measurement in each subject) were not interpretable, 13 because of a

186

  between time elapsed between measurements and differences in NPD values obtained.In 8 control 187 subjects, measurements were repeated 4 times over 4 to 8 days. Values obtained for total chloride 188 response are shown in figure 4. Mean coefficient of variation of the total chloride response was similar 189 with both methods (37 % on the nasal floor and 41 % under the inferior turbinate). 190 191 In 43 evaluated subjects, measurement was perceived as more convenient on the nasal floor by 17 192 subjects, under the turbinate by 11, and 15 reported no difference between techniques (p=0.5). 193 194 Total chloride response means and standard deviations in controls and CF subjects were used to 195 calculate the number of patients needed to include in each arm of a placebo controlled study with a 196 CFTR modulator. Power calculations were assuming correction of chloride response to 10, 35, 50 or 100 197 % of the values observed in control subjects with power of 80% at an alpha-level of 0.05 (

  of transepithelial nasal potential difference has gained interest because of its 201 potential to evaluate responses to CFTR modulators in vivo. This study aimed at exploring the influence 202 of the site of the measurement in the nostril as one of the major methodological differences between 203 techniques in use. The setup was identical for both techniques, except for the different site of 204 measurement and catheter. Main results are that mean NPD values were similar with both techniques,

Page

  222towards using the endpoints with the highest ability to detect changes in CFTR function. A few trials 223 of CFTR modulators by their effect on nasal potential. Applying topical gentamicin 224 on the nasal epithelium, mean total chloride response increased from 0±3.6 to -5±2.7mV in treated CF

230

  Patients' and physicians' preference should be taken into account. For measurements under the 231 turbinate, more technical skills are needed to visualize the turbinate and to place the catheter at the 232 right location in the nose. Displacement of the catheter is also more likely while securing the catheter to 233 the nose. None of the techniques is accessible without experience and learning time. Our centre has 234 several years of experience measuring NPD on the nasal floor. More recently, measurement under the 235 turbinate was introduced to anticipate participation to international studies. More than 20 236 measurements per technique were performed by each operator before the start of the study, to ensure 237 expertise in both techniques. Measured values and success rate did not change over time, showing 238 absence of a 'learning effect' during the data collection.

239

  This head-to-head comparison illustrates that simultaneous measurements taken from the nasal floor 240 and from the inferior turbinate do not differ much. Mean NPD values are similar, as is their 241 discriminating power between CF and controls and their repeatability. From the patients' perspective, 242 both techniques cause similar discomfort. Measurement taken from the nasal floor have the advantages 243 of an easier technique, less risk of catheter displacement during the test, and a lower number of patients 244 to include in trials of CFTR modulators, but more frequent flat tracings could obscure this advantage.

Figure 1 : 16 Figure 2 :Figure 3 :Figure 4 : 1 1 2 7

 116234117 Figure 1: NPD tracings in a healthy (panel a) and CF (panel b) subject. CF subjects have a more 259

11 Results:

 11 More interpretable tracings were obtained under the turbinate (120/124) than on the nasal 12 floor (109/124), p=0.015. Within each subject group, mean values obtained were similar for maximal 13 basal potential, response to amiloride and total chloride response. Both techniques discriminate well 14 between CF and controls. Repeatability was similar with both methods: mean differences between 2 15 measurements approximated zero for most values. 16 Also after correction for different number of interpretable tracings, simulation of sample size 17 calculation for use in CFTR corrective trials was slightly in favour of measurements obtained on the 18 nasal floor.

19 Conclusion:23

 19 NPD measurements under the inferior turbinate and on the nasal floor have similar 20 discriminative power for diagnostic use. Measurements under the turbinate result in a slightly higher 21 proportion of interpretable tracings but sample size calculation slightly favours the nasal floor method.Most patients with CF have a typical clinical picture of multi-organ disease, and the diagnosis is 24 confirmed by high sweat chloride concentration. In addition, genetic testing shows 2 CFTR mutations 1 .

