Maxim Kazarian
email: kazarian@mccme.ru

Sergei Lando
email: lando@hse.ru

Dimitri Zvonkine
email: dimitri.zvonkine@gmail.fr.

Double Hurwitz numbers and multisingularity loci in genus 0

In the Hurwitz space of rational functions on CP 1 with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincaré dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.

Introduction

Double Hurwitz numbers count transitive factorizations of the identity permutation in the symmetric group S K into a product of two permutations with given cycle types and a given number of transpositions. In topological terms, they enumerate topologically distinct meromorphic functions f on Riemann surfaces C of a given genus g with prescribed orders of poles and zeroes and prescribed nonzero simple critical values. In the case when the genus of C equals g = 0 and one of the two distinguished permutations is the identity, a closed formula for these numbers was proposed by Hurwitz more than a century ago. In general, in genus 0 explicit formulas for the generating function of double Hurwitz numbers are known. In spite of the existence of such formulas and a variety of modern proofs, many natural questions concerning these numbers remain open.

Hurwitz's argument was algebraic, based on the study of combinatorics of the permutation group. On the other hand, Hurwitz numbers are related to the geometry of spaces of meromorphic functions and their compactifications, called Hurwitz spaces. We propose a new recursion for genus 0 double Hurwitz numbers that has a topological origin: it is derived from cohomological identities on loci of functions with a given multisingularity, that is, a given ramification profile over one branch point. We expect that variations of this approach could be adapted to other families of Hurwitz numbers for which effective formulas are not known at the moment, including those for higher genus curves.

This paper is a continuation of our project on the study of cohomology classes of strata in Hurwitz spaces initiated in [START_REF] Kazarian | On intersection theory on Hurwitz spaces[END_REF][START_REF] Kazarian | Thom polynomials for mappings of curves with isolated singularities[END_REF][START_REF] Lando | On multiplicities of the Lyashko-Looijenga mapping on strata of the discriminant[END_REF][START_REF] Lando | Counting Ramified Coverings and Intersection Theory on Spaces of Rational Functions I (Cohomology of Hurwitz Spaces)[END_REF][START_REF] Kazarian | Universal cohomological expressions for singularities in families of genus 0 stable maps[END_REF].

Consider a generic map f : X → Y between two smooth compact manifolds. Given a singularity type α, denote by X α the locus of points in X where f has a singularity of this type. Thom's principle states that Poincaré dual cohomology class of X α is a universal polynomial in Chern classes of the map f that only depends on α. Further, consider the locus Y {α 1 ,...,α k } of points in Y whose preimage contains k points of given singularity types. This locus is called a multisingularity locus. Kazarian's principle states that the Poincaré dual cohomology class of a multisingularity locus is the push-forward under f of a universal polynomial in Chern classes of f that only depends on α 1 , . . . , α k . The universal map over a Hurwitz space does not satisfy the genericity assumptions for these principles to apply. However it turns out that one can compute the corrections to the universal polynomials that appear in this case.

Double Hurwitz numbers

Given a partition λ = (λ 1 , λ 2 , . . . ) we denote by ℓ(λ) the number of its parts, by |λ| the sum of its parts, by |Aut λ| the number of permutations of its parts that preserve their values, and by p λ and q λ the monomials p λ i and q λ i , respectively. For instance, if λ = (4, 3, 3, 1, 1, 1), then ℓ(λ) = 6, |λ| = 13, |Aut λ| = 2! • 3!, p λ = p 3 1 p 2 3 p 4 , q λ = q 3 1 q 2 3 q 4 . Let λ and µ be two partitions of a positive integer K. Let g ≥ 0 be an integer and denote m = ℓ(λ) + ℓ(µ) + 2g -2. Thus g can be recovered from m and vice versa. A transitive factorization of genus g and ramification type (λ, µ) is a list of two permutations σ, ρ ∈ S K and m transpositions τ 1 , . . . , τ m ∈ S K in the symmetric group S K such that

• the cycle type of σ is λ, the cycle type of ρ is µ;

• the product ρτ m . . . τ 1 σ is the identity permutation;

• the subgroup of S K generated by σ, ρ, and the transpositions τ 1 , . . . , τ m is transitive.

The double Hurwitz number h g;λ,µ is the number of transitive factorizations as above divided by K!. Double Hurwitz numbers can be organized into a natural generating function H, which is an infinite power series in two infinite sets of variables, p = (p 1 , p 2 , . . . ) and q = (q 1 , q 2 , . . . ), and an additional formal variable β. The coefficient of the monomial

β m p λ q µ in H is 1 m! |Aut λ| |Aut µ| h g;λ,µ .
The generating function H can be determined, for example, by the so-called cutand-join equation [START_REF] Goulden | Transitive factorisation into transpositions and holomorphic mappings on the sphere[END_REF] ∂ ∂β e H = W e H , W = 1 2 i,j

(i + j) p i p j ∂ ∂p i+j + i j p i+j ∂ 2 ∂p i ∂p j .

This equation determines the function H uniquely from the initial conditions H(0; p, q) = ∞ n=1 pnqn n . This initial condition indicates that the only degree n map C → CP 1 unramified outside 0 and ∞ is the map CP 1 → CP 1 given by z → z n . More explicitly, the series e H can be written as e H = e β W e pnqn n .

The Schur polynomials s λ (p) form an eigenbasis for the operator W , and the corresponding eigenvalues are given by

w(λ) = 1 2 ℓ i=1 λ i -i + 1 2 2 --i + 1 2 2 ,
where λ = (λ 1 , λ 2 , . . . , λ ℓ ), λ 1 ≥ λ 2 ≥ • • • ≥ λ ℓ > 0, see details, for example, in [START_REF] Kazarian | Combinatorial solutions to integrable hirarchies[END_REF]. From this one deduces the so-called Frobenius formula for H, see [START_REF] Lando | Graphs on surfaces and their applications[END_REF][START_REF] Okounkov | Toda equations for Hurwitz numbers[END_REF]:

e H = λ e w(λ) β s λ (p)s λ (q),
where the summation runs over the set of all Young diagrams (partitions) λ and s λ is the corresponding Schur polynomial.

The generating function H possesses interesting integrable properties. In particular, it satisfies equations of the Toda lattice hierarchy, see [START_REF] Okounkov | Toda equations for Hurwitz numbers[END_REF][START_REF] Kazarian | Combinatorial solutions to integrable hirarchies[END_REF].

The genus expansion for this function can be obtained by the substitution 2 H( ; -1 p; -1 q) = H (0) (p; q) + 2 H (1) (p; q) + . . . .

Here the term H (g) enumerates genus g coverings.

Let us collect genus 0 Hurwitz numbers with fixed ramification type λ = (λ 1 , . . . , λ r ) over zero and arbitrary ramification types over infinity into a single series h λ (q). More precisely, define the series h λ (q) by the expansion

H (0) (p 1 + 1, p 2 , . . . ; q)| β=1 = ∞ r=1 1 r! λ 1 ,...,λr h λ (q) p λ 1 . . . p λr . (1)
Thus the coefficient of q µ in h λ counts the connected genus 0 ramified coverings of the sphere with

• r marked and numbered zeros of orders λ 1 , . . . , λ r ;

• any number of unmarked simple zeros (this is consequence of the shift of variable p 1 by 1);

• a ramification profile µ over ∞;

• only simple branch points outside 0 and ∞.

This coefficient is equal to this number divided by |Aut (µ)| and by the factorial of the number of simple branch points. By definition, the function h λ (q) does not depend on the order of the entries λ 1 , . . . , λ r .

In this paper we derive a differential equation on the functions h λ and a broader set of generating functions to be defined later. The equation is of topological origin: it is derived from cohomological information contained in the stratification of the Hurwitz space by the multisingularity types of the functions. This differential equation allows one to compute all the double Hurwitz numbers recursively. As far as we can see, this kind of recursion has never appeared before.

Note that at the moment our method meets a serious obstacle in higher genus, due to the fact that, contrary to the genus 0 case, higher genus Hurwitz spaces are smooth orbifolds no longer.

Hurwitz spaces

The Hurwitz spaces we consider are spaces of meromorphic functions on a rational curve, with marked poles of prescribed orders. The orders of zeroes are not specified. Functions with given orders of zeroes form certain subvarieties in the Hurwitz spaces; these subvarieties are the strata of the stratification of these spaces with respect to the multisingularity type. Since zeros and poles play different roles, the ramification types over 0 and ∞ enter our formulas not in a symmetric way.

The derived relations between the singularity strata are independent of the specific Hurwitz space whose stratification we study; they are totally determined by the local degeneration types of the functions involved. In particular, the q-variables, that keep track of the orders of the poles, are not involved explicitly in our equations for the generating function.

For any tuple of positive integers κ = (k 1 , . . . , k n ), we denote by H κ = H (k 1 ,...,kn) the space of rational functions f (z) with n marked poles of orders k 1 , . . . , k n , respectively. Two rational functions are equivalent and correspond to the same point of the Hurwitz space if they can be obtained from each other by a homography z → az+b cz+d . This Hurwitz space is introduced and discussed in detail in [START_REF] Ekedahl | Hurwitz numbers and intersections on moduli spaces of curves[END_REF]. Since we only consider the case when the source curve has genus zero, we do not indicate the genus in the notation.

We'll make use of a generalization of these spaces, namely, for any integer r ≥ 1 we denote by H r|κ = H r|(k 1 ,...,kn) the space that parameterizes functions with n marked poles of prescribed orders and r extra marked points in CP 1 . The marked points that are not poles are called supplementary marked points. We number the marked points in the following order: the supplementary marked points are numbered from 1 to r, while the poles have numbers r + 1, . . . , r + n.

We denote by H r|κ a natural completion of the space H r|κ . This is the moduli space of stable maps from genus 0 curves to CP 1 with k preimages of ∞ of prescribed orders and r more marked points. For more details see [START_REF] Ekedahl | Hurwitz numbers and intersections on moduli spaces of curves[END_REF].

For r + n ≥ 3, the Hurwitz space is related to the moduli space M r+n of possibly singular stable genus zero curves with r + n marked points via the natural projection π : H r|κ → M r+n that takes a rational function to the stabilization of its source curve.

This projection is an orbifold cone (see [START_REF] Fulton | of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF], Chapter 4 for the definition of a cone, [START_REF] Ekedahl | Hurwitz numbers and intersections on moduli spaces of curves[END_REF], [START_REF] Sauvaget | Cohomology classes of strata of differentials[END_REF] for the orbifold generalization). The group C * of nonzero complex numbers acts on H r|κ by multiplying functions by constants. Its projectivization P H r|κ , that is, the space of non-constant C * -orbits, is a compact orbifold of dimension dim P H r|κ = ℓ(κ) + |κ| + r -3. It carries a natural (rational) fundamental homology class

[P H r|κ ] ∈ H 2(ℓ(κ)+|κ|+r-3) (P H r|κ )
and the characteristic cohomology class of the C * -action

ξ = c 1 (O(1)) ∈ H 2 (P H r|κ ),
where O(1) is the dual of the tautological line bundle of the cone. Here and below we always consider cohomology groups with coefficients in Q.

Remark 1.1 A particular case that we will encounter in our recursion is r = n = 1. In this case the moduli space M r+n = M 2 does not exist, and we consider H r|k as a cone over the orbifold BZ k = {point}/Z k . The projectivization P H 1|k is a weighted projective space. Similarly, one can view H 0|k as a cone over BZ k , but we will actually never encounter this space.

Strata and their degrees

Definition 1.2 For every pure dimension subcone Z ⊂ H r|κ , its degree is defined to be the intersection number of the subvariety P Z ⊂ P H r|κ with the suitable power of the class ξ,

deg r|κ (Z) = P Z ξ dim P Z = P H r|κ 1 1 -ξ ⌢ [P Z].
More generally, for any cohomology class α ∈ H * (P H r|κ ), we define its degree as the number deg r|κ (α) =

P H r|κ α 1 -ξ .
The degree is a Q-valued linear function on the total cohomology space of the projectivized Hurwitz space.

