Thomas Dedieu

Geometry of logarithmic Severi varieties at a general point

This is a set of notes based on the results

By a curve on S, we always mean a closed subscheme of S of pure dimension 1. The geometric genus of a reduced curve C is the (arithmetic) genus of its normalization C, namely

1 -χ(O C ) = n i=1 g i -n + 1
where g 1 , . . . , g n are the respective genera of the connected components C 1 , . . . , C n of C.

For every ξ ∈ NS(S) (i.e. ξ is a homology class on S that can be represented by a divisor) and integer g, we consider M ξ,bir g (S) the space parametrizing morphisms φ : C → S from a smooth genus g curve C (projective, but possibly disconnected) that are birational on their image, and such that φ * [C] = ξ.

We also consider V ξ g (S) the locally closed subscheme of Curve(S) consisting of those points [C] such that C is reduced and has geometric genus g and homology class ξ, where Curve(S) denotes the Hilbert scheme of curves on S.

(1.2) We denote by N the set of sequences α = [α 1 , α 2 , . . .] of non-negative integers with all but finitely many α i non-zero. In practice we shall omit the infintely many zeros at the end.

For α ∈ N, we let

|α| = α 1 + α 2 + • • • ; Iα = α 1 + 2α 2 + • • • + nα n + • • • .
For α, α ′ ∈ N, we say that α α ′ if α i α ′ i for all i 1. By a set Ω of cardinality α ∈ N, we mean a sequence of sets Ω = (Ω 1 , Ω 2 , . . .) such that each Ω i has cardinality α i .

(1.3) Definition. Let g ∈ Z, ξ ∈ NS(S), α, β ∈ N such that

Iα + Iβ = ξ • R,
and consider a general set Ω = {p i,j } 1 j αi i 1 of α points on R. 1We define M ξ g (α, β)(Ω) as the locally closed subset of M ξ,bir g (S) consisting of those [φ : C → S] such that the intersection φ(C) ∩ R is proper and contained in the smooth locus of R, and there exist α points q i,j ∈ C, 1 j α i , and β points r i,j ∈ C, 1 j β i , such that ∀ 1 j α i : φ(q i,j ) = p i,j and (1.3.1)

φ * R = 1 j αi i q i,j + 1 j βi i r i,j . (1.3.2)
Remark. The above definition is functorial. In other words, M ξ g (α, β)(Ω) represents a certain functor, see (2.10).

Correspondingly, we let V ξ g (α, β)(Ω) be the locally closed subscheme of V ξ g (S) consisting of those points [C] such that the normalisation ν : C → C ⊂ S belongs to M ξ g (α, β)(Ω). We call logarithmic Severi variety of the pair (S, R) its Zariski closure V ξ g (α, β)(Ω) scheme of curves on S.

(1.3.1) Notation, examples, and comments. In practice we will try to find a balance between rigorous and decipherable notation. For instance we will frequently drop the Ω, and replace ξ with an adequate shorthand.

Let us consider the emblematic case when S = P 2 and R is a line. Let ξ = d [H] with [H] the hyperplane class ; we shall simply denote ξ by d. For all g, V d g [0, 1], d -2 is the family of plane d-ics of genus g and tangent to the line R at some prescribed general point p ∈ R (in this case, Ω = {p}).

V d p(d)-1 [0, 1], d -2 is a divisor in V d p(d) [0, 1], d -2 )
, and its general member is a curve with one node at a general point of P 2 .

On the other hand the family W of plane d-ics with a node at the point p has codimension 3 in |dH| as well, and its closure W is a divisor in the "closed" Severi variety (c) the points p i,j and s i,j = φ(r i,j ) on C are pairwise distinct ;

V d p(d) [0, 1], d - 2 . However W is not part of V d p(d)-1 [0, 1], d -2 , as Definition (1.3) requires that curves in V d p(d)-1 [0, 1], d -2 have a local branch tangent to R at p, as illustrated below. not in V d p(d)-1 [0, 1] in V d p(d)-1 [0, 1] in V d p(d)-1 [0, 1] (1.4) Theorem. Let g ∈ Z, ξ ∈ NS(S), α, β ∈ N, Ω = {p i,j } 1 j αi ⊂ R be as in Def. (1.3), and consider an irreducible component V of V ξ g (α, β)(Ω). Let [C] be a general member of V , φ : C → C ⊂ S its normalization, q i,j (1 j α i ), r i,j (1 j β i ) points on C such that (1.3.1) and (1.3.2) hold. Set D = 1 j αi i q i,j + 1 j βi (i -1) r i,j . (1.4.0) If -K S • C i -deg φ * D| Ci 1 for every irreducible component C i of C, then dim V = -(K S + R) • ξ + g -1 + |β|. (1.4.1) If -K S • C i -deg φ * D| Ci 2 for every irreducible component C i of C, then (a ♭♭ )
(e) the curve C is smooth at its intersection points with R.

