Bertrand G Giraud
email: bertrand.giraud@cea.fr

Robi Peschanski
email: robi.peschanski@cea.fr

On the positivity of Fourier transforms

Keywords: numbers: 12.38.-t, 24.85.+p, 25.75.Dw, 05.70.-a

Characterizing in a constructive way the set of real functions whose Fourier transforms are positive appears to be yet an open problem. Some sufficient conditions are known but they are far from being exhaustive. We propose two constructive sets of necessary conditions for positivity of the Fourier transforms and test their ability of constraining the positivity domain. One uses analytic continuation and Jensen inequalities and the other deals with Toeplitz determinants and the Bochner theorem. Applications are discussed, including the extension to the two-dimensional Fourier-Bessel transform and the problem of positive reciprocity, i.e. positive functions with positive transforms.

I. INTRODUCTION

In a previous paper [START_REF] Giraud | On positive functions with positive Fourier transforms[END_REF], the following questions were asked:

• What are the constraints for an even, real function ψ(r) ensuring that its Fourier transform satisfies,

ϕ(s) ≡ 2 π +∞ 0 dr cos(sr) ψ(r) ≥ 0 , (1)
that is, ϕ be real and positive?

• What are the constraints on even Fourier partners such that ϕ and ψ(r) ≡ 2 π +∞ 0 ds cos(sr) ϕ(s) ≥ 0 ,

be both positive?

The motivation for such questions came from various areas of mathematics, notably [START_REF] Lévy | Fonctions caractéristiques positives [Positive characteristic functions[END_REF], and from physics, for which a good example is the Fourier-Bessel relation [3] between two physically observable quantities which are necessarily positive (for recent references, see e.g. [START_REF] Jaming | On the extremal rays of the cone of positive, positive definite functions[END_REF] for maths, and [START_REF] Pigolotti | Species clustering in competitive Lotka-Volterra models[END_REF] for various physics domains). In the paper [START_REF] Giraud | On positive functions with positive Fourier transforms[END_REF] an approach based on a Gaussian times a combination of a few, low order Hermite polynomials, allowing to consider eigenfunctions of the Fourier transforms [START_REF] Giraud | On positive functions with positive Fourier transforms[END_REF][START_REF] Lévy | Fonctions caractéristiques positives [Positive characteristic functions[END_REF], has shown that the structure of the positivity space in the manifold of the components may be quite involved. In the present work we aim at finding results on a more general basis.

Despite the mathematical interest of the problem, one might argue on practical grounds that the power of modern computers legitimates a brute force approach, of the kind, given ψ for instance, numerically calculate ϕ and "just look at it". We strongly feel, however, that, despite the existence of subtle methods for the integration of oscillatory integrands, finer analytical methods are in demand. In particular, direct criteria for ψ, for instance, without any numerical estimates for ϕ, can provide more elegant and more rigorous approaches.

The main optimal issue for finding out the set of functions solutions would be an overall constructive, sufficient and necessary condition satisfying the above-mentioned positivity. It seems that one is still far from that goal in the literature, since it appears to be a rather involved mathematical problem [6]. Indeed some general theorems exist on the positivity of Fourier transforms [START_REF] Scharnhorst | For a comprehensive list of references[END_REF], if only the well-known Bochner theorem [START_REF] Gel | Generalized Functions[END_REF] on the "positive-definiteness" of the function 1 which we will usefully recall later on. They possess a fundamental interest, and will help our studies.

However, up to our knowledge, they still gave only limited practical tools allowing to get information on the functional set of solutions. Some general useful sufficient constructive conditions are known, such as convexity [START_REF] We | Lafforgue for a fuitful private communication on convexity properties[END_REF] for the function in the integrand (i.e. ψ in [START_REF] Lévy | Fonctions caractéristiques positives [Positive characteristic functions[END_REF]), but it can be easily shown on specific examples [START_REF] Giraud | On positive functions with positive Fourier transforms[END_REF] that it gives only a limited part of the space of solutions. Hence necessary constructive conditions, namely a set of practical constraints on functions whose Fourier transforms are positive, would be helpful.

In the present work, we derive two separate sets of such constructive and necessary conditions obeyed by a function, say ψ(r), if its Fourier transform ϕ(s) is positive. One method (called in short "matrix method") is based on the positivity of a hierarchy of determinants of Toeplitz matrices2 . The second method (called "analytic method") is based on an analytic continuation onto the imaginary axis associated with Jensen inequalities [START_REF] Jensen | Sur les fonctions convexes et les inégalités entre les valeurs moyennes[END_REF]. They are constructive, meaning that they will be expressed directly (and in a sense "simply") in terms of ψ(r). They will be formulated as a set of necessary inequalities, for all values of the variable r most often. This we find much constraining, and thus, hopefully, making a step towards a determination of the set of solutions of the positivity condition [START_REF] Giraud | On positive functions with positive Fourier transforms[END_REF]. If starting with positive functions ψ, the same constraints give useful indications on the solution of the condition (2), which is our final goal.

Let us illustrate one typical constraint from the matrix method. Using a generalization of the well-known consequence of Bochner's theorem, namely

ψ(0) ≥ ψ(r) , ∀r (3)
coming from the application of positive definiteness to the 2-matrix

ψ(0) ψ(r) ψ(r) ψ(0) , (4)
one obtains

ψ(2r) ≥ 2 ψ(r) 2 ψ(0) -ψ(0) , ∀r , (5)
with one more dimension of the matrix. Indeed, considering a three-by-three "equidistance" symmetric Toeplitz matrix,

  ψ(0) ψ(r) ψ(2r) ψ(r) ψ(0) ψ(r) ψ(2r) ψ(r) ψ(0)   , (6)
and applying Bochner's theorem, the positive definiteness of (6) gives [START_REF] Pigolotti | Species clustering in competitive Lotka-Volterra models[END_REF]. Noting that ( 6) is a symmetric Toeplitz matrix whose generalization to higher dimensions is straightforward, we will take advantage of the hierarchy of inequalities obtained by increasing the dimensionality. Note also that the positive definiteness of the matrix (6) implies that of (4) using test vectors of the form (a, b, 0). Ψi,cosh FIG. 1: Left: Full line, positive ψ(r); dashed line, its partly negative transform ϕ(s). Center: Full line, ψi ≡ ψ(ir); dashed line, its expected cosh lower bound according to (7) fails, thus detecting the non-positivity of ϕ. Right: Same detection with the matrix method at order 4 (dashed line), while order 3 (full line) is insufficient.

To illustrate the "analyticity method", one of our results reads,

ψ(ir) ≥ ψ(0) cosh 2 r π ∞ 0 dr ′ r ′2 1 - ψ(r ′ ) ψ(0) ∀ r ∈ [0, ∞[ , (7)
where ψ(ir) is the analytic continuation of ψ(r) onto the imaginary axis. It is a real, convex function, since, by definition (2), ψ(r) is a function of r 2 and it is assumed here that both integrals, ∞ 0 ds cosh(sr)ϕ(s), and ∞ 0 ds s 2 cosh(sr)ϕ(s), are convergent. As will be shown in the following, the coefficient of r in the cosh seen in formula (3) is the average value of s in Fourier space, but the formula, obviously written in the ψ space, introduces no oscillating cosine term. Figure 1 illustrates a typical case3 where the "cosh test" [START_REF] Scharnhorst | For a comprehensive list of references[END_REF] eliminates a candidate ψ, if only because ψ(ir) is blatantly non convex, and the same detection is obtained provided we use a Toeplitz matrix of order 4 or more. The plan of the paper is as follows. The next section Sec. II, is devoted to the description of the two sets of constraints, namely those coming from the matrix method, and then from the analyticity method. Section III provides a numerical investigation of the compared efficiency of both methods based on the approach initiated in Ref. [START_REF] Giraud | On positive functions with positive Fourier transforms[END_REF]. Section IV extends our reasoning and numerical evaluation to the 2-dimensional case of the Fourier-Bessel transform. Section V is a summary and conclusion.

II. NECESSARY CONDITIONS

A. Simple positivity and maximality tests

In the following, ψ(r) is an even function, positive, (ψ(r) ≥ 0, ∀r), and we further assume that ψ has as many derivatives as might be needed. If its Fourier partner ϕ is also positive, then, necessarily, i) The function ψ is "positive definite", hence ψ is maximum at the origin, namely ψ(0) ≥ ψ(r), ∀r, see (3). This may be seen from the Bochner theorem for the pair of points, x 1 = 0, x 2 = r, and the test vector, u = (1, -1), leading to i,j=1.2

u i ψ(x i -x j ) u j = 2[ψ(0) -ψ(r)] ≥ 0 . (8)
ii) For every positive integer q every product, ϕ q ≡ s q ϕ, is positive, hence the Fourier partner, ψ 2q ≡ (-) q d 2q ψ/dr 2q , taking into account the change of sign in front of the cosine in formula (2) verifies the same maximality condition i.e. ψ 2q has its maximum at the origin.

iii) Every moment is positive, µ q ≡ ∞ 0 ds s q ϕ(s) > 0, hence in particular, for even moments and thus from every even derivative of ψ, the condition ψ 2q (0) > 0. The somewhat more involved expression for odd moments is derived in Subsection C, see further. iv) Every average value, such as µ q /µ 0 or µ q /µ q-1 , is positive. We show in the following how, while even moments, see iii), are easily observed in the r-space, odd average values are also available from the r-space.

v) The positivity, for any width b in momentum space, of a product, e -s 2 /(2b 2 ) ϕ, of ϕ by a Gaussian, induces the "positive-definiteness" of the result of the corresponding convolution,

ψ b (r) ≡ b √ 2π ∞ 0 dr ′ e -b 2 (r-r ′ ) 2 /2 + e -b 2 (r+r ′ ) 2 /2 ψ(r ′ ) , (9)
with, for ψ b and its derivatives, all the consequences already seen under i)-iv).

B. Matrix method

Elaborating on the positive definiteness of the function ψ following from the Bochner theorem, let us introduce the hierarchy of symmetric real Toeplitz n-matrices

T n    ψ(0) ψ(r) ψ(2r) ... ψ[(n-1)r] ψ(r) ψ(0) ... ... ψ[(n-2)r] ... ... ... ... ... ψ[(n-1)r] ψ[(n-2)r] ... ψ(r) ψ(0)    . (10)
Application of the Boechner theorem can be shown to lead to the following set of inequalities on Toeplitz determinants of positive-definite functions (thus whose Fourier transforms are positive):

||T n || > 0, ∀n . (11)
It is known that the matrices T n-1 and T n have intertwined eigenvalues, Hence the smaller eigenvalue of T n-1 is always larger than the one of T n , which is itself positive from positive definitiveness. Hence verifying [START_REF] Jensen | Sur les fonctions convexes et les inégalités entre les valeurs moyennes[END_REF] for any given n, ensures positivity4 of all determinants ||T k || for k < n. Indeed, the matrix method will make a practical use of ( 11) for finite n. Notice, however, that ||T n || vanishes like ∝ r n(n-1) when r → 0, creating numerical accuracy problems for its sign. It may be more efficient to track the first change of sign of the lowest eigenvalue.