In our recursion we will encounter a particular case when Z is included into the zero section of the cone. In this case we set

deg Z =      z∈Z 1 stab(z) if dim Z = 0, 0 else.
For any tuple of non-negative integers λ = (λ 1 , . . . , λ r ), we denote by X λ = X (λ 1 ,...,λr) ⊂ H r|κ the set of functions having a zero of order exactly λ i at the i th supplementary marked point for i = 1, . . . , r. Thus, if λ i = 0, then the function f does not vanish at the i th marked point; if λ i = 1 then the function must have a simple zero at the corresponding point, and so on. Further, we denote by X λ the closure of X (λ 1 ,...,λr) in H r|κ . In particular, with this notation we have H r|κ = X (0,...,0) , and in general X λ is a closed suborbifold of H r|κ called a multisingularity stratum.

Note that we may consider the same type λ of degeneration of functions in Hurwitz spaces H r|κ for different κ = (k 1 , . . . , k n ).

Proposition 1.3 For each tuple λ = (λ 1 , . . . , λ r ) of positive integers, the generating function for the degrees of the singularity stratum X λ in different Hurwitz spaces coincides with the generating function h λ (q) for the corresponding Hurwitz numbers defined by Eq. (1):

h λ (q) = ∞ n=1 1 n! k 1 ,...,kn deg r|(k 1 ,...,kn) (X λ ) q k 1 . . . q kn .
This proposition will be proved in Section 2.1. It shows that computing double Hurwitz numbers is equivalent to computing the degrees of multisingularity strata in Hurwitz spaces.

The Hurwitz space H r|κ carries natural "tautological" classes ψ 1 , . . . , ψ r . Namely, the cotangent line at the ith supplementary marked point to the source curve of a rational function defines a line bundle over H r|κ denoted by L i , and we set ψ i = c 1 (L i ) ∈ H 2 (H r|κ ), i = 1, . . . , r. These classes are the pull-backs under the natural projection π : P H r|κ → M r+n of the corresponding ψ-classes defined in a similar way on M r+n .

To each singularity stratum X λ one can assign the fundamental class [P X λ ] of its projectivization and its "descendants", obtained by multiplying [P X λ ] by monomials in the ψ-classes ψ 1 , . . . , ψ r . Definition 1.4 For any two tuples of non-negative integers λ = (λ 1 , . . . , λ r ) and ν = (ν 1 , . . . , ν r ), of the same length ℓ(λ) = ℓ(ν) = r, we define the function

x λ,ν (q) = ∞ n=1 1 n! κ=(k 1 ,...,kn) deg r|κ (ψ ν 1 1 . . . ψ νr r [P X λ ]) q k 1 . . . q kn .
The function x λ,ν (q) is invariant under simultaneous permutations of the parts in λ and ν. We have h λ (q) = x λ,(0,...,0) (q).

The recursion

To formulate the recursion, let us introduce a new set of variables t i,j , i ≥ 0, j ≥ 0, and collect the functions x λ,ν (q) into the following descendant genus zero Hurwitz potential

X(t, q) = ∞ r=1 1 r!
λ=(λ 1 ,...,λr) ν=(ν 1 ,...,νr)

x λ,ν (q) t λ 1 ,ν 1 . . . t λr,νr .

The original generating function for the genus zero double Hurwitz numbers is related to the series X(t, q) as follows:

H (0) (p 1 + 1, p 2 , . . . ; q)| β=1 = X(t, q) t i,0 =p i (i≥1), t i,j =0 (j>0 or i=0) . Proposition 1.5 We have x (0,...,0),(ν 1 ,...,νr) (q) = |ν| ν 1 , . . . , ν r z |ν|,r (q)
where

z d,r (q) = K,n 1 n! n + r -3 d K n+r-3-d k 1 +•••+kn=K n i=1 k k i i k i ! q k i .
In [START_REF] Zvonkine | An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere[END_REF], an algebra of power series in a single variable has been introduced. This algebra is generated by two power series, namely,

∞ n=1 n n-1 x n n! , ∞ n=1 n n x n n! .
This algebra contains all the generating functions for Hurwitz numbers enumerating ramified coverings of the sphere with fixed degenerate ramification, provided the degree n of the coverings varies. For ramified coverings of the torus, a similar role is played by the algebra of quasimodular forms. The power series z d,r (q) play a similar role in computing double Hurwitz numbers. Namely, all the series h λ (q 1 , q 2 , . . . ) and x λ,µ (q 1 , q 2 , . . . ) belong to the algebra generated by the series z d,r . Note that the series z d,r are not algebraically independent, they obey certain polynomial relations. We will give a description of this algebra in a separate paper.

The expression in Proposition 1.5 gives the initial conditions for our recursion. To formulate the recursion itself, we need the following explicit series in the t-variables:

Ψ a,ℓ = ∞ k=0 1 k! ν 1 +•••+ν k =ℓ+k-3 λ 1 +•••+λ k =a |ν| ν 1 , . . . , ν k k i=1 t λ i ,ν i . Remark 1.6 The coefficients of Ψ are equal to |ν| ν 1 , . . . , ν k = M ℓ+k ψ ν 1 1 • • • ψ ν k k .
This is the form in which this series appears in the proof of the recursion.

Theorem 1.7 The series X obeys the following differential equations valid for any s ≥ 0 and m ≥ 0:

∂X ∂t s+1,m = ∂X ∂t s,m + s ∂X ∂t s,m+1 - ℓ≥1,a≥0 1 ℓ! ∂Ψ a,ℓ ∂t s,m σ 1 +•••+σ ℓ =a ℓ i=1 σ i ∂X ∂t σ i ,0 .
These differential equations provide a recursion for the coefficients of the series X. The recursion expresses each series x λ,ν (q) with ℓ(λ) = ℓ(ν) = ℓ as a polynomial in the known series z d,r (q) of Proposition 1.5 with r ≤ ℓ.

The geometric meaning of the recursion will be explained in Section 3.

Examples

The recursion of Theorem 1.7 translates into polynomial relations between the Taylor coefficients x λ,ν of the t-expansion of X. Each series x λ,ν is expressed as a polynomial in similar series of smaller degrees with respect to the grading defined by deg x λ,ν = |λ|. Thus, due to the initial conditions of the recursion, the series x λ,ν are expressed as polynomials in the series z d,r of Proposition 1.5. Here are a few first examples for the Hurwitz generating series h λ (q) = x λ,(0,...,0) (q):

h (2) = z 0,1 + z 1,1 , h (3) = z 0,1 - z 2 0,1 2 + 3 z 1,1 + 2 z 2,1 , h (4) = z 0,1 - 5 z 2 0,1 2 + 6 z 1,1 -2 z 0,1 z 1,1 + 11 z 2,1 + 6 z 3,1 , h (5) = z 0,1 - 15 z 2 0,1 2 + 5 z 3 0,1 6 + 10 z 1,1 -15 z 0,1 z 1,1 -2 z 2 1,1 + 35 z 2,1 -6 z 0,1 z 2,1 + 50 z 3,1 + 24 z 4,1 , h (2,2) = -6 z 0,1 + z 2 0,1 + z 0,2 -11 z 1,1 + 2 z 1,2 -6 z 2,1 + 2 z 2,2 , h (3,2) = -10 z 0,1 + 9 z 2 0,1 + z 0,2 -z 0,1 z 0,2 -35 z 1,1 + 6 z 0,1 z 1,1 + 4 z 1,2 -z 0,1 z 1,2 -50 z 2,1 + 8 z 2,2 -24 z 3,1 + 6 z 3,2 , h (2,2,2) = 85 z 0,1 -40 z 2 0,1 -18 z 0,2 + 6 z 0,1 z 0,2 + z 0,3 + 225 z 1,1 -24 z 0,1 z 1,1 -51 z 1,2 + 6 z 0,1 z 1,2 + 3 z 1,3 + 274 z 2,1 -84 z 2,2 + 6 z 2,3 + 120 z 3,1 -54 z 3,2 + 6 z 3,3 .

Acknowledgements

The first two authors are grateful to the participants of the seminar "Characteristic classes and intersection theory" at the Department of mathematics, NRU HSE. The third author is partly supported by the ANR-18-CE40-0009 ENUMGEOM grant.

Degrees of strata in Hurwitz Spaces

Degrees of strata are double Hurwitz numbers

In this section we prove Proposition 1.3, i.e., the equality

h λ (q) = ∞ n=1 1 n! k 1 ,...,kn deg r|(k 1 ,...,kn) (X λ ) q k 1 . . . q kn .
Proof. The argument is similar to that of [START_REF] Ekedahl | Hurwitz numbers and intersections on moduli spaces of curves[END_REF]. Let m be the number of simple critical points of a generic rational function in the stratum X λ , those where the critical values differ from zero and infinity. This number can be determined by the Riemann-Hurwitz formula. Then there is a natural map from X λ to C m that associates to a rational function f ∈ X λ the unordered collection of its m critical values. This map extends to an algebraic map X λ → C m called the Lyashko-Looijenga map, or branching morphism. The target space C m is the mth symmetric power of the complex line C. By definition, the Hurwitz number is the degree of this map, that is, the number of preimages of a generic point. The Lyashko-Looijenga map is, of course, C * -equivariant, where C * acts on the target C m with weights 1, . . . , m. It follows that the Lyashko-Looijenga map descends to a map of projectivizations Λ : P X λ → P C m , where P C m is the weighted projective space with weights 1, . . . , m. The map Λ has the same degree as the Lyashko-Looijenga map itself. Moreover, we have Λ * O(1) = O(1) and Λ * ξ = ξ. Thus the degree of the Lyashko-Looijenga map or of Λ is equal to

deg(Λ) = P X λ ξ m-1 P C m ξ m-1 = m! deg ℓ(λ)|κ (X λ ).
The last equality is due to the fact that P C m ξ m-1 = 1 m! . We conclude that up to the factor m!, the Hurwitz number (the left-hand side of the equality) is equal to the degree of the singularity stratum X λ . The factorial factor is accounted for automatically when the numbers are collected into the appropriate generating series. As a result, the two generating series for the Hurwitz numbers and for the degrees of the singularity strata coincide.

In the exceptional case n = 1, r = 1, k 1 = λ 1 = k the stratum X λ is composed of only one map f = x k and we have m = 0. The corresponding Hurwitz number is equal to 1/k, which is also the degree of the cone according to Remark 1.1.

The initial conditions of the recursion

In this section we compute the degree of the cohomology class ψ ν 1 1 . . . ψ νr r [P X (0,...,0) ]. Using this, we determine the generating function x (0,...,0),ν (q) and thus prove Proposition 1.5.

Recall that P X (0,...,0) = P H r|κ . Applying the projection formula to the forgetful map π : P H r|κ → M r+n we get

deg r|κ (ψ ν 1 1 . . . ψ νr r [P X (0,...,0) ]) = P H r|κ ψ ν 1 1 . . . ψ νr r 1 -ξ = Mr+n ψ ν 1 1 . . . ψ νr r π * 1 1 -ξ .
The push-forward class π * 1 1-ξ is the so-called total Segre class of the cone H r|κ → M r+n . It is computed in [START_REF] Ekedahl | Hurwitz numbers and intersections on moduli spaces of curves[END_REF]:

π * 1 1 -ξ = n i=1 k k i i k i ! (1 -k 1 ψ r+1 ) . . . (1 -k n ψ r+n ) .
Therefore, in this case, the degree of the cohomology class is expressed as an intersection number of ψ-classes on M r+n :

M r+n ψ ν 1 1 . . . ψ νr r π * 1 1 -ξ = n i=1 k k i i k i ! M r+n ψ ν 1 1 . . . ψ νr r (1 -k 1 ψ r+1 ) . . . (1 -k n ψ r+n ) = n i=1 k k i i k i ! n + r -3 ν 1 , . . . , ν r (k 1 + • • • + k n ) n+r-3-|ν| .
The last equality is due to the well-known formula for the intersection numbers of ψ-classes over M n (see [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF]):

M n ψ k 1 1 . . . ψ kn n = n-3 k 1 ,...,kn if k 1 + • • • + k n = n -3, 0 if k 1 + • • • + k n = n -3.
It follows that

x 0,ν (q) = ∞ n=1 1 n! κ=(k 1 ,...,kn) deg r|κ (ψ ν 1 1 . . . ψ νr r [P X λ ]) q k 1 . . . q kn = |ν| ν 1 , . . . , ν r K,n 1 n! n + r -3 |ν| K n+r-3-|ν| k 1 +•••+kn=K n i=1 k k i i k i ! q k i .
This is exactly the formula announced in Proposition 1.5.