(1.4.3) If -K S • C i -deg φ * D| Ci 4 for every irreducible component C i of C, then (a) the curve C is nodal.
Note that if β = 0, then (b) holds under the weaker condition (1.4.0) for trivial reasons, see (3.4). the spaces H 0 (C, N φ ) and H 1 (C, N φ ) are respectively the Zariski tangent space and an obstruction space for the deformations of φ with fixed target S. In particular, we have

-Background from deformation theory 2.1 -Deformations of maps with fixed target

(2.2.2) χ(N φ ) dim R φ h 0 (N φ ).
(2.

3) The rank 1 sheaf N φ may have torsion. We denote by H φ its torsion part and by Nφ its maximal torsion-free quotient ; they fit in an exact sequence

(2.3.1) 0 → H φ → N φ → Nφ → 0.
The torsion sheaf H φ is supported on the divisor Z of zeroes of the differential dφ, and it is zero if and only if Z = 0 ; in this case, we say that φ is an immersion. Moreover, there is an exact sequence of locally free sheaves on C

(2.3.2) 0 → T C (Z) → φ * T S → Nφ → 0,
which readily implies the identification of line bundles on C

(2.3.3) Nφ ∼ = ω C φ * ω -1 S (-Z).
(2.4) Construction as Hom C g /M g , S × M g /M g ). Other possible notation : Mor Mg C g , S × M g ). With proviso. Indeed, the morphism ̟ := pr 2 : Φ( ŪM ) → M is a well-defined family of codimension 1 algebraic cycles of S in the sense of [6, I.3.11]. Since M is normal, the claim follows from [6, I.3.23.2].

-Comparison of the spaces of maps and curves

It follows that there is a morphism from M to the Hilbert scheme of curves on S. It obviously factorizes through V ξ g (S), and actually through its normalization V ξ g by the universal property of the normalization. Since by definition the semi-normalization morphism M → M ξ,bir

g is 1 : 1, two points [φ : C → S], [φ ′ : C ′ → S] ∈ M are mapped to the same point [Γ] ∈ V ξ g if and only if there exists an isomorphism ι : C ∼ = C ′ such that φ = φ ′ • ι.
To draw a precise conclusion about the morphism M → V ξ g from this, it is necessary to specify precisely who is M ξ,bir g ; we will need and state less, see Prop. (2.7).

(2.6) From curves to maps. On the other hand, consider the universal family U V → V ξ g of curves gotten from the universal family over the Hilbert scheme of curves on S. Let V be the normalization of V ξ g , and ŪV the normalization of U V × V ξ g V . Teissier's résolution simultanée theorem [START_REF] Teissier | Résolution simultanée I, II, in Séminaire sur les singularités des surfaces[END_REF] asserts that ŪV → V is a family of smooth genus g curves ; it comes with a morphism of V -schemes

ŪV → U V × V ξ g V ⊂ S × V .
It follows that there is a morphism from V to the space M ξ,bir g . It is generically injective, because the universal family of curves over V ξ g is nowhere isotrivial.

From the considerations in §(2.5) and (2.6), one deduces the following.

(2.7) Proposition. Let [φ : C → S] ∈ M ξ,bir g (S) be a general point (i.e., a general point of any irreducible component of M ξ,bir g (S)). Let Γ := φ(C), ξ ∈ NS(S) the homology class of Γ, and g its geometric genus. Then [Γ] belongs to a unique irreducible component of V ξ g and

dim [Γ] V ξ g = dim R φ .
(Recall that R φ0 is the complete local C-algebra that prorepresents Def φ/S ).

(2.7.1) Remark. If we take M ξ,bir g as the moduli space of maps, then dim

R φ = dim [φ] M ξ,bir g , whereas if one defines it as in [5] one has dim R φ = dim [φ] M ξ,bir g -dim(Aut C).
The next result provides a sharper upper bound on the dimension of the Severi varieties than that given by the inequality (2.2.2) dim R φ h 0 (N φ ).