C. Analyticity method

Recall that µ 0 ≡ ∞ 0 ds ϕ(s) = π/2 ψ(0). Consider the probability distribution, P(s) = 2/π ϕ(s)/ψ(0). By analytic continuation of (2) in r-space we obtain,

ψ(ir) = ψ(0) ∞ 0 ds cosh(sr) P(s) = ψ(0) cosh(sr) , (12)
where we have already assumed that ϕ(s) decreases fast enough when s → ∞ to ensure convergence of the integral.

Then our key ingredient is to make use of the well-known Jensen inequality [START_REF] Jensen | Sur les fonctions convexes et les inégalités entre les valeurs moyennes[END_REF], f (X) P ≥ f ( X P ), valid for any probability distribution P(X), and any convex function f , the hyperbolic cosine being obviously convex. The following lower bound holds, ∀r, ψ(ir) ≥ ψ(0) cosh( s r) .

The calculation in r-space of s = µ 1 /µ 0 is explained in the next subsection D.

An extension of the inequality (13) can be obtained as follows. For notational simplicity, use a short notation σ for s and, temporarily, set ψ(0) = 1, an inessential normalization. More inequalities are easily found if one notices that

ψ(r) + ψ(ir) = [cos(rs) + cos(irs)] ≥ [cos(rσ) + cos(irσ)], (14) ψ(r) -ψ(ir) = [cos(rs) -cos(irs)] ≤ [cos(rσ) -cos(irσ)], (15)
since the sum, [cos x + cos(ix)] and the difference, [cos x -cos(ix)], are obviously a convex and a concave real function of x, respectively. Equivalently, consider now the first complex eighth root ω of unity,

ω 2 = i. It is trivial to find that the combinations, [cos x + cos(ωx) + cos(ω 2 x) + cos(ω 3 x)] and [cos x -cos(ωx) + cos(ω 2 x) -cos(ω 3 x)] are real convex and that [cos x - i cos(ωx) -cos(ω 2 x) + i cos(ω 3 x)] and [cos x + i cos(ωx) -cos(ω 2 x) -i cos(ω 3 x)] are real concave. The following inequalities result, ψ(r) + ψ(ωr) + ψ(ir) + ψ(ωir) ≥ cos(σr) + cos(ωσr) + cos(iσr) + cos(ωiσr), (16) ψ(r) -ψ(ωr) + ψ(ir) -ψ(ωir) ≥ cos(σr) -cos(ωσr) + cos(iσr) -cos(ωiσr), (17) ψ(r) -iψ(ωr) -ψ(ir) + iψ(ωir) ≤ cos(σr) -i cos(ωσr) -cos(iσr) + i cos(ωiσr), (18) ψ(r) + iψ(ωr) -ψ(ir) -iψ(ωir) ≤ cos(σr) + i cos(ωσr) -cos(iσr) -i cos(ωiσr), (19)
Further roots of unity clearly give further inequalities.

The bounds found in (13-19) obviously hold, mutatis mutandis, for ψ b and even derivatives of ψ and ψ b .

D. Odd Moments

Odd average values, appearing as inputs in inequalities (13-19), are obviously also required to be positive, as previously noted as condition iii). It is important to obtain them as simply as possible in terms of ψ(r). However, this is less obvious since odd moments of ϕ cannot be trivially observed from mere derivatives of ψ.

Let us derive them using the Laplace transform of ϕ,

χ(r) ≡ ∞ 0 ds e -rs ϕ(s) ⇒ ∞ 0 ds s ϕ(s) = - dχ dr (r = 0), (20)
which allows a useful expression of the first moment of ϕ. Indeed, starting with the expression,

χ(r) = 2 π ∞ 0 ds e -rs ∞ 0 dr ′ ψ(r ′ ) cos sr ′ , (21)
for r = 0 one can switch the order of integrations in (21) and write,

χ(r) = 2 π ∞ 0 ψ(r ′ ) dr ′ ∞ 0 ds e -rs cos sr ′ = 1 2π ∞ 0 dr ′ 1 r + ir ′ + 1 r -ir ′ ψ(r ′ ) . ( 22
)
By integration by part we then obtain,

χ(r) = i 1 2π ∞ 0 dr ′ log r + ir ′ r -ir ′ dψ(r ′ ) dr ′ , (23)
since the boundary term,

-i 1 2π log r + ir ′ r -ir ′ ψ(r ′ ) r ′ =∞ r ′ =0 , (24)
obviously vanishes for all r > 0.

Actually, the expression (23) of gthe function χ, well defined also at r = 0 by continuity, is the correct analytic continuation of formula (22) at r = 0 and thus the representation of the regular Laplace transform (21) for all r ≥ 0.

Finally, one gets the following formula for the first moment displayed in (20),

∞ 0 sds ϕ(s) ≡ - dχ dr (r = 0) = - 2 π ∞ 0 dr ′ r ′ dψ(r ′ ) dr ′ . ( 25
)
It is also possible to make a "reverse" integral by part, using an a priori arbitrary integration constant C, namely,

∞ 0 sds ϕ(s) = - 2 π 1 r ′ [ψ(r ′ ) -C] r ′ =∞ r ′ =0 - 2 π ∞ 0 dr ′ r ′2 [ψ(r ′ ) -C] . (26)
In fact, the boundary term is nonsingular (and finally zero) only when choosing the constant C = ψ(0), leading to the final formula,

∞ 0 s ds ϕ(s) = 2 π ∞ 0 dr ′ r ′2 [ψ(0) -ψ(r ′ )] . (27)
Inserting the result (27) into the inequality (13) gives rise to formula [START_REF] Scharnhorst | For a comprehensive list of references[END_REF], a self-content functional expression depending only on the knowledge of ψ. We stress again that the formula involves no oscillating terms, except those which might be present in ψ.

It will be noticed, incidentally, that any numerical result from Eq. ( 27) that returns a negative value, µ 1 < 0, for this moment, negates at once the expected positivity of ϕ. A detailed comparison between ψ(ir), ψ(r) ± ψ(ir), etc and their respective bounding functions, as detailed above, becomes useless in such a case of negative moment.

Higher order odd moments, µ are obviously obtained upon substituting ψ 2q for ψ in Eq. ( 27). And the generalization of Eq. ( 27) to ψ b and its moments is trivial.

In the following Section III we show numerical, illustrative cases, the existence of so-called "rebels", where, although ϕ contains negativity, several among the known criteria described previously are not violated. Then we test and compare in a systematic way the efficiency of both methods in creating an efficient "filter" for positivity.

III. NUMERICAL INVESTIGATIONS A. 1-d test functions with two parameters; maximality and cosh tests efficiency (Subsecs. II A and C)

We first consider combinations, ψ = c 0 ξ 0 + c 2 ξ 2 + c 4 ξ 4 , of the following three harmonic oscillator eigenstates,

ξ 0 = π -1 4 e -r 2 /2 , ξ 2 = π -1 4 e -r 2 /2 (2r 2 -1)/ √ 2, ξ 4 = π -1 4 e -r 2 /2 (4r 4 -12r 2 + 3)/(2 √ 6), (28)
with c 0 = cos α, c 2 = sin α cos β, c 4 = sin α sin β, 0 ≤ α < π/2 and 0 ≤ β < 2π. We explored the corresponding half sphere via a grid of parallels and meridians, separated from each other by steps of π/90 and π/45 respectively. Only those functions ψ that are positive are retained for further studies. This selects 951 grid points, out of which 520, shown as tiny dots in all parts of Figures 234, correspond to simultaneous ψ > 0 and ϕ > 0. The remaining 431 points, shown with larger dots in the left part of Fig. 2, correspond to those cases where ϕ has negative parts.

Preselection by conditions of maxima at the origin : As stated in Subsec. II A, it is easy to eliminate 271 out of these 431 cases, via a double maximality condition, namely that ψ(0) and ψ 2 (0) be maxima for ψ and ψ 2 , respectively. The 160 remaining "rebels" satisfy this double maximality condition.

The right part of Fig. 2 shows, besides the 520 "fully acceptable cases" (tiny dots), 160 cases where the nonpositivity of ϕ is not detected by the double maximality condition (moderate size dots) and 271 cases where the same condition is efficient for the detection (big dots). Further screening by a combination of maximality and cosh tests, including convolution : Figure 3 shows, for those 160 functions ψ for which ϕ is non-positive although ψ(0) and ψ 2 (0) are maxima, the result of an analysis of their "convoluted" ψ b , see Eq. ( 9), with b = 2, then b=1, for the left, then right, part of the Figure, respectively. The following eight conditions are used, simultaneously : a) ψ b (0) > 0, ψ b (0) maximum of ψ b , b) the same two conditions for ψ b2 ≡ -d 2 ψ b /dr 2 , c) both average moments for ψ b and ψ b2 must be positive, d) both analytic continuations ψ b (ir) and ψ b2 (ir) must be larger than their respective "cosh" lower bound. (Note, incidentally, that preselected functions ψ, so that ψ(0) > 0 and ψ 2 (0) > 0 make maxima, do not extend necessarily such a positivity and maximality to ψ b (0) and ψ b2 (0).) The big dots in Fig. (3) indicate when one, or several violations, eliminate some among the 160 non-positive ϕ's. It turns out that the choice, b = 1, is here more efficient, with 58 functions rejected out of 160, that the choice, b = 2, with only 23 detections. An even better result is obtained if b = 1/2, with 83 detections. Clearly, with a more general set of test functions, it will be advisable to use a multiplicity of values of b and more criteria to make the detection very efficient, the price of further calculations following this method having to be paid.

Moderate efficiency of the "cosh bound" criteria (cf.13-19) : The efficiency of the criterion, ψ(ir) ≥ ψ(0) cosh ( s r), even when it is reinforced by its generalizations for ψ 2 , ψ b and ψ b2 , turns out to be slightly disappointing for the present set of 160 "rebel" functions. Most detections can already come from the simpler positivity and maximality criteria for ψ b , ψ b2 and so on. It seems, however, that the criterion is systematically quite useful for somewhat subtle cases where the negative parts of ϕ are very weak, but extend to high momenta. To attempt an improvement of the criterion, define then the "complement" function, ψ c ≡ ψ -ψ b . the Fourier transform of which is, obviously, (1 -exp[-s 2 /(2b 2 )]) ϕ. Distinct average momenta, s b and s c correspond to ψ b and ψ c , respectively. A more precise bound results,

ψ(ir) ≥ ψ b (0) cosh [ s b r ] + ψ c (0) cosh [ s c r ] . (29)
It is then trivial to extend this process to more components for the bound.

ψ(ir) ≥ w ψ b (0) cosh [ s b r ] + w ′ ψ b ′ (0) cosh [ s b ′ r ] + ψ c (0) cosh [ s c r ] . (30)
obviously samples the "probability distribution" P(s) better.

Criteria from convolution and even derivatives : Alternately, a multiplication of ϕ by a product of the form, s 2q exp[-s 2 /(2b 2 )], obviously combines derivatives and convolution, to provide further probes of ϕ in adjustable domains, and better detect possible defects of positivity.