From a cohomological relation to the recursion

As explained in the introduction, our recursion formula is a consequence of an identity in the cohomology ring of the Hurwitz space. In this section we introduce and prove this identity, then deduce the recursion formula.

The cohomological identity

Consider the multisingularity variety P X λ 1 ,λ 2 ,...,λr ⊂ P H r|κ . Meromorphic functions forming this variety vanish at the first marked point together with their first λ 1 -1 derivatives. The subvariety P X λ 1 +1,λ 2 ,...,λr ⊂ P X λ 1 ,λ 2 ,...,λr is distinguished by the condition that the next derivative, of order λ 1 , also vanishes. We will now show that the derivative of order λ i is actually a section of some specific line bundle and its vanishing locus can be related to the first Chern class of this line bundle.

Let C be a curve, f : C → CP 1 a meromorphic function, and x 1 ∈ C a marked point that is not a pole of f . The quotient of the space of s-jets at x 1 by the space of (s -1)-jets at x 1 is the line T ⊗s

x 1 C. Thus, if f (x 1 ) = • • • = f s-1 (x 1 ) = 0, then f (s) (x 1 ) is a linear map f (s) (x 1 ) : T ⊗s x 1 C → T 0 CP 1 .
Note that this map is well defined only if all the derivatives of order less than s vanish at x 1 .

As the function f varies in the Hurwitz space, the tangent lines T x 1 C to the source curves at x 1 form the line bundle L ∨ 1 whose first Chern class is -ψ 1 . Similarly, for the projectivized Hurwitz space, the tangent lines T 0 CP 1 to the target curve at 0 form the line bundle O(-1) with the first Chern class -ξ. As a corollary, we obtain the following statement.

Lemma 3.1 Consider the line bundle

Hom(L ∨⊗λ 1 1 , O(1)) over P H r|κ , whose first Chern class is c 1 (Hom(L ∨⊗λ 1 1 , O(-1))) = λ 1 ψ 1 -ξ. Then f (λ 1 ) (x 1
) (the derivative of the rational function of order λ 1 at the first marked point) is a well-defined holomorphic section of this line bundle over the multisingularity stratum P X λ 1 ,λ 2 ,...,λr .

The zeroes of the holomorphic section f (λ 1 ) (x 1 ) form a divisor in P X λ 1 ,λ 2 ,...,λr . The multisingularity stratum P X λ 1 +1,λ 2 ,...,λr is one of the components of this divisor. We will see that the section f (λ 1 ) (x 1 ) has a simple vanishing along this component. In general, however, the zero divisor has many other components formed by functions with singular source curves. We call these components the boundary components of the zero divisor. Denoting these components by D α and the vanishing order of the section along D α by m α , we obtain a cohomological relation

(λ 1 ψ 1 -ξ)[P X λ 1 ,λ 2 ,...,λr ] = [P X λ 1 +1,λ 2 ,...,λr ] + m α D α . ( 2
)
Our goal is now to list all the components D α of the zero divisor and determine the vanishing orders m α . The irreducible components D α of the zero divisor correspond to certain singularity types of the functions in the family H r|κ . We start by recalling the description of singularity types from [START_REF] Kazarian | Universal cohomological expressions for singularities in families of genus 0 stable maps[END_REF] adapted to our situation, the one in the presence of supplementary marked points. Definition 3.2 Let J ⊂ {1, . . . , r} and ℓ be a positive integer such that |J| + ℓ ≥ 3. Let σ 1 , . . . , σ ℓ be positive integers. We denote by X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr the locus of meromorphic functions f ∈ H r|κ satisfying the following conditions. (i) The source curve C has an irreducible component C 0 , called the central component, containing the supplementary marked points of the set J and no other marked points; the function f vanishes identically on C 0 . (iii) The component C 0 meets the other irreducible components at ℓ points; the function f has zeros of orders exactly σ 1 , . . . , σ ℓ at these points. (iii) The function f has zeros of orders exactly λ i at the supplementary marked point that are not in J. We denote by X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr the closure of this locus.

The closure X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr is a suborbifold of P H r|κ of pure codimension 1 + σ i . It has many irreducible components corresponding to all possible distributions of the poles and the remaining supplementary marked points among the ℓ components of the curve different from the central one. If a function f lies in X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr , then its derivatives of all orders vanish on the central component. Therefore, functions with nonisolated singularities can appear as the components D α of (2) if 1 ∈ J. The precise statement is given in the following theorem. Denote |σ| = ℓ i=1 σ i , |λ J | = i∈J λ i .

Theorem 3.3 a) The stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr lies in the closure of P X λ 1 ,...,λr iff |σ| ≥ |λ J |. b) The stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr has codimension 1 in P X λ 1 ,...,λr iff |σ| = |λ J |. c) Assuming this condition is satisfied, the divisor P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr is a component of the zero locus of f (λ 1 ) (x 1 ) iff 1 ∈ J. d) In this case the vanishing order of f (λ 1 ) (x 1 ) along P X

J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr equals σ 1 • • • σ ℓ . e)
The vanishing order of f (λ 1 ) (x 1 ) along P X λ 1 +1,λ 2 ...,λr equals 1. f) The section f (λ 1 ) (x 1 ) has no other zeros.

Corollary 3.4

The following relation holds in the cohomology ring of P H r|κ :

[P X λ 1 +1,λ 2 ,...,λr ] = (λ 1 ψ 1 -ξ)[P X λ 1 ,λ 2 ,...,λr ] - J, 1∈J⊂{1,...,r} σ⊢|λ J | σ 1 . . . σ ℓ(σ) [P X J;σ 1 ,...,σ ℓ(σ) λ 1 ,λ 2 ,...,λr ]. ( 3
)
This relation is the precise version of (2).

3.2 Vanishing loci of f λ 1 (x 1 ): proof of Theorem 3.3

The statement and the proof are similar to [START_REF] Kazarian | Universal cohomological expressions for singularities in families of genus 0 stable maps[END_REF], Section 2.1, but the presence of supplementary points introduces some extra complications. To prove the theorem we first need to discuss deformations of stable maps in order to give a precise description of the closure P X λ 1 ,λ 2 ,...,λr in P H r|κ .

Let f ∈ H r|κ be a stable map. For any point y ∈ C, a connected component X of f -1 (y) that is not a regular point of f is called a ramification locus. Thus a ramification locus is either a ramification point of f , or a node of the source curve C of f , or a stable genus 0 subcurve of C contracted to y. In [START_REF] Kazarian | Universal cohomological expressions for singularities in families of genus 0 stable maps[END_REF] we introduced standard deformation spaces for these three types of ramification loci, parametrizing all functions with the same monodromy as the ramification locus. Deformations of ramification points are given by polynomials, deformations of nodes by Laurent polynomials, and deformations of contracted curves by rational functions on contracted curves, respectively. We will recall more precise definitions later. Proposition 3.5 The neighborhood of f in H r|κ is canonically decomposed into a direct product of deformation spaces of the ramification loci of f .

Proof. Draw a disk D y around every branch point y ∈ C of f so that the disks do not intersect. A sufficiently small deformation f of f has all of its branch points in the union of the disks D y , while over C \ ∪D y the maps f and f determine the same nonramified covering. Each connected component of f -1 (D y ), except disks that map biholomorphically to D y , is a small deformation of the corresponding ramification locus X ⊂ f -1 (y).

Conversely, choose a deformation of every ramification locus of f sufficiently small for all branch points to lie inside D y . Then the monodromies around D y of a ramification locus of f and of its deformation coincide. Thus the deformations can be glued into the graph of f instead of the preimage f -1 (D y ), and we obtain a stable map in the neighborhood of f . Let X be a ramification locus of f over 0, that is, X ⊂ f -1 (0). Denote by λ(X) the sum of integers λ i over the supplementary marked points i contained in X. Further, denote by σ(X) the total degree of f in the neighborhood of X. More precisely, if X is an isolated zero of order λ, then σ(X) = λ. If X is a node and f has ramification orders σ 1 , σ 2 at the branches, then σ(X) = σ 1 + σ 2 . If X is a union of irreducible components of C that meet other components at ℓ points at which f has zeros of orders σ 1 , . . . , σ ℓ , then σ(X) = ℓ j=1 σ j .

Definition 3.6 A ramification locus X ⊂ f -1 (0) is acceptable if λ(X) ≤ σ(X).
Lemma 3.7 A stable map f lies in the closed stratum P X λ 1 ,λ 2 ,...,λr iff (i) we have f (x i ) = 0 for all i such that λ i ≥ 1, and (ii) all ramification loci in f -1 (0) are acceptable.

Proof. First suppose that f lies in P X λ 1 ,λ 2 ,...,λr . Take a smooth map f ∈ P X λ 1 ,λ 2 ,...,λr close enough to f . The connected components of f -1 (D y ) are in a one to one correspondence with those of f -1 (y) for any branch point y. Moreover, if X is a connected component of f -1 (0) and X the corresponding connected component of f -1 (D 0 ), then X contains the same set J ⊂ {1, . . . , n} of supplementary marked points as X and the degree of f on X is equal to σ(X). It follows that σ(X) ≥ λ(X). Indeed, the sum of orders of zeros of f over X cannot exceed its degree. This proves the "only if" part. To prove the "if" part, we will use Proposition 3.5. First suppose that every ramification locus X of f is either a point or an irreducible curve. If X ⊂ f -1 (0) is a curve, it carries a function g X with zeros of orders λ i at the supplementary marked points contained in X (other zeros are allowed) and with poles of orders σ j at the intersection points of X with other components of the curve C (no other poles are allowed). The acceptability condition ensures that such a function exists. The functions g X provide a deformation of X. If X ∈ f -1 (0) is a supplementary marked point, it can be deformed by a polynomial with a root of order exactly λ i at the marked point. For all other ramification loci (those that correspond to branch points y = 0 and those that do not contain supplementary marked points) choose any deformation. By Proposition 3.5, this family of deformations determines a deformation of f into the stratum P X λ 1 ,λ 2 ,...,λr .

Finally, if a ramification locus X is not irreducible, we can consider a family of smooth curves X s that tend to X as s → 0. The stable map obtained from f by replacing X with X s lies in the closure P X λ 1 ,λ 2 ,...,λr for any nonzero s, hence f lies in the closure, too.

Proof of Theorem 3.3. a) The stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr lies in the closure of P X λ 1 ,...,λr iff |σ| ≥ |λ J |. Consider a stable map f ∈ P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr . It follows immediately from the definition of P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr that the source curve C of f has only one contracted component, namely the central component of the stratum. This component is acceptable iff |σ| ≥ |λ J |. Thus (a) follows from Lemma 3.7.

b) The stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr has codimension 1 in P X λ 1 ,...,λr iff |σ| = |λ J |. Once we know that P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr lies in P X λ 1 ,...,λr , a simple dimension count provides the codimension. c) Assuming this condition is satisfied, the divisor P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr is a component of the zero locus of f (λ 1 ) (x 1 ) iff 1 ∈ J. This is because f has a zero of order exactly λ 1 at x i when i ∈ J, while it vanishes identically in the neighborhood of x i when i ∈ J. d) In this case the vanishing order of f (λ 1 ) (x 1 ) along P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr equals σ 1 • • • σ ℓ . The proof repeats that of Lemma 2.1 in [START_REF] Kazarian | Universal cohomological expressions for singularities in families of genus 0 stable maps[END_REF] with a slight modification due to supple-mentary marked points. We give here a rather concise presentation, referring to [START_REF] Kazarian | Universal cohomological expressions for singularities in families of genus 0 stable maps[END_REF] for more details.