Let φ : C → S be a morphism from a smooth projective curve C, birational onto its image Γ. Let ξ ∈ NS(S) be the homology class of Γ, and g its geometric genus. We consider Φ : C → S × B a deformation of φ over a pointed normal connected scheme (B, 0). Then Φ(C) ⊂ S × B is a deformation of Γ over (B, 0), see (2.5). There are thus two classifying morphisms κ and γ from (B, 0) to M ξ,bir g (S) (or R φ ) and Curve(S) respectively, with differentials

dκ : T B,0 → H 0 (C, N φ ) and dγ : T B,0 → H 0 (Γ, N Γ/X ).

(2.8) Lemma. The inverse image by dκ of the torsion H

0 (C, H φ ) ⊂ H 0 (C, N φ ) is contained in the kernel of dγ.
Proof. Given a non-zero section σ ∈ H 0 (C, N φ ), the first order deformation of φ defined by σ can be described in the following way : consider an affine open cover {U i } i∈I of C, and for each i ∈ I consider a lifting

θ i ∈ C(U i , φ * T X ) of the restriction σ |Ui . Each θ i defines a morphism φi : U i × Spec(C[ε]) → S extending φ |Ui : U i → X.
The morphisms φi are then made compatible after gluing the trivial deformations U i × Spec(C[ε]) into the first order deformation of C defined by the coboundary ∂(σ) ∈ H 1 (C, T C ) of the exact sequence (2.2.1). In case σ ∈ H 0 (C, H φ ), everyone of the maps φi is the trivial deformation of σ |Ui over an open subset. This implies that the corresponding first order deformation of φ leaves the image fixed, hence the vanishing of dq 0 (σ). ✷

(2.9) Corollary. Let g ∈ Z, ξ ∈ NS(S). Let [C] be a general point of V ξ g , and φ : C → C ⊂ S its normalization. Then dim V ξ g h 0 ( C, Nφ ).
Proof. By generality we may assume that

[C] is a smooth point of V ξ g . Then dim V ξ g = dim T [C] V ξ g , and by (2.6) there is a map dκ [φ] : T [C] V ξ g → H 0 ( C, N φ ).
It is injective because to every tangent vector θ ∈ T [C] V corresponds a non-trivial deformation of C. On the other hand, it follows from Lemma (2.8) that Im

dκ [φ] ⊂ H 0 ( C, Nφ ). ✷ Lemma (2.8
) is a crucial observation (and indeed, the cornerstone of the proof of Theorem (1.4)) that was made by Arbarello and Cornalba [1, p. 26], who deemed it a fenomeno assai curioso. They write : « nel caso in cui φ sia una birazionalità tra C e Γ, la presenza di "cuspidi" su Γ, comporta l'esistenza, dal punto di vista infinitesimo, di più di un modello liscio della curva Γ, se così ci possiamo esprimere. » 1Next, paraphrasing them, in order to use this phenomenon constructively they establish [1, Cor. 6.11] : in the above notation, if B is the complex unit disk and if the family of curves is equisingular, then dκ(∂/∂t) belongs to H 0 (C, H φ ) if and only if it is zero. Later, Caporaso and Harris (together with J. de Jong, they write) state and prove [START_REF] Caporaso | Counting plane curves of any genus[END_REF]Lem. 2.3]. They add the remark that this is linked to the notion of equisingularity, even though they make absolutely no use of this, neither in the statement nor in its proof.

The treatment I give here is that of Sernesi and myself in [START_REF] Dedieu | Equigeneric and equisingular families of curves on surfaces[END_REF]. Although essentially equivalent to that of [START_REF] Caporaso | Counting plane curves of any genus[END_REF], it slightly differs in its formulation. This formulation, I hope, sheds some light on what is actually going on and in particular displays that equisingularity has very little to do in the argument for Corollary (2.9).

-Tangency conditions with respect to a fixed curve

In general we consider R a fixed reduced curve on S. In this subsection we study deformations of curves on S satisfying some tangency conditions with R ; it follows from our Definition (1.3) that it suffices to treat the case when R is smooth.