Recall that ψ

b2 (r) = -b/ √ 2π d 2 /(dr 2 ) ∞ 0 dr ′ [e -b 2 (r-r ′ ) 2 /2 +e -b 2 (r+r ′ ) 2 /2
]ψ(r ′ ), with a similar formula for ψ b4 . The left part of Figure 4 shows, with b = 1/2, how the bound, ψ b2 (ir) ≥ ψ b2 (0) cosh [ s b2 r ], taken alone, converts 79 among the 160 "moderate size" dots into "big dots". Its right part shows the same, with 80 detections, for the bound, ψ b4 (ir) ≥ ψ b4 (0) cosh ( s b4 r ). These two sets of big dots, while much overlapping, show significant differences. If both criteria are used simultaneously, the detection rate increases to 94/160 ≃ 59%.

Returning to combining criteria already used in this numerical study, most of them positivity and maximality criteria, and taking into account those earlier 271 cases where elimination was very easy, a typical best detection rate reads, (98 + 271)/(160 + 271) ≃ 85%.

B. 1-d test functions with four parameters; more statistics; efficiency of the matrix method (Subsec.II B)

We now consider normalized combinations of the first five even eigenstates of the harmonic oscillator. Components c 0 , c 2 ,..., c 8 are random and we have a set of 4388 cases with both ψ and ϕ positive and, as a test set for our criteria, 21988 cases where ϕ is partly negative while ψ remains positive.

For our statistics, we consider in the following ten "associated functions", namely ψ, its sign weighted derivatives, (-) q d 2q /dr 2q ψ, q = 1, 2, 3, 4, then, with b = 1, its convoluted form [START_REF] We | Lafforgue for a fuitful private communication on convexity properties[END_REF] and its sign weighted derivatives, (-) q d 2q /dr 2q ψ b , q = 1, 2, 3, 4, to be simultaneously tested. Recall that any of these associated functions is just a candidate for a positive Fourier transform related to the same ϕ.

Bochner test with the 3-matrix : A first test consists in the positiveness of the spectrum of the three equidistant point Toeplitz matrix (6). Typically, we let r run with steps .025 until r = 3.5; this is sufficient in the present case. As soon as any of the found eigenvalues for any of the associated functions is negative, the non positivity of ϕ is exposed. This detects 19653 among the 21988 cases of interest, hence an efficiency of ≃ 89%. If one looks at the remaining 2335 "rebel" cases to see whether the the bound, ψ(r) ≥ ψ(0) cosh( s r), holds, only 14 more cases show their violation.

Weaker efficiency of the "cosh test" : A second test consists in omitting the "Bochner test" and rather starting by seeing whether the same bound, ψ(r) ≥ ψ(0) cosh( s r), is violated by any among the ten associated functions. Out of 21988 cases, we find 15887 violations, hence an efficiency of ≃ 72% for this mode of negativity detection. This leaves 6101 "rebel" cases, to be filtered, in turn, by the Bochner test. We find 3780 additional eliminations. As expected, 19653+14=15887+3780=19667, the latter number being the number of detections when both test modes are used.

Sensitivity to the convolution parameter : We also considered different values of the parameter b, ranging from b = 1/5 to b = 5. Detections rates are sensitive to b. Best rates correspond to b values between ≃ 1/2 and ≃ 1, and the choice, b = 1, guided by the exponential decrease in our harmonic oscillator basis, is quite acceptable.

Weak efficiency of the "cosh ± cos" test : For the sake of completeness, the remaining, undetected 2321 cases when b = 1 were submitted to the test described by Eqs. (14-15). No new detection was brought by such bounds.

Further Bochner tests : Then, for the same 2321 cases, the function ψ alone, without its nine associates, was submitted to the Bochner test, by an "equidistance" matrix again, of order 5 now. A gain of 20 more detections was obtained.

Returning to the initial basis of 21988 cases, this test of ψ alone by the 5th order Bochner matrix yields 18003 detections, to be compared with the 19653 results obtained with the third order matrix applied to all ten associates. A simultaneous test of ψ and ψ 2 reaches 18687 detections and a simultaneous test of ψ and ψ b , with again b = 1, returns 19247 detections. This seems to show that a Bochner test on the convoluted ψ b is more useful than a test on the derivative ψ 2 , ψ 2 ,... provided that b is well chosen. The same fifth order Bochner test applied to ψ, ψ 2 , ψ b and ψ b2 yields 19941 detections. When all ten associates are involved, the result reaches 20468, i.e. ≃ 93%. Combining this 5th order Bochner test for the ten associates with a convexity bound test for the same hardly gives ≃ 10 more detections.

Preliminary conclusions on the efficiency of criteria : At this stage with this basis of test functions three conclusions stand, namely i) Bochner tests are more efficient than the cosh tests and similar ones obtained from the Jensen inequality, ii) "associate functions", by derivations and convolutions, do improve results and iii) good detection rates, of order 90% at least, are rather easily available.

IV. TWO-DIMENSIONAL GENERALIZATION

We are interested in the same problematics described in the introduction, applied to the 2-dimensional Fourier case, in the form of the Fourier-Bessel transform. Indeed, one consider here rotationally invariant partners ψ(x) and ϕ(k), where x and k are the lengths of two-dimensional vectors x and k, respectively. Typically [3], in particle physics, ϕ(k) corresponds to a gluon distribution in transverse momentum space k ≡ | k| and ψ(x) is the quark-antiquark dipole distribution in transverse size x ≡ | x|, both distributions being physically required to be positive real functions.

The Bessel-Fourier transforms analogous to formulas (1,2) read,

ϕ(k) = +∞ 0 xdx J 0 (kx) ψ(x) , (31) ψ
(x) = +∞ 0 kdk J 0 (kx) ϕ(k) (32)
Conditions i)-v) extend without difficulty, mutatis mutandis, to this two-dimensional, and more precisely radial, situation. The following Subsections are therefore restricted to only those considerations which reflect the change of dimensionality.

A. Matrix method

The Bochner theorem is known to apply to higher dimensional Fourier transforms. Hence, applying it on the abscissa axis x = (x, 0) for the rotational invariant function ψ(x), one is led to consider the matrix method applied to the following set of Toeplitz matrices

   ψ(0) ψ(x) ψ(2x) ... ψ[(n-1)x] ψ(x) ψ(0) ... ... ψ[(n-2)x] ... ... ... ... ... ψ[(n-1)x] ψ[(n-2)x] ... ψ(x) ψ(0)    . (33)
Looking for positive definiteness of matrices (33), or its failure, then follows the same path as in the one-dimensional case.

B. Analyticity method

Basic lower bound

Following the same method as for the one-dimensional Fourier transform, one defines the probabilistic formulation of a positive Fourier-Bessel transform as kdk P(k) ≡ kdk ϕ(k)/ψ(0) ⇒ ψ(x) = ψ(0) J 0 (kx) ,

where ψ(0) = ∞ 0 kdk ϕ(k) provides the normalization. For the same reasons as for the 1-d case, the analytic continuation x → ix leads to the following Jensen inequality,

ψ(ix) = ψ(0) I 0 (kx) ≥ ψ(0) I 0 (x k ) = ψ(0) I 0 x ∞ 0 kdk kϕ(k)/ψ(0) , ∀x ∈ [0, ∞[ , (35)
where the convex function I 0 (u) ≡ J 0 (iu) is the Bessel function of the second kind. As in the previous section, the next step is to make from Eq. (35) a self-content equation in terms of ψ. For this sake, one introduces the Laplace transform,

χ(x) = ∞ 0 kdk e -kx ϕ(k) = ∞ 0 kdk e -kx ∞ 0 x ′ dx ′ ψ(x ′ ) J 0 (kx ′ ) . ( 36
)
Exchanging the integrals, while keeping x > 0, strictly positive in order to preserve the convergence of the integration first over k which can be done analytically, one finds,

χ(x) = ∞ 0 x ′ dx ′ ψ(x ′ ) ∞ 0 kdk e -kx J 0 (kx ′ ) = x ∞ 0 x ′ dx ′ (x 2 + x ′2 ) 3/2 ψ(x ′ ) . (37)
An integration by part, similar to that used in subsection II C, returns the derivative of χ for all strictly positive x,

dχ dx (x) = ∞ 0 x ′ dx ′ 2x 2 -x ′2 (x 2 + x ′2 ) 5/2 ψ(x ′ ) = - ∞ 0 dx ′ dψ dx ′ (x ′ ) x ′2 (x 2 + x ′2 ) 3/2 , ( 38
)
where the last expression, obtained by another integration by part, is now regular at x = 0 and thus a faithful representation of the function dχ/dx for all x ≥ 0. One then gets for the required moment,

∞ 0 kdk kϕ(k) = - dχ dx (x = 0) = - ∞ 0 dx ′ x ′ dψ dx ′ (x ′ ) = ∞ 0 dx ′ x ′2 (ψ(0) -ψ(x ′ )) . (39)
Interestingly enough, this is the same expressions, except for a coefficient, as (27) obtained in the 1-d case.

Inserting the result (39) into the inequality (35) yields,

ψ(ix) ≥ ψ(0) I 0 x ∞ 0 dx ′ x ′2 1 - ψ(x ′ ) ψ(0) , ∀x ∈ [0, ∞] . (40)

Hierarchy of further bounds

Let us introduce the hierarchy of positive functions,

ϕ p (k) ≡ k p ϕ(k) ; ϕ 0 (k) ≡ ϕ(k) . (41)
Following the same approach as in section II, we introduce the probability distributions,

kdk P p (k) ≡ kdk ϕ p (k) ∞ 0 kdk ϕ p (k) . ( 42
)
We also introduce the 2-dimensional radial Laplacian operator ∆ 2 acting on ψ(x). It leaves the Bessel functions J 0 (kx), I 0 (kx) invariant up to factors, namely,

∆ 2 [ψ(x)] ≡ - 1 x d dx x d dx [ψ(x)] ⇒ ∆ 2 [J 0 (kx)] = k 2 J 0 (kx) . (43)
This operator ∆ 2 plays in our 2-dimensional problem the role of the ordinary double derivative in 1-d. Then, for even p = 2q, we now define a new function ψ 2q (x) from the multiple ∆ 2 -derivatives of ψ(x),

ψ 2q (x) ≡ [∆ 2 ] q ψ(x) = +∞ 0 kdk J 0 (kx) k 2q ϕ(k) = +∞ 0 kdk J 0 (kx) ϕ 2q (k) , (44)
by acting on the Bessel-Fourier transform formula (32). In particular, we get for an even moment of ϕ, of order 2q, namely the zeroth order moment of ϕ 2q ,

ψ 2q (0) = [∆ 2 ] q ψ(x)| x=0 = +∞ 0 kdk ϕ 2q (k) . (45)
Now, extending the Jensen inequality (13) to the probability distributions (42), we get for the positive functions ψ 2q (ix) the following hierarchy of inequalities,

ψ 2q (ix) = ψ 2q (0) I 0 (xk) P2q ≥ ψ 2q (0) I 0 xk P2q = ψ 2q (0) I 0 x ∞ 0 kdk ϕ 2q+1 (k) ∞ 0 kdk ϕ 2q (k) , (46)
where,