83

  29 is dysfunction of a single protein, the Cystic Fibrosis Transmembrane Regulator (CFTR). The protein is a c-30 AMP activated chloride channel, located at the apical membrane of the respiratory epithelia. In the 31 airways, absence or dysfunction of the CFTR protein leads to decreased chloride secretion and increased 32 sodium absorption. The latter is caused by loss of inhibition of the epithelial sodium channel (ENaC) by 33 the absent or dysfunctional CFTR 4 . This disturbed salt transport results in a dehydrated surface liquid 34 layer and viscous secretions, defective mucociliary transport, chronic infection and inflammation and 35 eventually lung destruction and respiratory insufficiency 5 . 36 37 CFTR function can be assessed in vivo by measurement of transepithelial nasal potential difference 38 (NPD). The electrical potential difference between the surface and the subepithelial layer arises from ion 39 transport through the epithelia 6 . This potential is measured in vivo between a surface electrode placed 40 on the nasal epithelium and an electrode in connection with the subcutaneous space. Under basal 41 conditions, net absorption of sodium through the ENaC channel results in a negative basal potential at 42 the epithelial surface. Addition of amiloride to the perfusion fluid inhibits ENac, making the surface 43 potential less negative. Perfusion with a chloridefree solution promotes chloride secretion through CFTR, 44 thereby generating a more negative surface potential. Adding isoprenalin to the perfusion fluid further 45 activates protein-kinase A regulated chloride channels, mainly CFTR, leading to a more negative Nasal potential difference was measured simultaneously under the nasal turbinate and on the nasal floor

86A 6 MgS0 4 .

 64 PE90 tubing exploring bridge is inserted under the inferior turbinate under direct visualisation by a 87 nasal speculum plus a frontal light. In the opposite nostril, a modified 8 Fr Foley bladder catheter is 88 inserted along the nasal floor, close to the septum, with a side hole facing down. Both exploring bridges 89 are connected to an Ag/AgCl electrode through a continuous perfusion with Ringer solution (NaCl 833 g/l, KCl 0.3 g/l, K2HPO4 0.42 g/l, KH2PO4 0.05 g/l, MgCl2.6H20 0.24 g/l) at 0.1 91 ml/minute. The reference bridge is a 23 G subcutaneous needle flushed with identical Ringer solution, 92 connected to an Ag/AgCl electrode. Two high impedance voltmeters (Knick Portamess®, Elektronische 93 Meβgeräte, Berlin, Germany) connected to a computer recorded the potential at a frequency of 1 Hz.94The nasal mucosae is perfused with different solutions through a PE50 tubing attached to the PE90 95 tubing with a silicone sheath (under the turbinate), or via the second channel of the Foley catheter (nasal 96 floor) (figure1, panel c).97Sequence of test procedures was standardized as follows: maximal basal potential at the inferior surface 98 of the inferior turbinate was measured, followed by the maximal basal potential on the nasal floor in the 99 other nostril. The Foley catheter is then inserted to the place of maximal potential, and secured with 100 tape on the nose. The procedure is repeated in the other nostril under the turbinate, with the PE50/PE90 101 catheter. Continuous recording is done during perfusion with Ringers solution (solution 1 at 5 ml/min) 102 during 1-3 minutes, until a stable value is reached (<1 mV drift over 30 seconds). Sequential perfusion is 103 then started with Ringers + amiloride 100 µM (solution 2), chloridefree sodium gluconate solution (Na 104 gluconate 32.26 g/l, Ca gluconate 0.97 g/l, K gluconate 0.95 g/l, K 2 HPO 4 0.42 g/l, KH 2 PO 4 0.05 g/l, 7H20 0.30 g/l) + amiloride 100 µM (solution 3), chloridefree solution + amiloride 100 µM + 106 isoprenalin 10 µM (solution 4), each at 5 ml/minute during 3 minutes and until a stable value is reached.107The final chloridefree solution containing amiloride, isoprenaline and ATP 100 µM (solution 5) is 108 perfused during 1 to 2 minutes and until a peak value is reached. The perfusion line passes through a 109 thermostatic bath set up at a temperature of 43°C, to heat the solutions to 35-37°C, measured at the tip 110 of the exploring catheters. The amiloride response is the potential difference between measurement at 111 the end of perfusion with solution 1 and the end of solution 2. Total chloride response is the difference 112 between potential at the end of perfusion with solution 2 and at the end of solution 4. ATP response is 113 the potential difference at the end of perfusion of solution 4 and the peak potential during solution 5, 114 and is used as positive control. Ratio between amiloride response and total chloride response is 115 calculated as e (total chloride response/amiloride response)10 . TransPD is the difference between potential at the end of 116 perfusion with solution 1 and at the end of perfusion with solution 4 (figure 1, panel a and b) 117 At the end of the measurement, position of the catheters is checked visually. For quality control, skin 118 potential and offset of the electrodes and of the closed circuit are measured at the start and the end of 119 the procedure. Only skin potentials more negative than -30 mV and electrode or circuit offsets below 5 120 mV are allowed. Tracings were discarded if the response to amiloride and total chloride response were 121 both less negative than -5 mV ('flat tracings') or if technical problems occurred (drift, large artefacts, 122 position of the catheter changed during the procedure).