Let f be a generic stable map in P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr . In particular, the central component C 0 of the source curve of f is smooth, the supplementary marked points are zeros of orders exactly λ i , the other zeros are simple, and all branch points of f except 0 are simple. According to Proposition 3.5, there is a local chart on the Hurwitz space H r|κ in the neighborhood of f that is a product of standard deformation families for the ramification loci of f . We are going to determine a parametrization of P X λ 1 ,λ 2 ,...,λr in this chart. By the genericity assumption, any ramification locus over y = 0 is just a simple ramification point. Its deformation family has one parameter corresponding to moving y in C. Any such deformation keeps f in P X λ 1 ,λ 2 ,...,λr . Every supplementary marked point x i with λ i ≥ 2 and i ∈ J is a ramification locus. The corresponding deformation family is the space of polynomials of degree λ i . Any such deformation is transversal to the stratum P X λ 1 ,λ 2 ,...,λr , since it does not preserve the zero of order λ i at x i . Thus, in the parametrization of the stratum P X λ 1 ,λ 2 ,...,λr , all the parameters of the deformations must be set to 0. Finally, the most important ramification locus is the central component X = C 0 . Its deformation family has moduli parameters, corresponding to local coordinates in the moduli space M |J|+ℓ on a neighborhood of the curve C 0 , and smoothing parameters, describing the ways to smooth out the nodes where C 0 meets the rest of the curve. Changing the moduli parameters always keeps f in the stratum P X λ 1 ,λ 2 ,...,λr . The smoothing parameters were described in [START_REF] Ekedahl | Hurwitz numbers and intersections on moduli spaces of curves[END_REF]. They are u i , 1 ≤ i ≤ ℓ and a ij , 1 ≤ j ≤ ℓ i -1 if we write the stable map in the following form:

f (z) = ℓ i=1 u i z -z i k i + a i,1 u i z -z i k i -1 + • • • + a i,k i -1 u i z -z i . (4)
Introduce a coordinate z on the central component C 0 of f . Denote by α 1 , . . . , α ℓ the values of the z-coordinate at the intersection points with the branches. Denote by β j , j ∈ J the values of the coordinate z at the supplementary marked points. Then all functions g(z) with zeros and poles of prescribed orders at the points α i and β j are proportional to

g(z) = j∈J (z -β j ) λ j ℓ i=1 (z -α i ) σ i .
We rewrite g in the form (4) and denote by ūi , āi,j the values of the coefficients u i and a ij in this expansion. Let K = LCM(σ 1 , . . . , σ ℓ ) and r i = K/σ i for 1 ≤ i ≤ ℓ. Then the stratum P X λ 1 ,...,λr is parametrized in coordinates u i and a ij as follows:

u i = ζ i c r i ūi , a ij = ζ j i c jr i āij .
Here (ζ 1 , . . . , ζ ℓ ) is a collection of roots of unity,

ζ k i i = 1. The action of Z/KZ on Z/σ 1 Z × • • • × Z/σ ℓ Z has σ 1 • • • σ ℓ /K
orbits corresponding to local irreducible components of the stratum P X λ 1 ,...,λr , and we choose one collection of roots of unity in each orbit. In each component of the stratum, the λ 1 th derivative at x 1 of the function corresponding to parameter c is equal to c K • g (λ 1 ) (x 1 ). Thus the vanishing order of the λ 1 th derivative at c = 0 equals K for each component. Since there are σ 1 • • • σ ℓ /K components, we get the total vanishing order equal to

σ 1 • • • σ ℓ .
e) The vanishing order of f (λ 1 ) (x 1 ) along P X λ 1 +1,λ 2 ...,λr equals 1. This is similar to (d), but much easier. Once again, choose a generic f ∈ P X λ 1 +1,λ 2 ...,λr and a chart in H r|κ , containing f , that is a product of standard deformations of ramification loci. This time the important ramification locus is the supplementary marked point x i . The corresponding deformation space is

z λ 1 +1 + a 1 z λ 1 -1 + • • • + a λ 1 .
The stratum P X λ 1 ,λ 2 ...,λr in this coordinate is parametrized by one variable a as follows:

(z -a) λ 1 (z + λ 1 a).

The derivative of order λ 1 at a is just (λ 1 + 1)a, thus its vanishing order along a = 0 equals 1.

f) The section f (λ 1 ) (x 1 ) has no other zeros.

Let f be any stable map lying in the closed stratum P X λ 1 ,λ 2 ...,λr and satisfying

f (λ 1 +1) (x 1 ) = 0. Denote by X the connected component of f -1 (0) containing x 1 .
Let J be the set of marked point contained in X, ℓ the number of points at which X meets other components of the curve and σ ′ 1 , . . . , s ′ ℓ the orders of zeros of f at these points. If X = {x 1 } then we let J = {1}, ℓ = 1, σ ′ 1 = order of the zero of f at x 1 . It follows from Lemma 3.7 that all the connected components of f -1 (0), including X, are acceptable. In particular, we have j∈J λ j ≤ σ ′ i . Choose any list of integers σ i ≤ σ ′ i such that j∈J λ j = σ ′ i . If X = {x 1 } we let σ 1 = λ 1 + 1. We claim that f lies in the closure of the stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr (or of P X λ 1 +1,λ 2 ...,λr , if X = {x 1 }). To see that apply to f the following sequence of deformations. (i) If X is not smooth, deform it into a smooth curve. (ii) Deform generically all ramification loci that do not contain supplementary marked points x i with λ i > 0. (iii) Deform all ramification loci over 0 except X so that each supplementary marked point x i contained in these loci becomes a zero of order λ i . This is possible by Lemma 3.7. (iv) Deform the function f at the ℓ attachment points to X using the polynomial deformation families of degree σ ′ i to reduce the orders of zeros at these points from σ ′ i to s i . Thus we have constructed a stable map f ∈ P X λ 1 ,λ 2 ...,λr arbitrarily close to f that lies in the stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr or in P X λ 1 +1,λ 2 ...,λr . It follows that f itself lies in the closure of one of these above strata.

Proof of the recursion

The recursion formula of Theorem 1.7 is obtained by multiplying both sides of the cohomological relation (3) by a monomial in the ψ-classes and taking the degrees of the corresponding cohomology classes.

Lemma 3.8 The stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr decomposes into several irreducible components each of which is a direct product of the moduli space M |J|+ℓ and ℓ multisingularity strata in Hurwitz spaces. Such components are in a one-to-one correspondence with the ways to split the set of poles and the set of supplementary marked points outside J into ℓ unordered parts in such a way that each part contains at least one pole.

Proof. If we remove the central component from a curve in the stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr , the remaining part of the curve decomposes into ℓ connected components. Each of them contains at least one pole of the rational function f , since otherwise f would vanish identically on the connected component in question. The poles and the supplementary marked points lying on these connected components determine a splitting of the set of poles and supplementary marked points into ℓ parts.

Pick one splitting like that. Denote by K i , 1 ≤ i ≤ ℓ, the list of orders of the poles on the ith connected component. Similarly, denote by J i , 1 ≤ i ≤ ℓ, the set of supplementary marked points that lie on the ith connected component. Each element s of J i comes with the corresponding label λ s . Now, to each set J i we add one extra element corresponding to the point where the ith connected component meets the central component. This element carries the label σ i . Thus for each i we get the complete set of data that determines a multisingularity stratum in a Hurwitz space. Given ℓ functions in these ℓ Hurwitz spaces and one stable curve in M |J|+ℓ we can assemble a rational function f from the intersection P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr by gluing them together in the natural way. Thus we obtain one irreducible component of the stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr .

Proof of the recursion

Recall that we must prove the following differential equation holds for the generating series X:

∂X ∂t s+1,m = ∂X ∂t s,m + s ∂X ∂t s,m+1 - ℓ≥1,a≥0 1 ℓ! ∂Ψ a,ℓ ∂t s,m σ 1 +•••+σ ℓ =a ℓ i=1 σ i ∂X ∂t σ i ,0 .
Let's consider the coefficient of the monomial

r i=2 t s i ,m i |Aut {(s i , m i )}|
in this equality. The equality of these coefficients is obtained by multipying the cohomological identity (3) by ψ m 1 ψ m 2 2 . . . ψ mr r and taking the degree. The coefficient in the left-hand side is the degree of [X s+1,s 2 ,...,sr ]ψ m 1 ψ m 2 2 . . . ψ mr r , which is the left-hand side of (3) multiplied by ψ m 1 ψ m 2 2 . . . ψ mr r .

The sum of the coefficients in the first two terms of the right-hand side is the degree of [X s,s 2 ,...,sr ](ξ + sψ 1 )ψ m 1 ψ m 2 2 . . . ψ mr r , which is the first left-hand side term of (3) multiplied by ψ m 1 ψ m 2 2 . . . ψ mr r . Finally, the terms of the last sum are indexed by a choice of ℓ, a choice of a, a choice of σ 1 , . . . , σ ℓ , and a way to decompose the monomial t s 2 ,m 2 • • • t s k ,m k into a product of the ℓ + 1 monomials J, J 1 , J 2 , . . . , J ℓ . The choices of ℓ, a, σ 1 , . . . , σ ℓ , and J determine one term P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr in the summation of the cohomological identity [START_REF] Ekedahl | Hurwitz numbers and intersections on moduli spaces of curves[END_REF], where a = i∈J λ i . The choice of J 1 , . . . , J ℓ determines the choice of an irreducible component of the stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr , as explained in Lemma 3.8. Let us inspect the contribution to the coefficient of

r i=2 t s i ,m i |Aut {(s i , m i )}|
arising from each given choice of ℓ, a, σ 1 , . . . , σ ℓ , J, J 1 , . . . , J ℓ . This contribution is a product of several factors.

The first factor is

|Aut {(s i , m i )}| |Aut (J)| ℓ i=1 |Aut (J i )| .
This factor accounts for the fact that the supplementary marked point are numbered, while the corresponding variables t s i ,m i in the monomial are not. The factor transforms the number of ways to split the monomial into a product into the number of ways to split the supplementary marked points into ℓ + 1 groups.

The second factor is 1/ℓ!. It accounts for the fact that the irreducible components of the stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr are labelled by unordered set partitions into ℓ parts, while the terms of the product

ℓ i=1 σ i ∂X ∂t σ i ,0
are ordered. The third factor is ℓ i=1 σ i . This factor is present in the cohomological identity [START_REF] Ekedahl | Hurwitz numbers and intersections on moduli spaces of curves[END_REF], where it represents the vanishing order of the section.

The fourth factor is the coefficient of the series Ψ a,ℓ . As explained in Remark 1.6, this coefficient is the intersection number

M |J |+ℓ i∈J ψ m i i .
This factor appears as we integrate over the moduli of the central component the product of relevant ψ-classes.

As stated in Lemma 3.8, each irreducible component of the stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr is the product of the moduli space M |J|+ℓ and ℓ multisingularity strata in Hurwitz spaces. Moreover, each multisingularity stratum comes with a monomial in ψ-classes. So as not to multiply unnecessary notation, let's just denote the multisingularity strata by X (i) and the corresponding monomials in ψ-classes by ψ (i) for 1 ≤ i ≤ ℓ.

Then the fifth and last factor is the product of degrees

ℓ i=1 deg [X (i) ψ (i) ].
Now it remains to explain why the product of factor 4 (the integral over the moduli of the central component) and factor 5 (the product of degrees of ℓ multisingularity strata multiplied by monomials ψ (i) ) is equal to the degree of the corresponding irreducible component of the stratum P X J;σ 1 ,...,σ ℓ λ 1 ,λ 2 ,...,λr multiplied by ψ m i i . This follows from a general statement. The Segre class of a product of cones C i → B i is equal to the product of their Segre classes. Given a collection of cohomology classes α i ∈ H * (B i ), the degree of the product i α i is equal to the evaluation on

B i of α i • s( C i ).
The latter, in turn, is equal to the product of evaluations of α i • s(C i ) on B i . In our case, the cones C i are the multisingularity strata X

(i) and one trivial one over M |J|+ℓ . The classes α i are the monomials in ψ-classes.

We conclude that the recursion equality is obtained by multiplying the cohomological relation 3 by the monomial

ψ m 1 1 • • • ψ mr r
and taking the degrees of all classes. This completes the proof of Theorem 1.7.

A reduced form of the recursion

Let us set p i = t i,0 (i ≥ 1) and t j = t 0,j (j ≥ 0). The variables p 1 , p 2 , . . . ; t 0 , t 1 , . . . form the reduced set of the t-variables. Consider the restriction homomorphism Q[t 0,0 , t 1,0 , t 0,1 , . . . ] -→ Q[p 1 , p 2 , . . . ; t 0 , t 1 , . . . ], t i,0 → p i (i ≥ 1), t 0,j → t j , t i,j → 0 (i, j ≥ 1).

Denote by X and X s,m the image of the descendant Hurwitz potential X and its partial derivative ∂ ts,m X, respectively, under the reduction. Then one can observe that the collection of functions X and X s,m , s, m ≥ 0, is closed with respect to the recursion of Theorem 1.7.