ψ 2q (ix) = +∞ 0 kdk I 0 (xk) ϕ 2q (k) , (47)
is the analytic continuation of ψ(x) onto the imaginary axis. Since ψ 2q is just another function assumed to have a positive Fourier transform, we can at once take advantage of Eq. ( 40) and obtain,

ψ 2q (ix) ≥ ψ 2q (0) I 0 x ∞ 0 dx ′ x ′2 1 - ψ 2q (x ′ ) ψ 2q (0) . (48)

Convolution

For any width b the product, exp[-k 2 /(2b 2 )] ϕ(k), retains the assumed positivity of ϕ and converts the partner ψ into the result of a convolution,

ψ b (x) = b 2 /(2π) 2π 0 dθ ∞ 0 x ′ dx ′ exp[-(x 2 + x ′ 2 -2xx ′ cos θ) b 2 /2] ψ(x ′ ) , (49)
an obvious generalization of Eq. ( 9). The angle θ is that between r and r ′ and the angular integration yields,

ψ b (x) = b 2 exp[-b 2 x 2 /2] ∞ 0 x ′ dx ′ I 0 (b 2 xx ′ ) exp[-b 2 x ′ 2 /2] ψ(x ′ ) . (50)
Additional inequalities are easily obtained because the Taylor series of J 0 has the same alternating sign structure as that of cos. The combinations, J 0 (x) + J 0 (ix) and J 0 (x) -J 0 (ix), therefore, are convex and concave, respectively. Generalizations of inequalities ( 14)-( 19) to the present radial, 2-d situation are obvious. An extension of relations ( 14)-( 19) to this hierarchy of functions ψ 2q , and to the corresponding functions associated with ψ b , is trivial.

C. Numerical investigations for the Bessel transform

We generated 10127 test functions ψ(x) by randomly mixing the following 5 functions,

√ 2 e -x 2 /2 , √ 2 e -x 2 /2 (1-x 2 ), e -x 2 /2 (2 -4x 2 + x 4 )/ √ 2, e -x 2 /2 (6 -18x 2 + 9x 4 -x 6 )/(3 √ 2), e -x 2 /2 (24 -96x 2 + 72x 4 -16x 6 + x 8 )/(12 √
2). This is an orthonormal basis with the measure xdx, 0 ≤ x < +∞, using Laguerre polynomials. The basis states are eigenstates of the Fourier-Bessel transform, with eigenvalues {1, -1, 1, -1, 1}, respectively. Then our random mixtures were screened so that each ψ be square normalized, positive, with its maximum at the origin, while ϕ has some negativity, to be detected.

We implemented the "matrix test" at orders 5, 8, 9 and 10 for ψ alone, for simplicity, neglecting its associated derivative and convoluted functions. Detection rates were ≃ 29%, 39%, 41%, 43%, respectively, compared to ≃ 69%, 82%, 84%, 86%, for the 1-dimensional case again for ψ alone. These 2-d detection rates are less satisfactory than the detection rates obtained with our 1-d test set, but still acceptable, because of our screening for cases with a maximum at the origin only.

The analyticity test for ψ alone returned hardly a ≃ 3% detection rate, which is weak and similar to ≃ 5% for the 1-dimensional case. This is confirmed by combining the matrix test at order 10 with this analyticity test. Except for one case, those 14% functions undetected by the matrix test also escape the analyticity test.

V. CONCLUSION AND OUTLOOK

The constructive characterization of real functions with positive Fourier transforms in one and two-dimensional (radial) dimensions has been analyzed here with two different methods. One, the "analyticity method", makes use of analytic continuation combined with convexity properties (Jensen inequalities). The other, the "matrix method", deals with finite Toeplitz matrices testing the Bochner theorem through tests of positive definiteness.

We test these methods on a set of random mixtures of functions combining a Gaussian with polynomials forming a finite, but of sizable dimension, orthogonal basis. We estimate and compare the efficiencies of both methods.

In one dimension, the matrix method reaches a 86% efficiency for the test on the function ψ alone, and reaches easily more than 93% for the combined test including derivatives. This is to be compared with 43% for the analyticity method and 85% at most, after combining with suitable convolutions and differentiation preserving the positivity of the Fourier transform. All in all the matrix method seems at present quite successful and more directly efficient than the analyticity one.

In the two-dimensional (radial) case, the situation is less satisfactory for the matrix method. It reaches a 43% efficiency under the same conditions, but is still probably not too difficult to improve quantitatively. The analyticity method, by contrast, is at the present stage much ineffective, with only ≃ 3% detection for the same set of non positive Fourier-Bessel transforms. This method requires a qualitative improvement to be competitive; a good subject for future research.

By way of an outlook, there are possible directions for improvement.

Although our sets of test functions are reasonably big (10 4 typically) and diversified, we feel that our conclusions should be strengthened by the consideration of a larger basis and not necessarily weighted by Gaussians, with more allowed configuration mixing. However, we may already define quite a few new research directions on the same line, namely:

-In two dimensions, it is clear that a vectorial version of the matrix model is to be investigated, by comparison with the one-dimensional projection used in section IV.

-The analyticity method, despite its mathematical simplicity based on generalized convexity properties, seems at its present stage suffering from a deficit of efficiency. The question remains of improved convexity tests.

-Considering the matrix method and its growing accuracy with matrix dimension, it is tempting to consider the possibility of using Toeplitz matrices of large rank.

-At last but not least, the full constructive characterization of positive positive-definite functions is yet to be reached, hoping that the steps made in the present paper may lead to this goal.

FIG. 2 :FIG. 3 :

 23 FIG.2: Left: Tiny dots, simultaneous ψ > 0, ϕ > 0. Bigger dots, ψ > 0, but non-positivity of ϕ. Right: Same tiny dots. Big dots, cases eliminated by maximality conditions for ψ(0), ψ2(0). Moderate dots, non-positivity undetected by these conditions.

  For instance one could use two values b and b ′ simultaneously for the convolution parameter, hence test functions ψ b and ψ b ′ , and set a combination, ψ c = ψ -w ψ b -w ′ ψ b ′ , with positive weights w, w ′ such that w + w ′ ≤ 1, to retain the assumed positivity of the respective Fourier images ϕ b , ϕ b ′ , ϕ c . The corresponding bound,

FIG. 4 :

 4 FIG.4: Tiny dots, simultaneous ψ > 0, ϕ > 0. Big dots, ψ > 0, non-positivity of ϕ detected with b = 1/2 and just the hyperbolic cosine bound for ψ b2 , left, or for ψ b4 , right. Moderate dots, ψ > 0, non-positive ϕ undetected by such a bound alone.

Toeplitz matrices are defined, see e.g.[START_REF] Gray | Many textbooks on Toeplitz matrices are available. A convenient introduction is[END_REF] by coefficients c ij ≡ c i-j .

The function and its Fourier transform read, ψ(r) = e -r 2 2 (0.718081 -0.064879r 2 -0.0685793r

+ 0.0269736r 6 + 0.00119983r 8 ), ϕ(s) = e -s 2 2 (0.97805 -1.24138s 2 + 0.587989s 4 -0.0605688s 6 + 0.00119983s 8 ); the average of s is s = 0.836263.

Note also the convergence of ||Tn|| 1/n , n → ∞, to a well-defined Fourier series through the Szegö-Kaĉ theorem, see e.g.[START_REF] Gray | Many textbooks on Toeplitz matrices are available. A convenient introduction is[END_REF].

Acknowledgements

We want to thank Philippe Jaming (Mathematical institute, Bordeaux I U.) for some fruitful insights on positive positive-definite functions, and Bertrand Eynard for interesting mathematical remarks on the subject.


Graham Robert

David Mcgrann

Marten Thies

Heick

Andres Mäe

Sabine Fillinger

Pille Sooväli

Thies Marten Heick
email: thiesm.heick@agro.au.dk

Fungicide Sensitivity Shifting of Zymoseptoria tritici in the Finnish-Baltic Region and a Novel Insertion in the MFS1 Promoter

Keywords: azoles, MDR, SDHI, septoria tritici blotch, transposon, QoI

Septoria tritici blotch (STB) is caused by the ascomycete Zymoseptoria tritici and one of the predominating diseases in wheat (Triticum aestivum) in Europe. The control of STB is highly reliant on frequent fungicide applications. The primary objective of this study was to assess sensitivity levels of Z. tritici to different fungicide groups. The fungicides included in this study were epoxiconazole, prothioconazole-desthio, tebuconazole, and fluxapyroxad. A panel of 63 isolates from Estonia, Latvia, and Lithuania, and 10 isolates from Finland were tested. Fungicide sensitivity testing was carried out as a bioassay analyzing single pycnidium isolates on different fungicide concentrations. The average EC 50 value in Baltic countries and Finland to epoxiconazole was high ranging from 1.04 to 2.19 ppm. For prothioconazole-desthio and tebuconazole, EC 50 varied from 0.01 to 0.24 ppm, and 1.25 to 18.23 ppm, respectively. The average EC 50 value for fluxapyroxad varied from 0.07 to 0.33 ppm. To explain the range of sensitivity, the samples were analyzed for CYP51 and Sdh mutations, as well as cytb G143A, CYP51 overexpression, and multidrug resistance (MDR). Frequencies of ZtCYP51 mutations D134G, V136A/C, A379G, I381V, and S524T in the Finnish-Baltic region were lower than in other European countries, but have increased compared to previous years. The frequency of cytb G143A conferring strobilurin resistance also augmented to 50-70% in the Z. tritici populations from Estonia, Finland, Latvia, and Lithuania. No Sdh mutations were found in this study, and neither strains of MDR phenotypes. However, we found a strain harboring a previously unknown transposon insertion in the promoter of the MFS1 gene, involved in drug efflux and multi-drug resistance. This new insert, however, does not confer an MDR phenotype to the strain.