(2.10) Let m be a non-negative integer. Let φ : C → S be a non-constant morphism from a smooth projective curve C. A deformation of φ with fixed target preserving a tangency of order m with R over a pointed base (B, 0) is a deformation Φ : C → S × B of φ with fixed target as in (2.1), such that there exists a section Q of C → B such that the pulled-back divisor Φ * R contains Q with multiplicity m (i.e., Φ * R -mQ 0).

C π Φ / / S × B pr 2 { { ✈ ✈ ✈ ✈ ✈ ✈ B Q < <
The tangency is said to be respectively at a fixed point

p ∈ R if Φ(Q) = {p} × B, and at a variable point if pr 1 • Φ(Q) is a curve.
We say that a family of maps Φ : C → S × B preserves a tangency of order m with R if for all b ∈ B it is locally around b a deformation of maps preserving a tangency of order m.

The following result displays the additional conditions the class of a deformation of maps has to meet for this deformation to preserve a tangency with R. It is [START_REF] Caporaso | Counting plane curves of any genus[END_REF]Lem. 2.6]. Let B be a reduced scheme and Φ : C → S × B be a family of maps preserving a tangency with R of order exactly m. Let b ∈ B be a general point, and φ : C → S be the corresponding map. This comes with a classifying map κ, with differential dκ : T 0 B → H 0 (C, N φ ) ; we call dκ its composition with the projection H 0 (C, N φ ) → H 0 (C, Nφ ).

Let q := Q ∩ C, and l -1 be the order of vanishing of the differential dφ at q (i.e., l is the multiplicity of the point p := φ(q) in the local branch of φ(C) corresponding to q). Note that necessarily l m.

(2.11) Lemma. Let σ ∈ Im( dκ) be a non-zero section, and denote by v q (σ) its order of vani-

shing at q = Q ∩ C. (a) One has v q (σ) ∈ {m -l} ∪ [[m, +∞[[. 2 (b) If the tangency is at a fixed point of R, then actually v q (σ) ∈ [[m, +∞[[.
Proof. This is a local computation. Let (x, y) be (analytic) local coordinates on S at p = φ(q), such that R is defined by the equation y = 0. Then the vector fields ∂/∂x and ∂/∂y generate T S near p, and their pull-backs generate φ * T S near q ; by abuse of notation we shall denote them by ∂/∂x and ∂/∂y as well.

We may assume that B is a curve. Let ε be a local coordinate on B centered at b, and t be a local equation of the section Q near q. Thus (t, ε) are local coordinates on C at q. We may assume that t gives a local coordinate on C at q, in such a way that φ is given locally by

φ(t) = t l + a l+1 t l+1 + • • • , t m , if l < m t n + a n+1 t n+1 + • • • , t m for some n m, if l = m.
From now on we assume l < m and leave the other, similar, case to the reader. Then the differential of

φ at q is ∂ ∂t -→(lt l-1 + (l + 1)a l+1 t l + • • • ) • ∂ ∂x + mt m-1 • ∂ ∂y = t l-1 (l + (l + 1)a l+1 t + • • • ) • ∂ ∂x + mt m-l • ∂ ∂y .
We see that around q on C, the torsion part H φ of N φ is a skycraper sheaf of length l -1 concentrated at q, generated by the section

τ : t → l + (l + 1)a l+1 t + • • • • ∂ ∂x + mt m-l • ∂ ∂y .
The torsion-free quotient Nφ is an invertible sheaf, generated by ∂/∂y. Observe that modulo τ , ∂/∂x is t m-l • ∂/∂y times an invertible, hence the image of ∂/∂x in Nφ vanishes to the order exactly ml at q. In turn the family Φ is given locally by

Φ(t, ε) = (t l + a l+1 t l+1 + • • • ) + ε(u 0 + u 1 t + • • • ) + O(ε 2 ), t m , ε , as Φ * R contains Q with multiplicity m. By definition, the corresponding section dκ(∂/∂ε) of Nφ is dκ( ∂ ∂ε ) = (u 0 + u 1 t + • • • ) • ∂ ∂x mod τ.
Since ∂/∂x itself vanishes modulo τ to the order ml as we have seen, one has v q dκ(∂/∂ε) ml in any event.

Moreover, by generality of b ∈ B we may assume that all Φ(ε, •) have their differential vanishing to the order l at Q ∩ C ε , which translates into the fact that u 1 = • • • = u l-1 = 0. Then dκ(∂/∂ε) vanishes either to the order ml, if u 0 = 0, or to some order larger than m, if u 0 = 0. When the tangency is maintained at a fixed point we are necessarily in the latter case. The lemma is proved. ✷

Let C ⊂ S × B be a family of a reduced curves over a reduced base B. It is said to preserve a tangency of order m with R if the corresponding family of normalization maps over the normalization B of B (see (2.6)) does.