123 124 7 130

 1237 Patient preference for one of the methods was assessed by a visual analogue scale. Nasal potential 125 measurement was not done during periods of exacerbation of nasal symptoms. Nasal medication other 126 than irrigations with normal saline and topical steroids were not allowed in the 24 hours preceding the 127 test. All tests were performed by two experienced operators. The study was approved by the IRB of the 128 University Hospital of Leuven, and informed consent was obtained from all participants. Nasal potential difference was measured in 61 non-smoking healthy subjects, without significant 131 respiratory or upper airway morbidity (median age 30 years, range 20-57), 37 CF patients (median age 18 132 years, range 10-35) and 26 known or obligate heterozygotes (siblings or parents of CF patients with 133 median age 43 years, range 8-56). All but one of the CF patients carried two severe CFTR mutations, with 134 the exception of one patients compound heterozygote for G551R/3849+10 kB. The latter had a sweat 135 chloride of 30 mEq/l, while the other patients had a median sweat chloride of 105 (range 86 to 124 136 mEq/l). In 41 control patients and 16 CF patients, a second measurement was done after a median delay 137 of 57 days.

SPSS 15 (

 15 SPSS Inc) was used for statistical analysis. Number of successful tracings with each method is 140 compared with McNemar's test, difference of proportion of successful tracings between operators with 141 Fisher's exact test. Patient preference for one of the methods was assessed with a chi square test. NPD 142 variables are presented as median and interquartile range (IQR). Differences between the controls, 143 heterozygotes and CF subjects were assessed pairwise with the Mann-Whitney U test. Difference 144 between paired measurements on the nasal floor and under the nasal turbinate were assessed by the 145 Wilcoxon test, as were differences between repeated measurements, together with the Bland and 146 Altman method 20 . The ability to discriminate between normal subjects and CF patients was assessed 147 with the area under the curve(AUC) of the ROC-curves 21 . Power calculations were done using Statistica 8 148 (Statsoft, Inc), with a power of 80%, 2-tailed, p<0.05. Linear regression was used to explore trends over floor tracings (first measurement in each subject) were not interpretable, 13 because of a 153 'flat tracing', two because of a technical problem. Under the turbinate, 4 of 124 tracings were not 154 interpretable. Two tracings were discarded because of a technical problem and two because of catheter 155 displacement to the medial part of the nose. Less tracings taken from the inferior turbinate were 156 rejected (p=0.015). The proportion of rejected tracings was similar for both operators.