INTRODUCTION

Zymoseptoria tritici is the causal agent of septoria tritici blotch (STB), the most devastating leaf disease on wheat (Triticum aestivum) worldwide [START_REF] O'driscoll | The wheat-Septoria conflict: a new front opening up?[END_REF][START_REF] Fones | The impact of Septoria tritici blotch disease on wheat: an EU perspective[END_REF]. The disease thrives best under humid-temperate conditions and can have several disease cycles per season. STB is present throughout Europe but is the prevalent disease in regions that offer conducive climatic conditions such as Ireland, the United Kingdom, or Northern Germany [START_REF] Jørgensen | IPM Strategies and their dilemmas including an introduction to www[END_REF]. The disease incidence of STB decreases toward South and North-East Europe [START_REF] Garnault | Spatiotemporal dynamics of fungicide resistance in the wheat pathogen Zymoseptoria tritici in France[END_REF], where STB competes with other leaf blotch diseases, for instance, Parastagonospora nodorum and Pyrenophora tritici-repentis (Heick et al., 2017b). Agricultural practices have a direct impact on disease severity. Whereas early sowing of winter wheat and minimum tillage favor epidemics of STB, the use of varietal resistance and a delay of sowing help mitigate the primary inoculum at the beginning of the season, and thus disease severity in the following year. Despite recent achievements in breeding and a focus on non-chemical practices [START_REF] Gladders | Agronomic and meteorological factors affecting the severity of leaf blotch caused by Mycosphaerella graminicola in commercial wheat crops in England[END_REF][START_REF] Brown | Genetics of resistance to Zymoseptoria tritici and applications to wheat breeding[END_REF], STB control is highly reliant on frequent and timely applications of fungicides. Yield losses can amount to up to 30-50% if the disease is not efficiently controlled [START_REF] Jørgensen | IPM Strategies and their dilemmas including an introduction to www[END_REF]. Currently, three main fungicide groups are available for STB disease control: (1) quinone outside inhibitors (QoI), (2) 14α-demethylase inhibitors (DMI), and (3) succinate dehydrogenase inhibitors (SDHI). Compounds of those three groups have been used for many years now and have efficiently reduced the impact of STB. Nevertheless, field efficacies of many active ingredients belonging to those groups have reduced due to fungicide resistance in recent years [START_REF] Blake | Changes in field dose-response curves for demethylation inhibitors (DMI) and quinone outside inhibitor (QoI) fungicides against Zymoseptoria tritici, related to laboratory sensitivity phenotyping and genotyping assays[END_REF][START_REF] Kildea | A combination of target-site alterations, overexpression and enhanced efflux activity contribute to reduced azole sensitivity present in the Irish Zymoseptoria tritici population[END_REF].

Population genetic studies based on mark-release-recapture experiments conducted in the field indicated that pathogen populations might change significantly over a single growing season in response to host genotypes [START_REF] Zhan | Local adaptation and effect of host genotype on the evolution of pathogens: an experimental test in a plant pathosystem[END_REF]. The development of fungicide resistance has been attributed to several molecular mechanisms, including alterations and overexpression of the target-site and the pathogen's ability to lower the amount of fungicide in the cell through overexpression of efflux pumps [START_REF] Cools | Update on mechanisms of azole resistance in Mycosphaerella graminicola and implications for future control[END_REF]. Resistance to all single-site fungicides is present in most Z. tritici populations, to which degree, however, may greatly vary locally. Mutations in the target gene of a fungicide are the most common mechanisms of fungicide resistance in Z. tritici [START_REF] Mullins | Molecular modelling of the emergence of azole resistance Mycosphaerella graminicola[END_REF]. Fungicide resistance can occur in a disruptive manner or stepwise.

Resistance to QoI fungicides arose promptly after the introduction of this mode of action (MoA) in the early 2000s, associated with point mutations in the mitochondrial cytochrome b gene (cytb). Isolates carrying F129L or G137R express moderate (partial) resistance and are comparably somewhat sporadic in Z. tritici populations. In contrast, point mutation G143A confers full resistance and dominates in current Z. tritici populations. Consequently, QoI fungicides are no longer effective against Z. tritici in most European countries [START_REF] Fraaije | Role of ascospores in further spread of QoI-resistant cytochrome b alleles (G143A) in field populations of Mycosphaerella graminicola[END_REF][START_REF] Sierotzki | Cytochrome b gene sequence and structure of Pyrenophora teres and P. tritici-repentis and implications for QoI resistance[END_REF].

Since their introduction in the 1970s, azoles have become essential components of plant disease control in the fields because of their wide-ranging efficacy against many agriculturally important diseases [START_REF] Russell | A century of fungicide evolution[END_REF]. Prolonged and intensive usage of agricultural azole fungicides in crop protection has been the main driver in the emergence of azole resistance in fungi. Several molecular mechanisms play a role in reduced azole sensitivity. The most common mechanism is alterations in the CYP51 gene leading to amino acid changes of the CYP51 enzyme. To date, over 30 different amino acid alterations (substitutions and deletions) have been identified in the CYP51 protein of modern Z. tritici populations, and over 30 different genotypes have been registered so far [START_REF] Cools | Update on mechanisms of azole resistance in Mycosphaerella graminicola and implications for future control[END_REF][START_REF] Huf | Proposal for a new nomenclature for CYP51 haplotypes in Zymoseptoria tritici and analysis of their distribution in Europe[END_REF]. Mutations leading to exchanges D134G, V136A/C/G, A379G, I381V, S524T, and deletions or mutations at nucleotides coding for amino acids 459-461 are currently claimed to have the highest effect on the sensitivity to azoles [START_REF] Cools | Impact of recently emerged sterol 14a-demethylase (CYP51) variants of Mycosphaerella graminicola on azole fungicide sensitivity[END_REF][START_REF] Wieczorek | Impact of DMI and SDHI fungicides on disease control and CYP51 mutations in populations of Zymoseptoria tritici from Northern Europe[END_REF]. The emergence of S524T in field populations of Z. tritici has caused particular concern as prothioconazole, along with epoxiconazole, is one of the two remaining azole fungicides still highly effective against STB [START_REF] Cools | Impact of recently emerged sterol 14a-demethylase (CYP51) variants of Mycosphaerella graminicola on azole fungicide sensitivity[END_REF]. Mutations can occur in combination, and the presence or absence of a single mutation can render an isolates resistant. Furthermore, [START_REF] Cools | Overexpression of the sterol 14α-demethylase gene (MgCYP51) in Mycosphaerella graminicola isolates confers a novel azole fungicide sensitivity phenotype[END_REF] described an insertion of 120 bp in the CYP51 gene that leads to overexpression of the gene and, consequently, an increased azole tolerance.

After the rapid development of resistance against QoI and DMI fungicides, a newer generation of SDHI for cereal diseases was launched in the last few years that are very effective in controlling STB [START_REF] Sierotzki | A review of current knowledge of resistance aspects for the next-generation succinate dehydrogenase inhibitor fungicides[END_REF]. Several target mutations have already been described both in the laboratory and in the field, which can lead to SDHI resistance. For example, C-T79N can be found alone or in combination with C-I29V or with both C-N33T, C-N34T, and C-H152R [START_REF] Dooley | Detection of Zymoseptoria tritici SDHI insensitive field isolates carrying the SdhC-H152R and SdhD-R47W substitutions[END_REF]. Several point mutations in the Sdh subunits have been associated with high EC 50 values: B-N225I/T, B-H267X, B-T268I, B-I269V, C-N86S, C-N86K, C-T79N, C-T79I, C-W80S, C-G90R, and C-H152R [START_REF] Rehfus | Mutations in Sdh genes in field isolates of Zymoseptoria tritici and impact on the sensitivity to various succinate dehydrogenase inhibitors[END_REF]. Unlike the CYP51 mutations, haplotypes with more than one Sdh mutation have rarely occurred in nature yet.

In addition to the target-site-specific mutations, Leroux and Walker (2011) described a third mechanism, rendering Z. tritici phenotypes resistant to several MoA, a phenotype considered as multi-drug-resistant (MDR). Those strains were later associated with an enhanced active fungicide efflux [START_REF] Omrane | Fungicide efflux and the MgMFS1 transporter contribute to the multidrug resistance phenotype in Zymoseptoria tritici field isolates[END_REF]. Three different types of inserts in the promoter region of the "major facilitator gene" (MFS1) were identified in MDR field strains leading to overexpression of this gene. In the current Z. tritici population, the number of MDR strains are increasing and are frequently found in Western Europe [START_REF] Kildea | A combination of target-site alterations, overexpression and enhanced efflux activity contribute to reduced azole sensitivity present in the Irish Zymoseptoria tritici population[END_REF]. In Northern Europe, strains harboring one of the inserts are present, but in low frequency (Heick et al., 2017b).

In Baltic countries, a wider range of active ingredients in fungicides is available than in other Northern European countries. Over a long period now, the Baltic Z. tritici populations have been exposed to different MoA; however, a systematic resistance testing of Z. tritici has not been carried out. The overall objective of this study was to describe the current fungicide resistance situation of Z. tritici in the Baltic region.

MATERIALS AND METHODS

Isolate Collection

A total of 73 single pycnidium isolates of Z. tritici were produced, 29 of which from Estonia, 4 from Latvia, 30 from Lithuania, and 10 from Finland in 2018. Leaf samples were collected from the upper two leaves layers, mainly from untreated plots. The leaves were placed in a petri dish on moist filter paper without prior surface sterilization. After 24 h incubation at room temperature, cirrhi from single pycnidia were transferred onto potato dextrose agar, amended with 0.01% streptomycin, using a sterile needle. Single spore colonies appeared after 6 days of incubation at 20 • C and 12 h white light/12 darkness. Spores were conserved in 20% glycerol at -80 • C.

Determination of EC 50

Spore suspensions were produced by scraping off 6-day-old Z. tritici spores and transferring them into sterile, demineralized water. The suspensions were vortex-mixed in 10 ml Falcon tubes for 10 min for homogenization. Spore concentrations were adjusted to 2.5 × 10 4 spores ml -1 . Epoxiconazole, tebuconazole, and prothioconazole-desthio (all Sigma-Aldrich, St. Louis, MO, United States) were mixed separately with 2 × potato dextrose broth to obtain the following final microtiter plate fungicide concentrations (mg l -1 ): 10, 3.3, 1.0, 0.3, 0.1, 0.03, 0.01, and 0 for epoxiconazole and 90, 30, 10, 3.3, 1.0, 0.3, 0.1, and 0 for prothioconazole-desthio and tebuconazole. One hundred microliter of spore suspension and 100 µl fungicide solution were added to a nunc TM 96-deep well microtiter plate (Thermo Fisher Scientific, Roskilde, Denmark). Technical duplicates of each isolate were performed on the same plate, and Dutch isolate IPO323 (azole-sensitive) and the Irish isolate OP15.1 (moderately azole-resistant) were included as references. Microtiter plates were wrapped in aluminum foil and incubated in the dark at 20 • C for 6 days. The plates were visually checked for bacterial and fungal contamination before the analysis in an iMark TM Microplate Absorbance Reader (Bio-Rad, Copenhagen, Denmark) at wavelength 620 nm. The fungicide dose reducing growth in the microplate wells by 50% (EC 50 ) was determined by non-linear regression (curve-fit) using GraphPad Prism (GraphPad Software, La Jolla, CA, United States). Resistance factors were calculated based on the reference EC 50 values for IPO323. The EC 50 values for fluxapyroxad (Sigma-Aldrich, St. Louis, MO, United States) were determined correspondingly using a 10-fold dilution series starting at 3 ppm.

Sequencing of Target Site Mutations

All isolates were tested for target mutations for azole, SDHI, and strobilurin fungicides. The presence of CYP51 mutations L50S, D134G, V136A/C, G379A, I381V, and S524T and cytb mutation G143A conferring strobilurin resistance were determined using Kompetitive Allele Specific PCR (KASP) (LGC Genomics, Teddington, United Kingdom) genotyping previously described by [START_REF] Kildea | Changes in the frequency of Irish Mycosphaerella graminicola CYP51 variants 2006-2011[END_REF]. All reactions were carried out in an Applied Biosystems Viia TM 7 Real-time PCR system machine (Thermo Fisher Scientific, Denmark) according to the manufacturer's protocol.