(2.12) Corollary. Let V ⊂ Curve(S) be a family of curves of genus g having a tangency of order m with the divisor R. Let [C] be a general member of V , φ : C → C ⊂ S be the normalization of C, q ∈ C the tangency point, and l -1 the order of vanishing of the differential

dφ at q. Then dim V h 0 C, Nφ (-a • q) ,
where a ml. If the tangency is at a fixed point of R, then actually a m.

Proof. As in the proof of Cor. (2.9), we have dim

V = dim T [C] V by generality of [C],
and there is an injective map T [C] V → H 0 ( C, Nφ ). By Lem. (2.11) the image of this map is contained in H 0 C, Nφ (-a • q) with a as in the statement. This ends the proof. ✷

-Proof of the Main Theorem

In this section we prove Theorem (1.4). The proof itself is in Subsection 3.2, after we give some lemmas in Subsection 3.1.

-Applications of the Riemann-Roch formula

Lemma (3.1) below is standard, but we will also use the more clever Lemma (3.2).

(3.1) Lemma. Let X be a smooth (possibly disconnected) projective curve, and L a line bundle on X.

Let k ∈ N * . If deg L ω -1
X Xi > k for all irreducible component X i of X, then the linear system |L| separates any k points on X.

Proof. Let Z be a subscheme of X of length k, and let Z ′ be another subscheme of X such that Z ′ Z. The assumption on the degree of L ensures that both L(-Z) and L(-Z ′ ) are non-special, hence by the Riemann-Roch formula h 0 L(-Z) < h 0 L(-Z ′ ) . ✷ (3.2) Lemma. Let X be a smooth (possibly disconnected) projective curve of genus g = 1χ(O X ), and L, M two line bundles on X such that Let us assume from now on that M is special. Then,

∀ X i component of X : deg( M | Xi ) deg( L| Xi ). (a) If deg L ω -1 X Xi > 0 for every component X i of X, then (3.2.1) h 0 (X, M ) h 0 (X, L) = deg(L) -g + 1. (b) If deg L ω -1 X Xi > 1 for every component X i of X,
h 0 (X, M ) h 0 (X, ω X ) = g + n -1.
Under Assumption (a),

deg(L) = n i=1 deg( L| Xi ) n i=1 (2g i -1) = 2g -2 + n,
where n is the number of components of X and g i is the genus of X i for i = 1, . . . , n ; recall that g = g in + 1. In this case, one has

h 0 (X, L) = deg(L) -g + 1 g -1 + n hence h 0 (X, L) h 0 (X, M ).
Under Assumption (b) one has in the same fashion deg(L) 2g -2 + 2n, hence

h 0 (X, L) g + 2n -1 > h 0 (X, M ).
This ends the proof as the specialty of M implies deg(M ) < deg(L) under the general assumption of the Lemma. ✷

-Proof of Theorem (1.4) (3.3)

We start by proving that V has expected dimension

(3.3.1) expdim V ξ g (α, β)(Ω) = -(K S + R) • ξ + g -1 + |β|.
By (2.2.2) the expected dimension of V ξ g is χ(N φ ), which by the Riemann-Roch formula and the exact sequence (2.2.1) equals

(3.3.2) expdim V ξ g = deg ω C -deg φ * ω S + 1 -g = -K S • C + g -1.
Now requiring that a curve C have tangency of order m with R at a specified point p is m linear conditions on the coefficients of the equation of C, and if we let the point p vary along R the expected codimension of the corresponding locus of curves C is one less, i.e. m -1. We thus end up with

expdim V ξ g (α, β)(Ω) = expdim V ξ g - i 1 j αi i - i 1 j βi (i -1) = expdim V ξ g -Iα -(Iβ -|β|),
which together with (3.3.2) gives the required equality (3.3.1) after one remarks that Iα + Iβ = R • ξ.

Note that this proves that in any event

(3.3.3) dim(V ) -(K S + R) • ξ + g -1 + |β|.