Theorem 4.1 The functions X s,m considered as formal power series in the variables p 1 , p 2 , . . . ; t 0 , t 1 , . . . obey the following equations

∂ ps X = X s,0 (s ≥ 1), X s+1,m = X s,m + s X s,m+1 - ℓ≥1, a≥s 1 ℓ! ∂Ψ a-s,ℓ ∂t m σ 1 +•••+σ ℓ =a ℓ i=1 σ i X σ i ,0 (m, s ≥ 0), X 0,m = ∂ tm X (m ≥ 0),
where

Ψ a,ℓ = k,j≥0     1 j! λ 1 +•••+λ j =a λ i ≥1 p λ 1 . . . p λ j        1 k! ν 1 +•••+ν k =ℓ+k+j-3 ν i ≥0 |ν| ν 1 , . . . , ν k t ν 1 . . . t ν k   
Using the reduced set of variables means that, for the Hurwitz spaces H r|κ under consideration, we associate both degeneracy conditions and the ψ-classes to only one of the supplementary marked points, say, to the first one, while to the other supplementary marked points we associate either degeneracy conditions or the ψclasses, but not both.

Applying relations of this theorem repeatedly, one expresses each partial derivative ∂ ps X = X s,0 in the direction of the p-variables as a power series in the variables t i , p j , and the partial derivatives ∂ tm X = X 0,m in the direction of the t-variables. Therefore, these relations determine the series X uniquely from the initial conditions

X p j =0 = r≥0 1 r! ν 1 ,...,νr |ν| ν 1 , . . . , ν r z |ν|,r t ν 1 . . . t νr ,
where z d,r = z d,r (q) is the same as in Proposition 1.5. Theorem 4.1 provides a reduced and simplified form of the recursion sufficient for an independent computation of the Hurwitz numbers of Proposition 1.3. Some further reduction is discussed in the next section.

String and dilaton equations

A study of the forgetful map π : H r+1|κ → H r|κ given by forgetting the last supplementary marked point leads to partial differential equations called the string and dilaton equations. They express the partial derivatives ∂ t 0,0 X and ∂ t 0,1 X, respectively, in terms of partial derivatives in other variables.

Theorem 5. [START_REF] Albarello | Geometry of algebraic curves[END_REF] The descendant genus zero Hurwitz potential X(t, q) obeys the following string and dilaton equations

∂X ∂t 0,0 = λ,d≥0 t λ,d+1 ∂X ∂t λ,d + k≥1 k q k ∂X ∂q k , ∂X ∂t 0,1 = λ,d≥0 t λ,d ∂X ∂t λ,d + k≥1 q k ∂X ∂q k -2 X.
Proof. Denote the marked poles of f by p 1 , . . . , p n , and the supplementary marked points by x 1 , . . . , x r or x 1 , . . . , x r+1 . Consider the locus δ i ⊂ H r+1|κ composed by stable maps f : C → CP 1 of the following form. The source curve C has n disjoint irreducible components C 1 , . . . , C n containing the marked points p 1 , . . . , p n respectively; the restriction of f to C 1 , . . . , C n has the form z → z k 1 , . . . , z kn ; these n components are connected by a union X of one or several irreducible components on which the map f vanishes identically; the subcurve X contains the supplementary marked points x 1 , . . . , x r ; the marked point x r+1 belongs to the component C i . The reason we have introduced the loci δ i is that the forgetful map π maps them to the zero section of the cone H r|κ . Thus the forgetful map P H r+1|κ → P H r|κ is not defined on P δ i . To construct a well-defined forgetful map we introduce the blow-up P H r+1|κ of P H r+1|κ along the loci P δ i , and denote by ∆ i the exceptional divisor corresponding to P δ i . Thus we obtain the following commutative diagram. Note that the locus δ i is isomorphic to the quotient orbifold

M 0,r+n / Z/k 1 Z × • • • × Z/k i Z × • • • × Z/k n Z
with the trivial group action, where the hat means a skipped factor. The restriction of p r to the corresponding exceptional divisor ∆ i is a degree k i map onto P H r|κ .

We will work in the blown-up space P H r+1|κ . Denote by ψ 1 , . . . , ψ r+1 the pullbacks of the ψ-classes from M g,n+r+1 at the supplementary marked points to P H r+1|κ . Similarly, denote by ψ 1 , . . . , ψ r the pull-backs of the ψ-classes from M g,n+r at the supplementary marked points to P H r+1|κ . Further, denote by ξ the pull-back of the ξ-class from P H r+1|κ to P H r+1|κ and by ξ the pull-back of the ξ-class from P H r|κ to P H r+1|κ . Finally, denote by D j , 1 ≤ j ≤ r, the pull-back from M g,n+r+1 of the divisor of curves of the form

D j = x j x r+1
.

Note that the restriction of p r to D j is an isomoprhism onto P H r|κ .

Lemma 5.2 The following relations hold in H * ( P H r+1|κ ):

a) D j D j ′ = 0 for j = j ′ ; b) ∆ i D j = 0 for 1 ≤ i ≤ n, 1 ≤ j ≤ r; c) ∆ i ∆ i ′ = 0 for i = i ′ ; d) ξ∆ i = 0 for 1 ≤ i ≤ n; e) ψ j D j = 0 for 1 ≤ j ≤ r; f ) ψ r+1 D j = 0 for 1 ≤ j ≤ r; g) ψ r+1 ∆ i = 0 for 1 ≤ i ≤ n. h) ξ = ξ + n i=1 ∆ i ; i) ψ j = ψ j + D j for 1 ≤ j ≤ r;
Proof. Relations (e), (f), and (i) are pull-backs of the well-known analogous relations from M g,n+r+1 . Relations (a), (b), and (c) follow from the fact that the geometric intersections of the corresponding divisors are empty. Relations (d) and (g) express the vanishing of the intersection between the exceptional divisor of a blow-up and a pull-back class from the base. Relation (h) is slightly harder. The line bundles p * r+1 (O(1)) and p * r (O(1)) are naturally identified except over the exceptional divisors ∆ i . So we have ξ = ξ + c i ∆ i for some coefficients c i . Now, the normal line bundle to the exceptional divisor ∆ i is identified with O(-1). Thus we have the equation ξ∆ i = 0 = ξ∆ i -c i ξ∆ i . It follows that c i = 1 for every i.

It follows from the relations that

ψ d j = ψ d j + D j ψ d-1 , ξ d j = ξ d j + n i=1 ∆ i ξ d-1 .
This can be easily proved by induction on d. Now, consider a stratum P X λ 1 ,λ 2 ,...,λr ⊂ P H r|κ , and the stratum P X λ 1 ,λ 2 ,...,λr,0 ⊂ P H r+1|κ obtained by adding one supplementary marked point with λ r+1 = 0. The preimage p -1 r (P X λ 1 ,λ 2 ,...,λr ) is the proper transform of P X λ 1 ,λ 2 ,...,λr,0 under the blowup p r+1 . For shortness, we will denote the proper transform by X and the strata by P X r and P X r+1 , respectively. Now we can prove the dilaton relation. The integral

P X r+1 ξ d ψ d 1 1 . . . ψ dr r ψ r+1 ,
where d is determined by degree reasons, is the coefficient of r j=1 t λ j ,d j n i=1 q k i |Aut {(λ j , d j )}| • |Aut {k i }| in the power series ∂X/∂t 0,1 . Applying the projection formula to p r+1 we rewrite the integral as

X ξ d ψ d 1 1 . . . ψ dr r ψ r+1 .
Now we use the relations from Lemma 5.2:

P X r+1 ξ d ψ d 1 1 . . . ψ dr r ψ r+1 = X ξ d + k i ∆ i ξ d-1 ( ψ d 1 1 + D 1 ψ d 1 -1 1 ) . . . ( ψ dr r + D r ψ dr-1 r )ψ r+1 = X ξ d ψ d 1 1 ψ dr r ψ r+1 .
Applying the projection formula to the map p r , we get (n + r -2)

P X r ξ d ψ d 1 1 ψ dr r .
This is the right-hand side of the dilaton relation, where the multiplication of the coefficient of X by n + r -2 is realized by the differential operator λ,d≥0

t λ,d ∂ ∂t λ,d + k≥1 q k ∂ ∂q k -2.
Thus we have proved the dilaton relation. We now prove the string relation in a similar way. The integral

P X r+1 ξ d ψ d 1 1 . . . ψ dr r ,
where d is determined by the degree argument, is the coefficient of

r j=1 t λ j ,d j n i=1 q k i |Aut {(λ j , d j )}| • |Aut {k i }|
in the power series ∂X/∂t 0,0 . Applying the projection formula to p r+1 we rewrite the integral as

X ξ d ψ d 1 1 . . . ψ dr r .
Now we use the relations from Lemma 5.2:

P X r+1 ξ d ψ d 1 1 . . . ψ dr r = X ξ d + k i ∆ i ξ d-1 ( ψ d 1 1 + D 1 ψ d 1 -1 1 ) . . . ( ψ dr r + D r ψ dr-1 r ) = n i=1 ∆ i ξ d-1 ψ d 1 1 . . . ψ dr r + r j=1 D j ξ d ψ d 1 1 . . . ψ d j -1 j . . . ψ dr r .
Applying the projection formula to the map p r , and recalling that p r | D j has degree 1, while p r | ∆ i has degree k i , we get n i=1

k i P X r ξ d-1 ψ d 1 1 . . . ψ dr r + r j=1 P X r ξ d ψ d 1 1 . . . ψ d j -1 j . . . ψ dr r .
This is the right-hand side of the string relation.

The string and dilaton equations provide a further reduction of the recursion of Theorem 4.1. Namely, they allow one to eliminate the variables t 0 = t 0,0 and t 1 = t 0,1 from consideration by setting them to be equal to zero.

Note that the q-derivatives of the right-hand sides of these equations preserve the ring generated by the functions z d,r (q) of Proposition 1.5. Namely, their action on the functions z d,r is given by the following equations

k≥1 k q k ∂ ∂q k z d,r = z d,r+1 -z d-1,r , k≥1 q k ∂ ∂q k z d,r = (d + 1) z d+1,r+1 + (2 -r) z d,r . (5)
These relations allow one to simplify manipulations with the potentials X and X by keeping the Taylor coefficients of their t-expansions as polynomials in the functions z d,r (q). The proof of ( 5) is obvious: these relations are equivalent to the following identities between the binomial coefficients:

n + r -3 d = n + r + 1 -3 d - n + r -3 d -1 , n n + r -3 d = (d + 1) n + r + 1 -3 d + 1 + (2 -r) n + r -3 d .
We conclude this section with a simple remark. Since the coefficients of the series Ψ a,ℓ of Remark 1.6 are intersection numbers of ψ-classes over genus 0 moduli spaces, these series also satisfy a string and a dilaton equations. Proposition 5.3 The series Ψ a,ℓ of Theorem 1.7 obey the following string and dilaton equations

∂Ψ a,ℓ ∂t 0,0 = Ψ a,ℓ+1 + i,j≥0 t i,j+1 ∂Ψ a,ℓ ∂t i,j , ∂Ψ a,ℓ ∂t 0,1 = i,j≥0 t i,j ∂Ψ a,ℓ ∂t i,j + (ℓ -2) Ψ a,ℓ .

Computing residual polynomials

The goal of this section is to relate the statements of Theorems 1.7 and 3.3 to the general multisingularity principle. Let F : X → Y be a generic finite holomorphic mapping of two compact complex manifolds of the same dimension. The mapping is degenerate at a point x ∈ X if dF (x) is a degenerate linear mapping of the tangent spaces at x and F (x), respectively. For a given singularity type σ, denote by Σ σ (X) the closure of the subset of points in X where F attains a singularity of type σ. The Thom principle states that the cohomology class [Σ σ (X)] ∈ H * (X) Poincaré dual to the subvariety Σ σ (X) admits a universal expression as a polynomial in relative Chern classes c 0 (F ), c 1 (F ), . . . of the mapping F defined by the series expansion

c(F ) = c(F * (T Y )) c(T X) = c 0 (F ) + c 1 (F ) + . . . , c i (F ) ∈ H 2i (X).
The polynomial is quasihomogenous, of degree equal to the codimension of Σ σ (X). This principle has been extended to the case of multisingularities by M. Kazarian [START_REF] Kazarian | Multisingularities, cobordisms, and enumerative geometry[END_REF]. Namely, let σ 1 , . . . , σ m be a set of singularity types. Denote by Σ σ 1 ,...,σm (Y ) the closure of the locus of points in Y , whose preimages contain singularities of types σ 1 , . . . , σ m . Let [Σ σ 1 ,...,σm (Y )] ∈ H * (Y ) be the corresponding Poincaré dual cohomology class. Then there are universal polynomials R σ 1 ,...,σm in the classes c i (F ) such that the generating function [Σ σ 1 ,...,σm (Y )]t σ 1 . . . t σm , where the summation is carried over all tuples of singularity types, is the exponent exp F * R of the pushforward, under F , of the generating function R = R σ 1 ,...,σm t σ 1 . . . t σm .