157 158

 157 To compare values obtained with each technique, only paired acceptable measurements were assessed.159Median and IQR of NPD values are given in table1. No significant differences were found between values 160 measured on the nasal floor and under the nasal turbinate, with only a slightly more negative maximal 161 potential in heterozygotes and a trend towards a more negative maximal potential under the inferior 162 nasal turbinate in CF patients. CF patients had significantly different mean values for all NPD parameters 163 compared to control subjects and heterozygotes. Significant difference between heterozygotes and 164 controls were only found for ATP response (p=0.023) and e (total chloride response/amiloride response) (p=0.048) 165 measured under the turbinate. There was no consistent trend over time in the measurements, and no 166 systematic difference between operators.

172

  under the curve (AUC) of the ROC curve for total chloride response (figure2) was 0.99, both on the nasal 173 floor and under the nasal turbinate (p=0.72), making this parameter also very suitable for differentiating 174 of transepithelial nasal potential difference has gained interest because of its 201 potential to evaluate responses to CFTR modulators in vivo. This study aimed at exploring the influence 202 of the site of the measurement in the nostril as one of the major methodological differences between 203 techniques in use. The setup was identical for both techniques, except for the different site of 204 measurement and catheter. Main results are that mean NPD values were similar with both techniques,

  205

220 221Page

 220 If nasal potential difference is to be used in clinical studies of CFTR modulators, efforts should be made 222 towards using the endpoints with the highest ability to detect changes in CFTR function. A few trials 223 of CFTR modulators by their effect on nasal potential. Applying topical gentamicin 224 on the nasal epithelium, mean total chloride response increased from 0±3.6 to -5±2.7mV in treated CF

230

  Patients' and physicians' preference should be taken into account. For measurements under the 231 turbinate, more technical skills are needed to visualize the turbinate and to place the catheter at the 232 right location in the nose. Displacement of the catheter is also more likely while securing the catheter to 233 the nose. None of the techniques is accessible without experience and learning time. Our centre has 234 several years of experience measuring NPD on the nasal floor. More recently, measurement under the 235 turbinate was introduced to anticipate participation to international studies. More than 20 236 measurements per technique were performed by each operator before the start of the study, to ensure 237 expertise in both techniques. Measured values and success rate did not change over time, showing 238 absence of a 'learning effect' during the data collection.

239251Figure 1 : 16 Figure 2 :Figure 3 :Figure 4 :

 116234 Figure 1: NPD tracings in a healthy (panel a) and CF (panel b) subject. CF subjects have a more 261

Figure 3 :

 3 Figure 3: Left : Bland and Altman plot showing limits of agreement between the first and the second measurement for total chloride response on the nasal floor (left) and under the inferior turbinate (right) in controls and CF patients Right : comparison of total chloride response in the first and second measurement on the nasal floor (closed symbols) and under the inferior turbinate (open symbols) for control subjects (circles) and CF patients (triangles) 451x177mm (96 x 96 DPI)

Table 2 )

 2 .

249 250 Page 15 of 46 John Wiley & Sons, Inc.

		Pediatric Pulmonology
	248	define in CF patients than in controls. Incorrect placement of the catheter is more likely to occur, Aknowledgments :
		especially in non-experienced hands, resulting in increased variability of measurements.
		F o F o
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	247	could improve the data analysis for studies of CFTR modulators, as flat tracings are more difficult to
		13 14

Table 1 :

 1 Median (IQR) NPD values measured on the nasal floor and under the inferior turbinate in normal subjects, CF patients and heterozygotes.

	% correction of	n of subjects in each
	total chloride response	arm
		Floor	Turbinate
	100 %	5	5
	50%	14	16
	35%	32	37
	20%	95	114
	F o r
	P
	e
	e r
	R
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Table 2 :

 2 Number of patients to be included in each arm of a placebo controlled trial of a CF disease modifying treatment, taking the different number of rejected tracings into account. Power calculations (1-β at 80%, α at 0.05, 2-tailed) are based on mean and standard deviations of total chloride response in control subjects and CF patients with each method, assuming correction of total chloride response in CF patients to 20, 35, 50 and 100% of the values found in control subjects.