✷ (3.6)

We now prove that under Assumption (1.4.1) the assertions (a ♭♭ ), (b), (c ♭ ), and (d) hold. Suppose by contradiction that (b) doesn't hold. Then we argue as in (3.5). In this case, part (b) of Lemma (3.2) applies thanks to Assumption (1.4.1), and we get that the inequality (3.5.1) is strict, which is in contradiction with (3.3.3).

The same argument shows that none of the points φ(r i,j ) can be fixed on R. This implies in particular that (c ♭ ) holds.

The proof of (d) is similar : if C were tangent to G, then it would belong to an irreducible component W of some Severi variety of the pair (S, R + G). Assumption (1.4.1) implies that W is liable for part (1.4.0) of Theorem (1.4), hence dim(W ) < dim(V ), in contradiction with the fact that [C] is a general member of V . The same argument shows that C avoids Γ (pick some random curve on S containing Γ).

Eventually, we note that equality holds in (3.4.7) if and only if D) is globally generated by assumption (1.4.1). Now it is indeed the case that equality holds in (3.4.7) since we have proved that dim(V ) = expdim V ξ g (α, β) . We conclude that D ′ 1 = Z 0 = 0, which means that (i) φ is an immersion outside of the points r i,j (this is assertion (a ♭♭ )) and (ii) l i,j i for 1 j β i . ✷

D ′ 1 = Z 0 = 0 since the line bundle ω C φ * ω -1 S (-
Remark. It is not enough to assume that ω C φ * ω -1 S (-D) is non-special and globally generated because of the argument we made to assume that the points q i,j and r i,j are pairwise distinct. We actually need to know something about every possible ω C φ * ω -1 S (-D ′ ), where D ′ = iq ′ i,j + (i -1)r ′ i,j in the notation used for this argument.

(3.7) We now prove that, under the assumption (1.4.2), φ is an immersion also at the points r i,j , i.e. that l i,j = 1 for 1 j β i , thus completing the proof of assertion (a ♭ ).

Let i 1 and 1 j β i . It follows from the assumption that the linear series ω C φ * ω -1 S (-D) separates any two points, so there exists a section σ ∈ H 0 C, ω C φ * ω -1 S (-D) with vanishing order v ri,j (σ) = 1 at the point r i,j . Seen as a section σ ∈ H 0 ( C, Nφ ), it vanishes at r i,j with order v ri,j (σ) = 1 + (il i,j ) (see (2.3.3)). By Lemma (2.11) this implies

1 + i -l i,j ∈ {i -l i,j } ∪ [[i, +∞[[
and therefore l i,j = 1 as required. ✷

(3.8) Let us prove that Assumption (1.4.2) implies Assertion (c), i.e., the points p i,j and s i,j = φ(r i,j ) are pairwise distinct. By (3.6), we already know that (c ♭ ) holds, i.e., none of the s i,j = φ(r i,j ) belongs to Ω = {p i,j }. We thus only need to show that no two of the points s i,j coincide.

Suppose there exist (i, j) and (i ′ , j ′ ) distinct such that φ(r i,j ) = φ(r i ′ ,j ′ ). Assumption (1.4.2) implies that the series

|ω C φ * ω -1 S (-D)| = |T [C] V | separates any two points. Therefore there exists a section σ ∈ T [C] V ∼ = H 0 ( Nφ (-D)) such that v ri,j (σ) = 1 and v r i ′ ,j ′ (σ) = 0.
This implies the existence of a deformation of C in which the points φ(r i,j ) and φ(r i ′ ,j ′ ) no longer coincide, a contradiction to the generality of [C] in V . ✷ (3.9) Let us prove that Assumption (1.4.2) implies Assertion (e), i.e., C is smooth at its intersection points with R.

At this point we know that (a ♭ ) and (c) under Assumption (1.4.2), i.e., the curve C is immersed and the points p i,j and s i,j are pairwise distinct. Because the intersection C ∩ R is set-theoretically the union of all the points p i,j and s i,j , this implies that C is smooth at its intersection points with R. ✷

-A characterization of logarithmic Severi varieties

We consider as before a pair (S, R) consisting of a smooth surface S and a smooth divisor R. In this Section we give an upper bound for the dimension of families of (not necessarily reduced) curves in S with prescribed homology class and genus. To be moved to intro, probably... This is an easy corollary, but the proof is quite tricky.

Let V be an irreducible locally closed subset of the Hilbert scheme of curves on S with homology class ξ ∈ NS(S). Assume that

-(K S + R) • ξ 1
so that everything is OK ( ? ?).