We shall apply the latter principle to the case of families of meromorphic functions; its applicability in this case has been proved in [START_REF] Kazarian | Thom polynomials for mappings of curves with isolated singularities[END_REF]. Let H = H κ be one of the Hurwitz spaces studied in this paper. The points of this space represent certain meromorphic functions. This family of functions fits into a diagram of mappings

X F / / P ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ Y Q ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧

H

Here X is the universal curve over H that can be identified with H 1|κ in the notation of the present paper, and Y = H × CP 1 . The multiplicative group C * of nonzero complex numbers acts on the space H by multiplying functions by constants. This action extends naturally to the spaces X and Y . It preserves the singularities and their types. Denote the C * -quotient spaces by P H, X, and Y , respectively, and the corresponding quotient mappings by P : X → P H, Q : Y → H, F : X → Y , so that we have a commutative diagram of the form

X F / / P ! ! ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ Y Q } } ④ ④ ④ ④ ④ ④ ④ ④ P H
We say that F has a singularity of type A k-1 at a point x ∈ X if x is a nonsingular point of the fiber of P passing through x, and the restriction of F to this fiber admits, in an appropriate local coordinate z in the fiber of P at x and an appropriate local coordinate in the fiber of Q at F (x) the form z → z k . Since multiplication of a function by a nonzero constant does not change the type of a singularity at a point, the same definition remains valid for a point x ∈ X. The shift by 1 in the notation of the singularity type is due to the tradition coming from singularity theory.

A k-1 singularities are the only possible isolated singularity types at smooth points of the fibers of P . Since the singularity types A k-1 , k = 1, 2, 3, . . . are indexed by integer numbers, the multisingularity classes are labeled by tuples of integers: the locus of functions with ramifications of orders ν 1 , . . . , ν r over one point is denoted by Σ ν 1 ,...,νr (Y ). Respectively, the generating function R for the corresponding residual polynomials can be considered as a function in formal variables t 1 , t 2 , . . . , where the variable t k is in charge of the singularity type A k-1 ; in particular, t 1 corresponds to nonsingular points of the mapping F .

If the fibers of P are smooth, the universal polynomials are computed in [START_REF] Kazarian | Thom polynomials for mappings of curves with isolated singularities[END_REF]. We reproduce these computations below. Unfortunately, these computations do not provide an answer in the general case when the fibers of P are allowed to be singular. Moreover, the whole multisingularity formula is not applicable in its straightforward form since the genericity conditions required for its applicability break down in the presence of nonisolated singularities. Thus, the formulas obtained in [START_REF] Kazarian | Thom polynomials for mappings of curves with isolated singularities[END_REF] should be corrected by terms supported on the loci of nonisolated singularities.

Observe that the class of the multisingularity locus Σ ν 1 ,...,νr (Y ) can also be computed by pushing forward the class of the subvariety X(ν 1 , . . . , ν r ) ⊂ P H r|κ discussed in the previous sections of the present paper under the natural forgetful map P H r|κ → P H 1|κ → P H κ . In particular, our inductive computation of the classes [X(ν 1 , . . . , ν r )] undertaken in this paper implies implicitly the computation of all necessary correction terms for the classes R ν 1 ,...,νr participating in the general multisingularity formula. However, the explicit computation of the pushforward homomorphism is still to be done.

We show below that the part of the series R corresponding to the contribution of the smooth fibers of P satisfies the equations of the Kadomtsev-Petviashvili (KP) integrable hierarchy of partial differential equations. One can hope that the corresponding enriched generating functions involving the contribution of nonisolated singularities also are solutions to suitable integrable hierarchies.

It is proved in [START_REF] Kazarian | Thom polynomials for mappings of curves with isolated singularities[END_REF] that in the case where the fibers of P are smooth, all the relative Chern classes of F can be expressed as universal polynomials in just two classes, ξ, ψ ∈ H 2 (X). Namely, the class ψ = ψ 1 is the ψ class associated with the unique supplementary marked point on X = P H 1|κ . The class -ψ can be defined also as the relative first Chern class of the fibration P : X → P H. The class ξ is the first Chern class c 1 (O(1)), where the line bundle O(1) is inherited from the C * -action on X. Hence, we define R(ψ, ξ; t

∂ 2 R ∂t 2 2 = 2ψξ ∂ 2 R ∂t 2 1 2 + 4 3
∂ 2 R ∂t 1 ∂t 3 - 1 3 ψ 2 ∂ 4 R ∂t 4 1 .
The scaled KP equations are obtained from the ordinary ones by applying to them the following scaling: a partial derivation of order k is replaced by ξ(-ψ) k-1 times the same derivation. Since all the partial derivatives in the KP equations are of order at least 2, the scaled equations are divisible by ξψ, and we simplify them by dividing by this monomial.

Proof. All the solutions of the KP equations are known to be logarithms of tau functions for the KP-hierarchy. The tau function exp(-ξ ψ RA) is, in fact, the following one: exp(-ξ ψ RA) = 1 + ξs 1 + ξ(ξ + ψ)s 2 + ξ(ξ + ψ)(ξ + 2ψ)s 3 . . . ,

where sk is the kth scaled one-part Schur polynomial (written in the variables t i ), which are the homogeneous parts of the decomposition e -1 ψ (t 1 +t 2 +t 3 +... ) = s0 + s1 + s2 + . . .

As is well known, an arbitrary linear combination of one-part Schur polynomials is a tau function.

In order to check that these formulas indeed produce the correct universal expressions for the residual polynomials, it suffices to verify them on a sufficiently rich bunch of examples of Hurwitz spaces. For such a bunch, consisting of the spaces of polynomials (versal unfoldings of the singularities A k , k = 1, 2, . . . ) the verification has been done in [START_REF] Kazarian | Thom polynomials for mappings of curves with isolated singularities[END_REF].

p i x n+1

  1 , t 2 , . . . ) = R i 1 ,i 2 ,... (ψ, ξ)t i 1 t i 2 . . . . -2(ξ + ψ)t 1 t 2 + (ξ + ψ)(ξ + 2ψ)t 3 + . . . . Theorem 6.1 For a generic family of functions on smooth curves, the generating function R of residual polynomials is a solution to the scaled KP hierarchy of partial differential equations. In particular, it solves the first scaled KP equation of the form

	The first few terms of the function R are
	R = t 1 + -	1 2	t 2 1 + (ξ + ψ)t 2 +	1 3	t 3 1


Miriam Beneito-Cambra

Pierre Gareil

Bernard Badet

Marie-Ange Badet-Denisot

Nathalie Delaunay
email: nathalie.delaunay@espci.fr

First investigations for the characterization of glucosamine-6phosphate synthase by capillary electrophoresis

Keywords: capillary electrophoresis, enzyme, glucosamine-6-phosphate synthase, inhibitor screening ADGP: 2-amino-2-deoxy-D-glucitol 6-phosphate, A5PO: arabinose-5-phosphate oxime, background electrolyte, BTP: Bis Tris Propane, DTT: dithiothreitol,

The enzyme glucosamine-6-phosphate synthase (GlmS) is an important point of metabolic control in biosynthesis of amino sugar-containing macromolecules and is therefore a potential target in order to design antibacterial and antifungal drugs. It has two oligomerization states, which are the active dimer and the inactive hexamer. For the first time, the potential of CE to separate and quantify the two forms was studied. After incubating GlmS with the D-glucosamine 6-phosphate (GlcN6P) inhibitor, an electrolyte based on sodium phosphate at pH 7.2 and an ionic strength of 100 mM plus GlcN6P (either 2 or 20 mM) allowed the hexamer-dimer separation. However, the displacement of the dimer/hexamer equilibrium during the analysis time prevented any improvement of the resolution when varying the effective separation length or the temperature of the analysis. Therefore, the use of a short-end CE method allowed the decrease in the analysis time to about 1 min. Some parameters such as the temperature and the time of incubation and the ratio of the inhibitor and enzyme concentrations were studied. Then, it was also possible to test, very rapidly and with a very small amount, some molecules having an inhibition potential for the GlmS enzyme (arabinose-5phosphate oxime, 2-amino-2-deoxy-D-glucitol 6-phosphate, and glucose-6-phosphate).

Introduction

The enzyme glucosamine-6-phosphate synthase (GlmS) is an important point of metabolic control in the biosynthesis of amino sugar-containing macromolecules [START_REF] Milewski | Glucosamine-6-phosphate synthasethe multi-facets enzyme[END_REF][START_REF] Durand | Highlights of glucosamine-6P synthase catalysis[END_REF].

Indeed, it catalyzes the first step of the hexosamine metabolism since it irreversibly converts D-fructose-6-phosphate into D-glucosamine 6-phosphate (GlcN6P) using Lglutamine as a nitrogen source. The end product, uridine diphosphate Nacetylglucosamine-6-phosphate, is an essential building block of bacterial and fungal cell walls. Therefore, GlmS is a potential target in the design of antibacterial and antifungal drugs [START_REF] Milewski | Glucosamine-6-phosphate synthasethe multi-facets enzyme[END_REF][START_REF] Durand | Highlights of glucosamine-6P synthase catalysis[END_REF][START_REF] Flocquet | Discovering new inhibitors of bacterial glucosamine-6P synthase (GlmS) by docking simulations[END_REF], which is a major issue because invasive fungal infections have increased dramatically over the past two decades and are associated with an increase in morbidity and mortality [START_REF] Ramakrishnan | Review on fungal enzyme inhibitorspotential drug targets to manage human fungal infections[END_REF]. Moreover, as GlmS leads to compounds belonging to glycosylation pathways potentially implicated in the regulation of glucose, it is also a therapeutic target to prevent from complications in type-2 diabetes [START_REF] Milewski | Glucosamine-6-phosphate synthasethe multi-facets enzyme[END_REF][START_REF] Durand | Highlights of glucosamine-6P synthase catalysis[END_REF].

The active form of GlmS from Escherichia coli is a dimeric assembly of two identical monomers, each consisting of a 27 kDa glutaminase domain and a 40 kDa synthase domain [START_REF] Milewski | Glucosamine-6-phosphate synthasethe multi-facets enzyme[END_REF][START_REF] Durand | Highlights of glucosamine-6P synthase catalysis[END_REF]. A study using size exclusion chromatography, dynamic and quasi-elastic light scattering, native polyacrylamide gel electrophoresis, and ultracentrifugation demonstrated that the dimer is in equilibrium with a hexameric form, in vitro as well as in cellulo [START_REF] Mouilleron | Structural basis for morpheein-type allosteric regulation of Escherichia coli glucosamine-6-phosphate synthase -Equilibrium between inactive hexamer and active dimer[END_REF]. Moreover, the enzyme crystal structure studies allowed the identification of two enzyme conformations, which are the dimer and the hexamer. It was also observed that increasing the protein concentration results in a decrease in enzyme specific activity and that the presence of GlcN6P favours the hexameric form. Both effects strongly supported GlmS inhibition through hexamer stabilization, which would reflect a physiologically relevant allosteric regulation of E. coli enzyme. In addition to glucose 6-phosphate bound at the active site, the crystallographic hexameric organization of E. coli GlmS enables the binding of another linear sugar. Keeping this binding site occupied should then force the GlmS to remain in the inactive hexamer state, which should be useful for the development of specific inhibitors.