Sequences of the Sdh sub-units B, C, D were obtained using the protocol by [START_REF] Rehfus | Mutations in Sdh genes in field isolates of Zymoseptoria tritici and impact on the sensitivity to various succinate dehydrogenase inhibitors[END_REF]. PCR reactions were performed utilizing the GoTaq Flexi DNA Polymerase kit (Promega, Madison, WI, United States) in a 25 µl volume containing 10.9 µl Gibco water, 5.0 µl 5× GoTaqFlexi PCR buffer, 1.5 mM MgCl 2 , 125 µM of each dNTP, and forward primer and reverse primer (both 10 µM), 1 unit GoTaqFlexi DNA polymerase (Promega, Madison, WI, United States), and 1.0 µl DNA (approximately 5 ng µl -1 ). The amplification was performed using the following conditions: 95 • C for 5 min, followed by 35 cycles of 95 • C for 30 s, 62 • C for 30 s, 72 • C for 30 s, with a final extension of 7 min at 72 • C. The PCR products were purified and sequenced by Macrogen Europe BV (Amsterdam, Netherlands). Sequence alignment was performed using CLC WorkBench 7 (QIAGEN, Aarhus, Denmark).

Testing for Potential Overexpression of the CYP51 and MFS1 Genes

All isolates were investigated for the presence of inserts in the CYP51 promoter region, conferring CYP51 overexpression [START_REF] Cools | Overexpression of the sterol 14α-demethylase gene (MgCYP51) in Mycosphaerella graminicola isolates confers a novel azole fungicide sensitivity phenotype[END_REF]. PCR reactions were performed using the GoTaq Flexi DNA Polymerase kit (Promega, Madison, WI, United States) in a 25 µl volume containing 10.9 µl Gibco water, 5.0 µl 5× GoTaqFlexi PCR buffer, 1.5 mM MgCl 2 , 125 µM of each dNTP, and forward primer Mg51-proF and reverse primer Mg51-seqR (both 10 µM), 1 unit GoTaqFlexi DNA polymerase, and 1.0 µl DNA (approximately 5 ng µl -1 ). The PCR conditions were 95 • C for 5 min, followed by 35 cycles of 95 • C for 30 s, 62 • C for 30 s, 72 • C for 30 s, with a final extension of 7 min at 72 • C.

To screen for inserts in the promoter region of the MFS1 gene, PCR reactions were performed using primer pair MFS_2F/MFS_4R designed by [START_REF] Omrane | Fungicide efflux and the MgMFS1 transporter contribute to the multidrug resistance phenotype in Zymoseptoria tritici field isolates[END_REF]. PCR reactions were set up in a 25 µl volume containing 19.75 µl Gibco water, 2.0 µl buffer, 1.25 µl dNTP mix forward primer and reverse primer (both 10 µM), and 1.0 µl DNA. The reactions were run according to the following protocol: 35 cycles of 95 • C for 30 s, 62 • C for 30 s, 72 • C for 30 s, with a final extension of 7 min at 72 • C.

All PCR reactions in this study were performed in an Applied Biosystems 2720 Thermal Cycler. The samples were loaded on a 1.5% agarose gel containing SYBR R stain (Thermo Fisher Scientific, Denmark) and run for 45 min at 100 V.

MFS1 Insert Sequence Analysis

The PCR fragment of strain 18-Zt-EE-06-03 obtained with the primer couple MFS_2F/MFS_4R was Sanger sequenced (Eurofins Genomics) with the same primers and the additional sequencing primer Estland-sample_LF (5 TGGTGTTTCCATGCGTTTAG 3 ). Output sequences were trimmed and assembled using the CodonCode Aligner software v9.0.1 (CodonCode Corp., Centerville, MA, United States). The insert sequence and surrounding MFS1 promoter sequence have been deposited at Genbank under accession number MN813065. The insert sequence was used for database searches using blastn, blastx [START_REF] Altschul | Basic local alignment search tool[END_REF] at NCBI 1 against non-redundant databases. The peptide translation of the longest ORF was analyzed using PSI-blast and DELTA-blast at NCBI. Inverted repeats were identified using the einverted algorithm 2 . Searches for annotated transposable elements (TE) were performed as blastn-searches against the TE database Repbase [START_REF] Kapitonov | A universal classification of eukaryotic transposable elements implemented in Repbase[END_REF].

RESULTS

Azole Sensitivity Testing and CYP51 Sequence Analysis of Baltic Z. tritici Strains Seventy-three Z. tritici isolates were sampled in Estonia (29), Latvia (4), Lithuania (30), and Finland (10) during 2018. The individual pycnidium isolates were investigated for sensitivity to epoxiconazole, prothioconazole-desthio, and tebuconazole in order to calculate EC 50 value for each isolate. In 2018, the average EC 50 value for epoxiconazole was 1.73 ppm, with single isolates ranging from 0.35 to 8.29 ppm. The average EC 50 was highest in Estonia (2.19 ppm) but lower in Latvia (1.97 ppm), Lithuania (1.67 ppm), and Finland (1.04 ppm). The average resistance factor (RF; mean EC 50 of the resistant population)/(mean EC 50 of the reference isolate IPO323) for epoxiconazole was 69 (reference isolate IPO323: 0.02-0.03 ppm) ranging from 42 in Finland to 87 in Estonia, 67 in Lithuania, and 79 in Latvia (Table 1). The average EC 50 value for prothioconazole-desthio was 0.14 ppm, with single isolates ranging from 0.01 to 2.0 ppm. The average EC 50 was highest in Lithuania (0.24 ppm) but lowest Estonia (0.14 ppm), Latvia (0.13 ppm), and Finland (0.06 ppm). The average RF for prothioconazole-desthio was 14 (reference isolate IPO323: 0.01 ppm) ranging from 6 in Finland to 24 in Lithuania, 13 in Latvia, and 11 in Estonia. The average EC50 value for tebuconazole was 8.45 ppm, with single isolates ranging from 0.13 to 59 ppm. The average EC 50 was highest in Estonia 18.23 ppm (0.13-59 ppm) but lower Lithuania 9.18 ppm (0.8-48 ppm), Finland 5.13 ppm (0.3-18.2 ppm) and Latvia 1.25 ppm (0.34-2.4 ppm). The average RF for tebuconazole was 211 (reference isolate IPO323: 0.041 ppm) ranging from 31 in Latvia to 456 in Estonia, 128 in Finland, and 230 in Lithuania (Table 1).

Several point mutations in the CYP51 gene have been associated with elevated EC 50 values for azoles. Sequencing analysis of all selected isolates showed the presence of CYP51 point mutations D134G, V136A/C, I381V, and S524T in Z. tritici, associated with increased fungicide resistance to the currently widely used azoles (Figure 1). Mutation I381V continued to dominate throughout the region and is present in frequencies of 96-100%, except for Latvia, where this mutation was present at a lower rate, 25%, respectively. Grouping of Z. tritici isolates from different Baltic countries revealed 14 different genotypes (Table 2). The substitution S524T, found in Estonian and Lithuanian samples, was always combined with V136A and I381V and in some isolates, these three substitutions were combined with L50S and D134G (Table 2). The frequencies for mutations L50S, D134G, V136A/C, and A379G, all of which have recently emerged in the European Z. tritici population, varied considerably. The only single substitutions of CYP51 found in the in Baltic-Finnish isolates were V136A and I381V.

Identification of G143A Mutations Z. tritici Strains Isolated in Finnish-Baltic Region

So far, G143A has been found in all isolates with high resistance levels to strobilurins [START_REF] Fraaije | Role of ascospores in further spread of QoI-resistant cytochrome b alleles (G143A) in field populations of Mycosphaerella graminicola[END_REF][START_REF] Sierotzki | Cytochrome b gene sequence and structure of Pyrenophora teres and P. tritici-repentis and implications for QoI resistance[END_REF]. All the isolates collected during 2018 were subsequently screened using the qPCR assay. The allele A143 was found in 41 of the 73 isolates (56%; Table 3). Analysis of the results showed that the frequencies of this mutation were at a high level throughout the Finnish-Baltic region: Estonia, Finland, Latvia (50%), and Lithuania (73%) (Table 3).

Sensitivity of Baltic-Finnish Isolates to Fluxapyroxad

The sensitivity profiles of all 73 single-spore isolates from the Finnish-Baltic region to SDHI fluxapyroxad were measured. The average EC 50 value for fluxapyroxad was 0.2 ppm, with single isolates ranging from 0.07 to 0.33 ppm. The average EC 50 was highest in Lithuania 0.33 ppm (0.01-1.00 ppm) but lower in Latvia 0.23 (0.05-0.49 ppm), Estonia 0.18 ppm (0.04-1.12 ppm), and Finland 0.07 ppm (0.03-0.27 ppm; Table 4). The average RF for fluxapyroxad was 7 (reference isolate IPO323: 0.03 ppm) ranging from 11 in Lithuania to 8 in Latvia, 6 in Estonia, and 2 in Finland (Table 4). Several point mutations in the Sdh subunits B, C, and D have been associated with elevated EC 50 values. In this study, we did not detect any alterations in the Sdh subunits.

Insertion in the CYP51 Promoter Region

From the Finnish-Baltic region, in 18% of all isolates tested a PCR fragment of 334 bp was detected, indicating that these isolates had no insert (wild-type). From four Estonian isolates (14%), one Latvian isolate (25%), and five Lithuanian isolates (17%), a PCR fragment of approximately 450 bp was amplified, indicating that these isolates had 120 bp insert in the promoter region as described by [START_REF] Cools | Overexpression of the sterol 14α-demethylase gene (MgCYP51) in Mycosphaerella graminicola isolates confers a novel azole fungicide sensitivity phenotype[END_REF]. In Estonian, Finnish, Latvian, and Lithuanian isolates, the 866 bp insert, previously described by [START_REF] Chassot | Sensitivity of CYP51 genotypes to DMI fungicides in Mycosphaerella graminicola[END_REF], [START_REF] Omrane | Fungicide efflux and the MgMFS1 transporter contribute to the multidrug resistance phenotype in Zymoseptoria tritici field isolates[END_REF], and [START_REF] Kildea | A combination of target-site alterations, overexpression and enhanced efflux activity contribute to reduced azole sensitivity present in the Irish Zymoseptoria tritici population[END_REF], was present in 65-75% (Figure 2).