  Thank you for resubmitting Title : Nasal potential measurements on the nasal floor and under the inferior turbinate : does it matter ? to Pediatric Pulmonology.Your manuscript has again been reviewed. I am pleased to report that your paper has been provisionally accepted for publication in Pediatric Pulmonology pending possible minor revision. One of the reviewers has the following comments: " Many thanks for addressing all the comments. I still think that Reviewer 1 comment 2 needs to be addressed With all due respect to the Bland Altman graph (I have used this myself) I still think that the graph of chloride response with both turbinate and floor results ON THE SAME FIGURE is really convincing and should be in the final version " Please consider these remarks and if you wish to change the figure then you can submit a revision of the paper marked R2. If not please let me know and I will send the current version on to the publisher.To respond to the reviewers' comments, log on to Manuscript Central http://mc.manuscriptcentral.com/ppul , and enter the Author Center. From there, please click on "Manuscripts with Decisions" where you will find your manuscript title. Click on "Create a Revision." A new window will open with the reviewers comments and a box to enter your response. From this point, complete the revision submission by following all the steps. Please respond in a point-by-point fashion to each and every suggestion of the reviewers.In revising your manuscript. please update your references to ensure you discuss your work in the context of the most recent research.You will be unable to make revisions directly to your originally submitted manuscript. Instead, you should revise your manuscript in a word processing program and then resubmit two copies of the revised version (one clean copy, and one "marked-up" copy.Use red font instead of black font to indicate the revised portions of your manuscript in the marked-up version. The clean version should be in all black font. Save both copies of your revised document to your computer and ensure the saved file titles contain "R2." For example, save your clean version as "yourmanuscripttitleR2clean" and and your marked-up version as "yourmanuscripttitleR2marked." This will make it easier for the reviewers as they review your revision.When your revised manuscript is ready, log on to Manuscript Central (see above) and submit the clean and marked-up versions, ensuring that both files are marked R2.You must upload your figures and tables as separate files. Figures and tables MUST NOT be embedded in the main text of the document. Only TIF or EPS files are appropriate for figures. Tables should be uploaded as either DOC or RTF files. Comments to the Author Many thanks for addressing all the comments. I still think that Reviewer 1 comment 2 needs to be addressed With all due respect to the Bland Altman graph (I have used this myself) I still think that the graph of chloride response with both turbinate and floor results ON THE SAME FIGURE is really convincing and should be in the final version Answer from the authors :As suggested, both graphs give similar information, but the different visualizations show different aspects of the comparison between the two measurements. We thus included a third graph in the figure, as you'll see in the second revision Reviewer: 2 Comments to the Author The questions have been answered extensively.

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	Dear Dr. Vermeulen:
	The following items should be included with your revision:
	a) clean version of manuscript
	b) marked up version of manuscript
	F o r c) point by point response to reviewers d) figures (as EPS or TIF files) and tables (as DOC or RTF files), if F applicable o r ***
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97.1% under the nasal turbinate, and 94.1 % and 98.2 % on the nasal floor. To reach a sensitivity of

John Wiley & Sons, Inc.Pediatric Pulmonology

Part of this work was funded by the 'Belgische vereniging voor Strijd tegen mucoviscidose', the

Belgian CF patient association.

100%, the cut-off for total chloride response would be -8 mV for the floor (with a specificity of 82.1 %) 177 and -10 mV for the turbinate (with a specificity of 78.5 %).

179

To assess repeatability, the measurement was repeated in 41 controls and 16 e (total chloride response/amiloride response) 0,14 (0,07 ; 0,24) 0,3 (0,08 ; 0,41) p=0,144

Transnasal PD (mV) -7 (-10 ; -2) -4 (-11 ; 1) p=0,843