The product GlcN6P with weak inhibitory properties (K i = 0.38 mM) was the first studied GlmS inhibitor [START_REF] Milewski | Glucosamine-6-phosphate synthasethe multi-facets enzyme[END_REF]. Other reported inhibitors, such as arabinose-5-phosphate oxime (A5PO) (K i = 14 μM) and 2-amino-2-deoxy-D-glucitol 6-phosphate (ADGP) (K i = 25 μM), which exhibit better inhibition properties are good candidates for the hexamer stabilization [START_REF] Flocquet | Discovering new inhibitors of bacterial glucosamine-6P synthase (GlmS) by docking simulations[END_REF]. The potential use of GlmS as a therapeutic target is likely to explain the sustained interest in the synthesis of new GlmS inhibitors [START_REF] Pawlak | Synthesis and biological activity of novel ester derivatives of N 3 -4-metoxyfumaroyl)-(S)-2,3diaminopropanoic acid containing amide and keto function as inhibitors of glucosamine-6-phosphate synthase[END_REF][START_REF] Zong | Design, synthesis and bioactivity of novel glycosylthiadiazole derivatives[END_REF].

Different methods were described in the literature to determine the GlmS activity, such as the Morgan-Elson colorimetric assay, the continuous coupled assay or the radiometric assay employing [ 14 C]Fructose-6-phospate and involving an electrophoretic separation of formed [ 14 C]GlcN6P [START_REF] Milewski | Glucosamine-6-phosphate synthasethe multi-facets enzyme[END_REF]. They all involve several laborious steps. More recently, new alternatives were proposed such as isothermal titration calorimetry [START_REF] Valerio-Lepiniec | Analysis of the Escherichia coli glucosamine-6-phosphate synthase activity by isothermal titration calorimetry and differential scanning calorimetry[END_REF] and matrix-assisted laser desorption/ionization time-of-flight mass spectrometry [START_REF] Gaucher-Wieczorek | Evaluation of synthase and hemisynthase activities of glucosamine-6-phosphate synthase by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry[END_REF].

Separation techniques, such as high performance liquid chromatography (HPLC), could also be a valuable alternative. Indeed, HPLC allowed the study of the enzyme activity or the screening of inhibitors thanks to the separation of the substrates or the products followed by spectrophotometry [START_REF] Ma | On-line chromatographic screening of matrix metalloproteinase inhibitors using immobilized MMP-9 enzyme reactor[END_REF][START_REF] Chen | HPLC-UV method for determining phosphorylated peptide and for Abl1 tyrosine kinase inhibition study[END_REF] or mass spectrometry detection [START_REF] Greis | Mass spectrometry for enzyme assays and inhibitor screening: an emerging application in pharmaceutical research[END_REF].

In the last two decades, capillary electrophoresis (CE) has become a powerful technique for the study of enzymes [START_REF] Bao | Determination of minute enzymatic activities by means of capillary electrophoretic techniques[END_REF][START_REF] Glatz | Determination of enzymatic activity by capillary electrophoresis[END_REF][START_REF] Fan | Advances in-capillary electrophoretic enzyme assays[END_REF][START_REF] Wang | Recent advances in CE-mediated microanalysis for enzyme study[END_REF][START_REF] Liu | Application of capillary electrophoresis in enzyme inhibitors screening[END_REF]. Indeed, CE is a powerful miniaturized separation technique that combines a short analysis time, high resolutions, a low cost, and a low sample and reagent consumption. In this study, for the first time, the potential of CE is evaluated for the measurement of the GlmS activity via the simultaneous determination of its active dimer and inactive hexamer forms after a pre-incubation with an inhibitor.

The commercially available GlcN6P was used to develop the CE separation, as it was previously observed that the addition of 20 mM of GlcN6P to a 15 µM GlmS solution (about 1 mg mL -1 ) led to an equilibrium between the dimer and the hexamer [START_REF] Mouilleron | Structural basis for morpheein-type allosteric regulation of Escherichia coli glucosamine-6-phosphate synthase -Equilibrium between inactive hexamer and active dimer[END_REF].

GlcN6P was added to the GlmS sample before injection in the CE capillary, but also in the background electrolyte (BGE) to maintain the equilibrium between both forms. The CE method was also used to investigate the effect of the incubation time, the incubation temperature, and the concentration ratio of inhibitor and GlmS on the dimer/hexamer ratio. Finally, the method was evaluated for inhibitor screening with A5PO, ADGP, and glucose-6-phosphate (Glc6P).

Materials and methods

Chemical and reagents

1 M NaOH solution was from VWR (Fontenay-sous-Bois, France). Acetone (99.9%), HEPES (99.5%), GlcN6P monobasic (99%), sodium dihydrogen phosphate (98%), sodium hydrogen phosphate (99.0%), NaCl, Bis Tris Propane (BTP), ethylenediaminetetraacetic acid (EDTA), dithiothreitol (DTT), D-arabinose-5-phosphate disodium, hydroxylamine, p-toluenesulfonic acid (PTSA), and Glc6P were purchased from Sigma-Aldrich (Saint-Quentin-Fallavier, France). Ultrapure (UP) water was delivered by a Direct-Q3 system (Millipore, Molsheim, France).

GlmS was purified from E. coli cells at 4°C with an anion exchange column (Q Sepharose) with a linear gradient from 0 to 1 M NaCl in a 20 mM BTP, 1 mM EDTA, and 1 mM DTT buffer at pH 7.2 [START_REF] Mouilleron | Glutamine binding opens the ammonia channel and activates glucosamine-6P synthase[END_REF]. The fractions containing GlmS were pooled, concentrated by ultrafiltration (30 kDa, Amicon-Ultra Centrifugal Filter Unit, Millipore) and purified with a Superdex 200 gel filtration column (Sigma-Aldrich) and a 50 mM HEPES buffer at pH 7.2 containing 1 mM of EDTA. The collected fractions were concentrated by ultrafiltration (30 kDa Amicon-Ultra). The purity of the isolated GlmS was checked by SDS-PAGE with a protocol already described [START_REF] Mouilleron | Glutamine binding opens the ammonia channel and activates glucosamine-6P synthase[END_REF]. A dialysis was carried out in order to reduce the concentrations of HEPES and NaCl. Finally, a stock solution of GlmS at 25.5 g L -1 (0.38 mM) in 50 mM HEPES and 50 mM NaCl buffer at pH 7.2 was obtained and kept at -20°C until use. A5PO was obtained from the reaction of the commercially available arabinose-5-phosphate with hydroxylamine [START_REF] Lecamus | Arabinose-5-phosphate oxime vs its methylenephosphonate mimetic as high energy intermediate of the glucosamine-6P synthase catalyzed reaction[END_REF]. ADGP was obtained by sodium borohydride reduction of GlcN6P [START_REF] Badet-Denisot | Nitrogen transfer in E. coli glucosamine-6P synthase. Investigations using substrate and bisubstrate analogs[END_REF].

Instrumentation

The CE experiments were carried out with an Agilent Technologies 7100 CE system (Massy, France) equipped with a photodiode array absorbance detector. The UV detector was set at 270 nm for the neutral marker (acetone) and between 192 and 200 nm for the GlmS enzyme. The data acquisition and the instrument control were carried out with the Chemstation software. The separations were performed with bare fused-silica capillaries (Photonlines, Marly-le-Roi, France) of 50 μm internal diameter and a total length of 35 or 58.5 cm (effective length 8.5, 26.5 or 50 cm, respectively).

Electrophoretic conditions

New capillaries were activated by successive flushes under ca. 1 bar with 1 M NaOH, 0.1 M NaOH, and water (10 min each, about 300 capillary volumes). Before each run, the capillary was rinsed with 0.1 M NaOH, UP water (5 min each one), and the appropriate BGE for 10 min. The appropriate amounts of GlcN6P, sodium dihydrogen phosphate, and sodium hydrogen phosphate were used to obtain the composition, pH, and ionic strength of the different tested BGEs. The GlcN6P pK a values of 6.2 and 8.2 were used to make the calculations. The solutions of GlmS at concentration of 0.05 mg/mL (0.76 μM), 0.13 g L -1 (1.9 μM), 0.25 g L -1 (3.8 μM) and 0.51 g L -1 (7.6 μM) were prepared by dilution of the GlmS stock solution in the appropriate BGE.

A dilution of acetone in the BGE (4%, v/v) was used as neutral marker. The solutions of GlmS and neutral marker were separately hydrodynamically injected, in two successive plugs but analyzed in the same run, to avoid any enzyme denaturation or other interference phenomenon.

Results and discussion

Development of the CE method

A 100 mM phosphate buffer at pH 7.2 was selected as BGE in order to study the activity of the GlmS enzyme with a physiological environment. At pH 7.2, GlmS is expected to be negatively charged, as its isoelectric point value is 5.6 [START_REF] Gonzalez-Ibarra | Sporothrix schenckii: purification and partial biochemical characterization of glucosmamine-6-phosphate synthase, a potential antifungal target[END_REF]. First, the GlmS electrophoretic profile was obtained with this phosphate buffer-based BGE, without adding the GlcN6P inhibitor in the sample and in the BGE. As shown in Figure 1a, 4 peaks appeared in the electropherogram. The migration time of the neutral marker (see the dotted electropherogram) is about 4 min. Consequently, the main peak with a migration time of 5.9 min corresponds to an anion, as expected for GlmS at pH 7.2.

Since the enzyme is initially present in the dimer form [START_REF] Milewski | Glucosamine-6-phosphate synthasethe multi-facets enzyme[END_REF][START_REF] Durand | Highlights of glucosamine-6P synthase catalysis[END_REF], the main peak was assigned to the GlmS dimer. The first two peaks with migration times of about 4 min, corresponding to the migration time of the neutral marker, may correspond to neutral impurities present in the enzyme solution. The peak with a migration time of 4.8 min was assigned to HEPES, which is present in the GlmS stock solution. This hypothesis was confirmed by the CE analysis of an aliquot of 0.3 mM HEPES diluted in the phosphate-based BGE (data not shown). In order to obtain the GlmS hexamer form, the enzyme was incubated for a given time in a vial containing the phosphate-based BGE plus 20 mM of GlcN6P and next injected in the CE system, using the incubation media as BGE to maintain the hexamer forms during the analysis [START_REF] De Kort | Potential of capillary electrophoresis with wavelength-resolved fluorescence detection for protein unfolding studies using βlactoglobulin B as a test compound[END_REF]. As can be seen Figure 1b, the presence of GlcN6P led to the appearance of an additional peak with a slightly higher migration time than that of the dimer peak. In addition, the intensity of this new peak increases with the incubation time, while the peak assigned to the dimer decreases. The UV spectrum of this new peak was recorded and it was similar to the spectrum obtained for the dimer (data not shown). Accordingly this new peak was assigned to the hexamer form.

A higher migration time for the hexamer GlmS form than for the dimer form is consistent with the expected migration order due to an increase in charge to hydrodynamic size ratio from the dimer to the hexamer form due to a higher compactness, leading to a slight increase in the electrophoretic mobility (absolute value)

for the hexamer. In the counter-electroosmotic flow mode, this resulted in an increase in the migration time.

Optimization of the dimer-hexamer resolution

In order to improve the resolution between the dimer and the hexamer peaks, the influence of the BGE pH was studied (Figure 2a). This modification can affect both electrophoretic (conformation and charge of the enzyme) and electroosmotic mobilities.

When the pH increased from 7.2 to 8.2, an increase in the electroosmotic mobility was observed as expected, which induced a decrease in the migration times of the dimer and hexamer forms, despite the increase of the negative charge of the enzyme that would contribute to an increase in the absolute value of the electrophoretic mobility.

Unfortunately, a pH increase from 7.2 to 8.2 resulted in a gradual decrease in the dimerhexamer resolution. When a BGE with a pH of 6.2 was tested, there was a marked decrease in the peak intensities of the enzyme (data not shown), may be due to a partial precipitation of the enzyme at a pH near its pI value. The BGE pH was therefore set at 7.2 for the other experiments of the study. The ionic strength of the BGE can also affect the dimer-hexamer separation, since it modifies the GlmS effective charge, its potential adsorption to the capillary wall, and the electroosmotic mobility. Figure 2b shows the obtained electrophoretic profiles using the BGEs with an ionic strength of 50, 100, and 150 mM. Table 1 presents the resulting electroosmotic and electrophoretic mobilities, the effective selectivity, the resolution, and the efficiency for the dimer and hexamer peaks. As expected, when the separation was performed with a BGE having a higher ionic strength, an increase in migration times was observed due to a significant decrease in the electroosmotic mobility.