Detection of Inserts in the MFS1 Promoter as Potential Drivers of Multi-Drug Resistance

We then searched for potential MDR isolates as described by [START_REF] Omrane | Fungicide efflux and the MgMFS1 transporter contribute to the multidrug resistance phenotype in Zymoseptoria tritici field isolates[END_REF]. We, therefore, amplified in all Z. tritici isolates a promoter fragment of the MFS1 gene encoding a membrane transporter of the major facilitator superfamily [START_REF] Roohparvar | MgMfs1, a major facilitator superfamily transporter from the fungal wheat pathogen Mycosphaerella graminicola, is a strong protectant against natural toxic compounds and fungicides[END_REF][START_REF] Roohparvar | The drug transporter MgMfs1 can modulate sensitivity of field strains of the fungal wheat pathogen Mycosphaerella graminicola to the strobilurin fungicide trifloxystrobin[END_REF] involved in MDR in Z. tritici field strains [START_REF] Omrane | Fungicide efflux and the MgMFS1 transporter contribute to the multidrug resistance phenotype in Zymoseptoria tritici field isolates[END_REF][START_REF] Omrane | Plasticity of the MFS1 promoter leads to multidrug resistance in the wheat pathogen Zymoseptoria tritici[END_REF], using the primer couple published by [START_REF] Omrane | Fungicide efflux and the MgMFS1 transporter contribute to the multidrug resistance phenotype in Zymoseptoria tritici field isolates[END_REF]. All but one isolate gave rise to the "sensitive" allele of 500 bp. Only isolate 18-ZT-EE-06-03 led to an amplicon of >2 kb. Sensitivity assays to the squalene epoxidase inhibitors tolnaftate and terbinafine, used as monitoring molecules to detect MDR [START_REF] Leroux | Multiple mechanisms account for resistance to sterol 14α-demethylation inhibitors in field isolates of Mycosphaerella graminicola[END_REF], revealed that the 18-Zt-EE-06-03 isolate was not resistant to these molecules and therefore potentially not MDR (data not shown).

G V A V T EE S D A A V T LT S G A A V T EE, LT L D A A V T EE, FI S G A A V S LT L G A A V S EE, LT S D V G V S LT. FI L G A A V S EE, FI, LT, LV L D V G V S FI L G V A V S EE, FI S D V A V S LV L G A A V S EE, FI, LT L D V A V S LV L D A A I S
Sequence analysis of the promoter amplicon revealed an 1884 bp previously unknown insert at position-194 relative to the Start codon (Figure 3). Blastn searches at Genbank revealed that the sequence was highly similar to sequences found in other Z. tritici strains (three copies in strains 3D7 and 1A5, one copy in 3D1, and IPO323, respectively), although at other locations. Blastx searches at Genbank revealed a conserved protein domain, known as DDE superfamily endonuclease involved in DNA transposition. We identified an open-readingframe (ORF) of 1032 bp (343 amino acids) transcribed from the complementary strand with substantial sequence similarity (>40% identity according to the algorithm used) to transposases as revealed by DELTA-blast [START_REF] Boratyn | Domain enhanced lookup time accelerated BLAST[END_REF] and PSIblast [START_REF] Altschul | Gapped BLAST and PSI-BLAST: a new generation of protein database search programs[END_REF]

DISCUSSION

The recent decline in field efficacy against STB of azole and SDHI fungicides in Northwestern Europe (Heick et al., 2017b;[START_REF] Kildea | A combination of target-site alterations, overexpression and enhanced efflux activity contribute to reduced azole sensitivity present in the Irish Zymoseptoria tritici population[END_REF] led us to investigate the present resistance situation of Z. tritici in the Finnish-Baltic region. Up to now, azoles and SDHI have provided sufficient control in Estonia, Latvia, and Lithuania. In our investigation, we tested single pycnidium isolates of Z. tritici in vitro for their sensitivity toward commonly used azoles for STB control. The results confirmed different levels of sensitivity for all isolates tested for the different azoles. The EC 50 values for epoxiconazole were highest in Estonia and prothioconazole-desthio in Lithuania (Table 1). The average EC 50 values for epoxiconazole were >1 ppm throughout the region, indicating a general adaption of the Baltic-Finnish Z. tritici population to this active ingredient. Compared to the resistance status in 2014, described by Heick et al. (2017b), an apparent sensitivity shift has occurred over the last four years (Table 5). The resistance factors for prothioconazoledesthio were lower than for epoxiconazole and indicated a moderate adaption.

The resistance level of tebuconazole was significantly higher for all isolates tested (Table 1). Resistance to tebuconazole has been widely spread in Europe and attributed to CYP51 mutation I381V [START_REF] Fraaije | A novel substitution I381V in the sterol 14alpha-demethylase (CYP51) of Mycosphaerella graminicola is differentially selected by azole fungicides[END_REF]. The field performance of tebuconazole has thus been considerably low in many European regions. Jørgensen et al. (2018a) described, however, a regional variation for some countries where tebuconazole still performed well. Also, in the Baltic countries, tebuconazole is still widely used and provides reasonable STB control in the field. The results from this study, though, suggested an adaption to a more tebuconazole-resistant Z. tritici population. Compared to other countries, this adaption has reached the Finnish-Baltic region with a delay.

The first cases of fungicide-adapted strains are usually found in "high-risk" areas with conducive climatic conditions for a fungus. High disease incidence and high inputs of fungicide to control the pathogen lead to a rapid selection for resistant strains (Jørgensen et al., 2018b). Examples for azole resistance development of Z. tritici at different paces have been observed in Northwestern Europe and France (Heick et al., 2017b;[START_REF] Garnault | Spatiotemporal dynamics of fungicide resistance in the wheat pathogen Zymoseptoria tritici in France[END_REF]. In the United States and Australia, where fungicide inputs are traditionally lower than in Europe, the adaption of Z. tritici to azoles is first now occurring [START_REF] Sykes | Sensitivity variation and cross-resistance of Zymoseptoria tritici to azole fungicides in North America[END_REF][START_REF] Mcdonald | Rapid parallel evolution of azole fungicide resistance in Australian populations of the wheat pathogen Zymoseptoria tritici[END_REF]. In Europe, Jørgensen et al. (2018a) showed that a gradient of fungicide resistance from west to east exists. Resistance issues of epoxiconazole and prothioconazole were reported in the United Kingdom from 2008. In Denmark, resistance to epoxiconazole and prothioconazole, both measured as the decline in field efficacy and in vitro, had not occurred before 2015. From 2015, field efficacy for both compounds declined noticeably, coinciding with a drastic increase in the frequency of azole-resistant isolates (Heick et al., 2017b). The recent appearance of increasingly more azole-resistant isolates in the Baltic area suggests a similar trend, as seen in Denmark and Sweden. Intensified fungicide monitoring activities for Z. tritici in the region will be carried in the coming year to follow up on this development. The development appears to be consistent in all three Baltic countries and Finland. In a study, [START_REF] Vagndorf | Population structure and frequency differences of CYP51 mutations in Zymoseptoria tritici populations in the Nordic and Baltic regions[END_REF] compared the Z. tritici isolates from the different Scandinavian and Baltic countries in 2014 using AFLP markers, and found two distinct populations: a Danish-Swedish and a Baltic-Finnish. The existence of a single Baltic-Finnish Z. tritici population might explain a delayed and simultaneous change in fungicide sensitivity.

Several studies have previously demonstrated that applications of fungicide select for mutations both in vitro and in the field [START_REF] Leroux | Mutations in the cyp51 gene correlated with changes in sensitivity to sterol 14alphademethylation inhibitors in field isolates of Mycosphaerella graminicola[END_REF][START_REF] Wieczorek | Impact of DMI and SDHI fungicides on disease control and CYP51 mutations in populations of Zymoseptoria tritici from Northern Europe[END_REF]Heick et al., 2017a). Several studies identified single nucleotide polymorphisms in the CYP51 gene being the primary force behind azole resistance [START_REF] Cools | Impact of recently emerged sterol 14a-demethylase (CYP51) variants of Mycosphaerella graminicola on azole fungicide sensitivity[END_REF][START_REF] Leroux | Multiple mechanisms account for resistance to sterol 14α-demethylation inhibitors in field isolates of Mycosphaerella graminicola[END_REF][START_REF] Cools | Update on mechanisms of azole resistance in Mycosphaerella graminicola and implications for future control[END_REF]. A specific combination of the CYP51 mutations determines the final sensitivity of an individual Z. tritici strain. However, as not all alterations are equally important, single frequencies of specific CYP51 mutations give a proper indication for the selection status of a population.

In this study, we investigated single pycnidium Z. tritici isolates sampled in the Finnish-Baltic region in 2018 for the occurrence of most important CYP51 mutations D134G, V136A/C, A379G, I381V, and S524T [START_REF] Leroux | Mutations in the cyp51 gene correlated with changes in sensitivity to sterol 14alphademethylation inhibitors in field isolates of Mycosphaerella graminicola[END_REF][START_REF] Cools | Update on mechanisms of azole resistance in Mycosphaerella graminicola and implications for future control[END_REF] to determine the resistance stage of this population. The leaf samples derived primarily from untreated plots to determine fungicide resistance in the absence of fungicides. I381V remains the most predominant mutation in all Baltic countries (Heick et al., 2017b;[START_REF] Vagndorf | Population structure and frequency differences of CYP51 mutations in Zymoseptoria tritici populations in the Nordic and Baltic regions[END_REF]; only in Latvia, this mutation was present at low levels in 2018 (Figure 1 and Table 6). Point mutation A379G, known to occur exclusively in combination with I381V, was present in all countries and showed an minor increase compared to 2014 (Heick et al., 2017b). D134G and V136A, which were absent or present at a low level in the Finnish-Baltic region in 2014, have increased over five years up to a level of 50% in Lithuania (Figure 1 and Table 6). V136C and S524T, which, especially in combination, have a significant impact on azole sensitivity [START_REF] Cools | Impact of recently emerged sterol 14a-demethylase (CYP51) variants of Mycosphaerella graminicola on azole fungicide sensitivity[END_REF], were present in Estonia and Lithuania at low levels of less than 10% (Figure 1). All the isolates tested from the Finnish-Baltic region had at least one mutation in the CYP51 gene. The Z. tritici wild-type, which was still present in 2014, seems thus to be replaced (Heick et al., 2017b). The evolution of CYP51 mutations in the Baltic-Finnish Z. tritici population starts more to resemble that in Denmark and Sweden (Heick et al., 2017b;[START_REF] Vagndorf | Population structure and frequency differences of CYP51 mutations in Zymoseptoria tritici populations in the Nordic and Baltic regions[END_REF]. This development is an indication that the evolution of the CYP51 gene has reached the Northeastern parts of Europe. This assumption is supported by the fact that only 2 of 14 haplotypes found in the Finnish-Baltic region showed a single mutation in the CYP51 gene (Table 2). In addition to target site mutation, the overexpression of the target gene can confer tolerance toward fungicides. In the Baltic Z. tritici population, we detected the 120 bp insertion in the promoter region of the CYP51 gene for the first time (Figure 2). It was first described by [START_REF] Cools | Overexpression of the sterol 14α-demethylase gene (MgCYP51) in Mycosphaerella graminicola isolates confers a novel azole fungicide sensitivity phenotype[END_REF] and associated with an overexpression of the CYP51 gene, increasing a strain's tolerance to azole fungicides. This insertion has been detected in several CYP51 haplotypes [START_REF] Huf | Proposal for a new nomenclature for CYP51 haplotypes in Zymoseptoria tritici and analysis of their distribution in Europe[END_REF], and has increased in frequency in the European Z. tritici population in recent years. Presumably, due to a fitness advantage in the presence of fungicides. Furthermore, the 866 bp insertion in the CYP51 promoter region was found at high frequencies (>70%) in all countries [START_REF] Chassot | Sensitivity of CYP51 genotypes to DMI fungicides in Mycosphaerella graminicola[END_REF]. This high number is consistent with studies of other European Z. tritici populations (Heick et al., 2017b;[START_REF] Kildea | A combination of target-site alterations, overexpression and enhanced efflux activity contribute to reduced azole sensitivity present in the Irish Zymoseptoria tritici population[END_REF]. [START_REF] Kildea | A combination of target-site alterations, overexpression and enhanced efflux activity contribute to reduced azole sensitivity present in the Irish Zymoseptoria tritici population[END_REF] found an overexpression of the CYP51 gene in the presence of epoxiconazole in a specific CYP51 background. However, the impact of this insert on the CYP51 gene needs to be further investigated to be entirely understood. The primary mechanism of decreased sensitivity against strobilurin class fungicides is the alteration of cytb at location G143 (mutation G143A). Resistance toward QoI fungicide occurred at least four times simultaneously in Europe and spread rapidly throughout Europe [START_REF] Fraaije | Role of ascospores in further spread of QoI-resistant cytochrome b alleles (G143A) in field populations of Mycosphaerella graminicola[END_REF][START_REF] Torriani | QoI resistance emerged independently at least four times in European populations of Mycosphaerella graminicola[END_REF]. Our results showed that G143A has increased over five years, reaching over 70% in Lithuania and 50% in Estonia (not detected in Estonia in 2014) and Latvia. Only in Finland, the frequency of this mutation decreased slightly during this period (Table 3). G143A mutants are known to have a high level of cross-resistance between different strobilurins. Even though not recommended anymore against STB, QoI fungicides remain effective against other diseases (rust diseases) and are therefore still applied in the field. The continuous use of any strobilurin creates favorable conditions for the further spreading of Z. tritici strains carrying G143A.