Considering the absolute value of the electrophoretic mobility of the dimer, a decrease is observed when the ionic strength increases from 50 to 150 mM. This behavior may be explained by an increase in the condensation of cations on the surface of the enzyme, which results in a decrease in its effective charge [START_REF] Cordechi | Net proton charge of βand κcasein in concentrated aqueous electrolyte solutions[END_REF]. Both led to an improved resolution between the GlmS dimer and hexamer peaks when the BGE ionic strength increased, to reach a maximum around 100 mM. This ionic strength value was selected for the rest of the study, since the effective selectivity and the efficiencies are the same at 100 or 150 mM, but a lower ionic strength contributes to reduce the Joule heating and 100 mM provides maximum resolution. It is worthwhile to notice that for each tested condition in this manuscript, the voltage value was set in order to stay in the area of a linear increase in current with voltage. In that case, the rise in temperature in the capillary is not detrimental and a maximum difference value of 5°C was calculated between the set value and the temperature existing in the center of the capillary [START_REF] Lauer | High performance capillary electrophoresis, Agilent Technologies, A Primer[END_REF]. The separation temperature was reduced to 15°C, in an attempt to improve the oligomer resolution, but no change in resolution was obtained (data not shown). An increase in the effective capillary length from 26.5 to 50 cm was next tested, keeping constant the other parameters. These two lengths led to analysis time of 7 and 27 min, respectively. This significant increase in migration times results both from an increase in the effective length and a decrease in the electrical field, as the voltage was kept constant. As can be observed in Table 2, the efficiencies increased with the effective capillary length as it is expected by the theory. However, the resolution between the two peaks was not improved. Antagonistic effects between the separation of the dimer and the hexamer along the capillary thanks to their different electrophoretic mobilities and the displacement of the equilibrium between the two forms may explain this particular behavior. To study this hypothesis, the effect of the analysis time on the ratio of the corrected areas of the hexamer and dimer peaks was investigated. As can be seen in Figure 3, the proportion of the hexamer form increased with the incubation time and reached a plateau after several tens of minutes. However, the analysis time led to significant different values of the ratio of the two forms, whatever the pre-incubation time of the sample before the injection in the capillary: the longer the analysis time, the higher the ratio of hexamer to dimer. This behavior is consistent with a displacement of the dimerhexamer equilibrium during the electrophoretic separation, despite the presence of the inhibitor in the BGE since the same solution is used for the CE analysis and the incubation step. Considering the increase in the hexamer proportion with the incubation time and the analysis time, which can be considered as an extension of the incubation time as the inhibitor is also present in the BGE, we can suggest that the kinetics of the hexamer formation is faster than its dissociation into dimer. To reinforce this hypothesis, a sample incubated for 50 min at room temperature was also analyzed with the longer capillary (50 cm effective length) at a higher temperature of analysis (35°C),

but not so different from the incubation temperature value (room temperature, therefore about 10°C) and during a limited time (17 vs 50 min). In that case, the analysis time was reduced to 17 min and the ratio of the hexamer/dimer corrected areas was reduced as expected, from 1.9 to 0.7. The meaning of the error bar is the highest and the lowest obtained values. For other conditions, see Table 2.

With this first CE method, other characterizations of the enzyme-inhibitor system were carried out. As an example, using a GlcN6P inhibitor concentration fixed at 20 mM, the concentration of the GlmS enzyme was varied from 0.76 to 3.8 μM, which corresponds to an inhibitor-enzyme molar ratio between 26,300 and 5,300. Figure 4a shows the ratio of the corrected areas of hexamer to dimer peaks vs the incubation time for the different ratios of the inhibitor to enzyme concentrations. The proportion of the inactive hexamer form increases with the concentration of the dimer, which is consistent with the mass action law. 2. Each condition was tested 2 and 3 times (incubation + electrophoretic analysis) in a and b, respectively. The meaning of the error bar is the highest and the lowest obtained values.

The incubation temperature was also studied: two aliquots of the GlmS enzyme were incubated with GlcN6P at 4 and 25°C, and injected at different incubation times (Figure 4b). In both cases, if the hexamer/dimer ratio increases with the incubation time, the low incubation temperature seems to favor the displacement of the equilibrium to the inactive hexamer form. We can notice that the equilibrium between the dimer and hexamer forms is temperature-dependent and that some Joule heating during the separation occurs, but it may have a little impact on it since, as already said, a maximum difference value of 5°C was calculated between the set value and the temperature existing in the center of the capillary [START_REF] Lauer | High performance capillary electrophoresis, Agilent Technologies, A Primer[END_REF] and, in most cases, the value is even below 2°C.

Short-end CE method

Since the time scale of the dimer-hexamer rearrangement kinetics is close to the analysis times previously used (7 and 27 min), a reduction of the analysis time was searched using an injection by the short-capillary end to reduce the effective length to 8.5 cm. Using this CE configuration, the analysis time was reduced to less than 2.5 min (data not shown). The resolution between the dimer and hexamer peaks was slightly improved to a value of 1.2, which confirms the hypothesis of a displacement of the dimer-hexamer equilibrium during the CE separation. It is worthwhile to notice that the injected volume was reduced to 2 nL (30 mbar for 2 s) keeping a non-significant contribution of the injection step to the peak band broadening, since the injected plug length was about 1% of the effective separation length.

To speed up further the analysis, the temperature was increased from 15 to 25°C and the voltage was increased as much as possible while preserving the resolution to a value of 15 kV. These conditions allowed a decrease in the analysis time to 1.2 min (Figure 5).

The UV spectra of the peaks attributed to the dimer and the hexamer had a maximum of absorption around 200 nm. The detection wavelength was next optimized and an improvement of the signal to noise ratio of about 15% was obtained using 192 nm instead of 200 nm. Using the short-end approach, the influence of the concentration of the GlcN6P inhibitor on the ratio of the corrected areas of the hexamer to dimer peaks was studied with an incubation at 4°C (Figure 6). A decrease in the GlcN6P concentration from 20 to 2 mM led to a smaller conversion into the inactive hexamer form, i. e. a decrease in the inhibition effect, which agrees with the mass action law, since the GlcN6P is a product of the enzymatic reaction. At a given inhibitor concentration, as already previously observed in Figure 4b, a decrease in the incubation temperature induced an increase in the inhibition effect. Then, a smaller inhibitor effect has to be expected at physiological temperatures. It is important to note that here, even if the analysis temperature is different from the incubation temperature, it must not affect significantly the hexamer-dimer ratio as the analysis time is very short (1.2 min) compare to the incubation time (up to 300 min).

This method was finally used to test three inhibitors, ADGP, A5PO, and Glc6P, each at a concentration of 10 mM in the sample and in the phosphate-based BGE at pH 7.2 and with an incubation at 4°C, which correspond to values close to the values used in this study with GlcN6P. All electropherograms show only one peak. In order to make the most correct identification possible using the migration time and to take into account the small variations of the electroosmotic flow, an internal standard was used. PTSA was selected as it has a migration time of about 2 min, has a strong absorption in UV and does not affect the profile of the GlmS peaks whatever the tested inhibitor. Table 3 presents the average, the standard deviation, and the confidence interval values of the corrected electrophoretic mobilities of the dimer and the hexamer forms obtained with the introduction of 20 mM of GlcN6P in the sample and in the BGE. The corresponding values pertaining to the single peak obtained by replacing GlcN6P by another potential inhibitor are also given. If for ADGP, it is not possible to conclude which form was present without further experiments such as spiking, the other two inhibitors allow the formation of the inactive hexamer form. These preliminary results confirm the great potential of this approach for a rapid screening of potential inhibitors.

Conclusion

The GlmS enzyme is characterized by an allosteric regulation involving an equilibrium between two forms: the active dimer and the inactive hexamer. For the first time, it was demonstrated that the two forms can be separated by CE after introducing GlcN6P in the sample for pre-incubation and in the BGE. A sodium phosphate based BGE plus 20 mM GlcN6P at pH 7.2 and an ionic strength of 100 mM leads to the best apparent resolution between the hexamer and dimer forms. Nevertheless, the displacement of the dimer/hexamer equilibrium during the time frame of the analysis prevents any improvement of the resolution when varying the effective separation length or the analysis temperature. Therefore, the use of a short-end CE method allowed the decrease in the analysis time to about 1 min. Some parameters such as the temperature and the time of incubation and the ratio of the inhibitor and enzyme concentrations were studied. Then, it was also possible to test, very rapidly and using minute amounts, some molecules having an inhibition potential for the GlmS enzyme.

Figure 1 AbsorbanceFigure 1 :

 11 Figure 1

Figure 2 TimeFigure 2 :

 22 Figure 2

Figure 3 Figure 3 :

 33 Figure 3

Figure 4 :

 4 Figure 4

Figure 5 :

 5 Figure 5: Electropherogram of the GlmS enzyme (1.9 μM) obtained with a short-end injection. Electrophoretic conditions: bare-fused silica capillary, 50 μm × 35 cm (8.5 cm to the detection). BGE: sodium phosphate buffer at pH 7.2 plus 20 mM GlcN6P (100 mM ionic strength). GlmS was incubated in the BGE at 4°C for 30 min. Separation voltage: 15 kV. Analysis temperature: 25°C. Hydrodynamic injection: 30 mbar × 2 s. UV detection at 192 ± 4 nm.

Figure 6 :

 6 Figure 6: Plots of the ratio of the average corrected areas of the hexamer to the dimer peaks vs the incubation time for a concentration of the GlcN6P inhibitor of 2 mM or 20 mM with an incubation at room temperature or at 4°C (n=2). Diamonds, 2 mM GlcN6P and incubation at 4°C. Squares, 20 mM GlcN6P and incubation at 4°C. Triangles, 20 mM GlcN6P and incubation at room temperature. [GlmS] = 1.9 μM. Analytical conditions, see Figure 5.

Table 1 .

 1 Electroosmotic mobility (µ

		BGE ionic strength	50 mM 100 mM 150 mM
		µ EO (x 10 5 cm 2 V -1 s -1 )	41.5	38.7	36.4
		µ ep,dimer (x 10 5 cm 2 V -1 s -1 )	-14.7	-13.2	-12.8
		α (dimer -hexamer)	a			-	1.1	1.1
		Rs (dimer -hexamer)	b		0	1.0	0.8
		hexamer-dimer corrected area ratio	-	1.0	1.9
		N (dimer) c									-	8000	9000
		N (hexamer) c									-	11000	10000
	a	 		hexamer t		t	EOF	  /	dimer t		t	EOF		( i t = migration time)
	b	Rs		1.177			hexamer t		dimer t	  /  hexamer 	dimer		(  = width at half height peak)
	c	 Nt  i i 5.54 / 	i		2

EO ), electrophoretic mobility of the dimer (µ ep,dimer ), effective selectivity (α), resolution (Rs), hexamer-dimer corrected area ratio, and efficiency (N) for the dimer and hexamer peaks obtained at different ionic strengths of the BGE at pH 7.2. The aliquots of 0.76 μM GlmS were incubated at room temperature during 120 min. Other conditions, see Figure

1

.

Table 2 .

 2 Effective

	Effective capillary length (l)	26.5 cm	50 cm
	l/L a			0.76	0.85
	α (dimer -hexamer) a	1.1	1.1
	Rs (dimer -hexamer) b	1.1	1.0
	N (dimer)	c		8000	9000
	N (hexamer)	c	12000	17000

selectivity (α), resolution (Rs), and efficiency (N) for the dimer and hexamer peaks obtained with two different effective capillary lengths. BGE: sodium phosphate buffer plus 20 mM GlcN6P at pH 7.2 (100 mM ionic strength). Hydrodynamic injection: 30 mbar × 4 s or 80 mbar × 5 s for effective length capillaries of 26.5 and 50 cm, respectively. Separation voltage, 10 kV. Analysis temperature, 15°C. The aliquots of 0.76 μM GlmS were incubated in BGE at room temperature during 120 min. a L: total capillary length

Table 3. Average, standard deviation (SD), and interval values of the corrected electrophoretic mobilities (µ ep ) of the dimer and the hexamer, obtained with 20 mM of GlcN6P in the sample and in the BGE, or of the single peak obtained with 10 mM of ADGP, A5PO or Glc6P in the sample and in the BGE. [GlmS] = 1.9 μM. Incubation between 60 and 180 min, at 4°C. PTSA was used as internal standard at a concentration of 0.1 mM. Other conditions: see Figure 6.