Field isolates with reduced sensitivity to SDHI fungicide commonly carry a single amino acid substitution in one of the four Sdh subunits [START_REF] Scalliet | Mutagenesis and functional studies with succinate dehydrogenase inhibitors in the wheat pathogen Mycosphaerella graminicola[END_REF]. In this study, the average EC 50 values varied in the region from 0.07 to 0.33 ppm. Only a few strains surpassed the EC 50 of 1.0 ppm. In Ireland, reduced field efficacy of SDHI was especially correlated with a high frequency of C-T79N (Kildea, personal communication). In Scandinavia, only rare cases of C-T79N and C-N86S are reported each year. No Sdh mutations were detected in the isolate collection tested in this study. Sdh mutations are, thus, not drivers for increased tolerance in these isolates. These findings are similar to the results presented by [START_REF] Yamashita | Non-target site SDHI resistance is present as standing genetic variation in field populations of Zymoseptoria tritici[END_REF]. The appearance of resistant isolates in the populations can be the result of in planta degradation of SDHI caused either by the plant and/or the fungus itself. This study shows that adapted Z. tritici strains exist in the fields and should be considered a potential risk for the future.

Finally, we did not detect any MDR strain among our collected isolates. Interestingly, using the PCR diagnostic test to detect MDR strains as proposed by [START_REF] Omrane | Fungicide efflux and the MgMFS1 transporter contribute to the multidrug resistance phenotype in Zymoseptoria tritici field isolates[END_REF], we found one strain that harbors a previously unknown insertion in the MFS1 promoter. Since this strain does not display an MDR phenotype, we suspect that it does not overexpress the MFS1 gene. It is intriguing to notice that this is the fourth type of repeated insert detected in the MFS1 promoter. While the three other insertions are or may be putative relics of TE driving MFS1 overexpression, the insertion of a whole transposon observed in strain 18-Zt-EE-06-03 probably abolishes transcription of MFS1, especially since the transcription of the transposase is from the opposite strand. The frequent presence of repeated elements in the MFS1, but also in the CYP51 promoter [START_REF] Chassot | Sensitivity of CYP51 genotypes to DMI fungicides in Mycosphaerella graminicola[END_REF][START_REF] Cools | Overexpression of the sterol 14α-demethylase gene (MgCYP51) in Mycosphaerella graminicola isolates confers a novel azole fungicide sensitivity phenotype[END_REF][START_REF] Kildea | A combination of target-site alterations, overexpression and enhanced efflux activity contribute to reduced azole sensitivity present in the Irish Zymoseptoria tritici population[END_REF] raises the question of promoter plasticity. Additional population genomic studies are required to clarify if these types of the insert are distributed uniformly over the entire Z. tritici genome, or if some genomic regions, such as specific promoters, are preferred. Moreover, the role of fungicide selection pressure in promoter insertions remains to be evaluated.

CONCLUSION

In conclusion, the results presented in this study show an increase in EC 50 values of the Baltic-Finnish Z. tritici population of commonly used azoles. We showed that the frequencies of key CYP51 mutations and G143A of the Baltic-Finnish Z. tritici population have increased from 2014 to 2018. The results indicate that the same development has taken place in the region, as had been witnessed in other European regions. Resistant management strategies should be advocated to prolong the field efficacy of all MoA used against STB. Though no SDHI resistance-conferring mutations were detected, the possible existence of non-target site SDHI resistance should be considered for the design of resistance management strategies. Finally, we found a strain harboring a fourth transposon insertion in the promoter of the MFS1 gene, which, however, does not display an MDR phenotype.

FIGURE 1 |

 1 FIGURE 1 | CYP51 mutation frequencies (%) in Z. tritici populations from Estonia, Finland, Latvia, and Lithuania in 2018.

  searches at NCBI. We then searched the Repbase database of eukaryotic TEs for similarity to any known TE, without success. Manual and in silico analysis of the promoter insert sequence, following the guidelines proposed by Wicker et al. (2007) for TE analysis, allowed us to detect the target site duplication (TSD) 5 TA/CTCGTG 3 , inversed at the opposite end, and a rather long terminal inverted repeat of 62 bp (with four mismatches). In recent work, Badet et al. (2020) have annotated TEs in the Z. tritici pangenome. Sequences highly similar to the MFS1 promoter insert of strain 18-Zt-EE-06-03 were annotated as DTT_Birute transposon family of the Tc1-Mariner superfamily, class II DNA transposons (Oggenfuss and Croll, personal communication) according to the nomenclature proposed by Wicker et al. (2007). The sequence was uploaded to GenBank under accession number MN813065.

FIGURE 3 |

 3 FIGURE 3 | Promotor structure of the MFS1 locus in strain 18-ZT-EE-06-03. The promotor insert with its location is indicated by the blue triangle. The inverted target site duplications at the insertion site are indicated in red in 5 -3 direction. A terminal inverted repeat (TIR) sequence of 62 bp as detected by the "inverted" algorithm is indicated by the violet boxes. An ORF of 1032 bp with strong similarity to transposases is indicated by the green arrow.

TABLE 1 |

 1 Summary of average EC 50 (ppm) values and resistance factors (RF) a for epoxiconazole, prothioconazole-desthio, and tebuconazole assessed for Z. tritici in Estonia, Finland, Latvia, and Lithuania.

	Country	Epoxiconazole		Prothioconazole-desthio	Tebuconazole
		Average	RF	Average	RF	Average	RF
	Estonia (n = 29)	2.19	87	0.14	11	18.23	456
	Finland (n = 10)	1.04	42	0.06	6	5.13	128
	Lithuania (n = 30)	1.67	67	0.24	24	9.18	230
	Latvia (n = 4)	1.97	79	0.13	13	1.25	31
	Average	1.73	69	0.14	14	8.45	211
	Reference IPO323	0.02-0.03		0.01		0.04
	a Resistance factor: (mean EC 50 of resistant population)/(mean EC 50 of reference isolate IPO323).

TABLE 2 |

 2 Amino acid polymorphisms in the CYP51 gene from Z. tritici field isolates collected in the Finnish-Baltic region in 2018.

	Origin		Position of amino acid polymorphism
		L50	D134	V136	A379	I381	T524
	EE, LT	S

TABLE 3 |

 3 Frequency of G143A mutations in Z. tritici collected in 2018 across the Baltic-Finnish region.

	Country		Frequency of G143A mutation
		G143A	G143	% G143A
	Estonia	14	14	50
	Finland	5	5	50
	Latvia	2	2	50
	Lithuania	22	8	73

TABLE 4 |

 4 Summary of measured EC 50 (ppm) values and resistance factors (RF) for fluxapyroxad assessed for Z. tritici in Estonia, Finland, Latvia, and Lithuania.

	Country	Fluxapyroxad
		Average	RF
	Estonia (n = 29)	0.18	6
	Finland (n = 10)	0.07	2
	Lithuania (n = 30)	0.23	8
	Latvia (n = 4)	0.33	11
	Average	0.20	7
	Reference IPO323	0.03

FIGURE 2 | Promoter inserts frequency (%) of the CYP51 gene of Estonian, Finnish, Latvian, and Lithuanian Z. tritici strains. Frontiers in Plant Science | www.frontiersin.org 5 April 2020 | Volume 11 | Article 385

TABLE 5 |

 5 Comparison of average EC 50 values (ppm) and resistance factors (RF) of epoxiconazole assessed for Z. tritici in Estonia, Finland, Latvia, and Lithuania in 2014 and 2018.

	Country		Epoxiconazole
		Average		RF
		2014	2018	2014	2018
	Estonia	0.07	2.19	7	87
	Finland	0.23	1.04	4	42
	Lithuania	0.14	1.67	15	67
	Latvia	0.16	1.97	17	79
	Average	0.15	1.73	16	69

TABLE 6 |

 6 CYP51 mutations frequencies (%) in Zymoseptoria tritici samples from the Finnish-Baltic region in 2014 and 2018.

	Country	Year			Mutations (%)
			D134G	V136A	V136C	A379G	I381V	S524T
	Estonia	2014	0	0	0	0	100	0
		2018	19	8	3	16	98	8
	Finland	2014	17	8	8	10	77	5
		2018	30	20	0	28	100	0
	Lithuania	2014	10	12	5	12	80	5
		2018	20	20	0	23	98	10
	Latvia	2014	0	5	0	10	95	0
		2018	50	50	6	25	25	0

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April 2020 | Volume 11 | Article 385

https://blast.ncbi.nlm.nih.gov

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ACKNOWLEDGMENTS

The authors would like to thank Roma Semaskiene (LAMMC), Marja Jalli (LUKE), and Olga Treikale (LAAPC) for providing leaf samples. The authors are grateful to Cherif Diatta (INRAE-BIOGER) for sequence and phenotype analysis of strain 18-Zt-EE-06-03.

FUNDING

This work was supported by the ECRI grant PA1-RUP-031 and ASTRA grant "Sordiaretus."

DATA AVAILABILITY STATEMENT

The datasets generated for this study can be found in the https: //www.ncbi.nlm.nih.gov/nuccore/mn813065.

AUTHOR CONTRIBUTIONS

AM and TH performed the experiments and analyzed the data. SF analyzed the data on the MFS1 insert. AM, SF, TH, and PS drafted the manuscript. All authors read and approved the final manuscript.

Conflict of Interest:

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