We suppose moreover that V parametrizes genus g curves in the following sense : let [X] be a general member of V ; there exists a smooth curve C and a morphism φ : C → X, not constant on any component of C and such that the push-forward in the sense of cycles φ * [C] equals the fundamental cycle of X.

(4.1) Proposition. Let [X] be a general member of V , and φ : C → X a morphism as above. For every finite subset Ω ⊂ R, one has

(4.1.a) dim V -(K S + R) • ξ + g -1 + Card (X ∩ R) \ Ω
(note that the last number is defined purely set-theoretically). If equality holds, then V is a dense subset of a component of a log-Severi variety V ξ g (α, β)(Ω) if and only if

(4.1.b) Card φ -1 (R) = Card(X ∩ R).
(4.1.1) Remark. (a) This is really a result about families of embedded curves in S, not families of maps. Indeed, if X is not reduced then φ involves multiple covers, and there is in general a positive dimensional family of maps giving the same X.

In the equality case, the map φ is necessarily a birational isomorphism on each component of C. (b) A straightforward consequence of Proposition (4.1) which may be useful for the applications is that the inequality (4.1.a) still holds if we only assume C to be reduced and replace g by the arithmetic genus of C ; this alternative inequality is always strict when C is not smooth.

-(K S + R) • ξ =2[H]•3[H]=6 + g =0 -1 + Card (X ∩ R) \ Ω =2 with Ω = ∅.
It is a divisor in the log-Severi variety V

3[H] 0 (0, 3)(∅), but it is not a component of the family V

φ -1 (R \ Ω) = Card (X ∩ L) \ Ω ,
the families that we get that are not log-Severi varieties may be considered as "log-Severi varieties with Ω containing multiple points" (these are not log-Severi varieties according to Definition (1.3)).

Example. Set (S, R) = (P 2 , L) as above, and ξ = 4[H], and fix p ∈ L. The family of plane quartics with one triple point at p has dimension 6 2 -1 -6 = 8, which equals

-(K S + R) • ξ =2[H]•4[H]=8 + g =0 -1 + Card (X ∩ L) \ Ω =1
with Ω = {p}.

One may wish to consider it as V Proof of Proposition (4.1). We divide it into several steps which correspond to paragraphs (4.2)-(4.4). We first treat the case when X is irreducible ; the general case is taken care of by induction in (4.4). In the case when e > 0, we consider the map ρ : V → Sym e R sending a curve to its reduced intersection scheme with R \ Ω ; this may not be well-defined everywhere since e may drop along certain closed subschemes of V , but it is in a neighbourhood of [X].

Then we can apply the e = 0 case of the Proposition to the fibres of ρ ; for a general Σ ∈ Sym e R, setting Ω ′ = Ω ∪ Supp Σ one gets that the fibre ρ -1 (Σ) has dimension at most -(K S + R) • ξ + g -1. Inequality (4.1.a) follows, and the equality case of the Proposition as well, again applying the e = 0 case to the fibres of ρ.

Remark. Note that as a byproduct of the above reasoning, one gets that when V is dense in a suitable irreducible component of a log-Severi variety, the map ρ : V → Sym e R is dominant.

(4.3) Let us now consider the case when X is non-reduced, but still irreducible ; we shall show that inequality (4.1.a) holds, and that it is always strict.

We have to consider φ : C → X where X is non-reduced but irreducible, and C may be reducible. We let m be the degree of φ, i.e. the sum of the degrees of the various φ i : C i → X. The key is to write "Riemann-Hurwitz" correctly : we have

2g -2 = deg ω C m deg ω X red = m(2h -2),
which gives g mhm + 1.

(4.4) It remains to consider the case when X is reducible. Proceeding by induction on the number of irreducible components, we may assume that X = X 1 ∪ X 2 where X 1 and X 2 move in two families V 1 and V 2 such that V ⊆ V 1 × V 2 and the Proposition holds for V 1 and V 2 . Adding the two corresponding inequalities readily gives

(4.4.1) dim V dim V 1 + dim V 2 = -(K S + R) • ξ + g -1 + Card (X 1 ∩ R) \ Ω + Card (X 2 ∩ R) \ Ω . If (X 1 ∩ X 2 ∩ R)
\ Ω is empty then this is the required inequality (and the equality case follows).

If not, let us assume for simplicity that it consists of only one point p (the general case is strictly similar). If p is a fixed point of either one of the two families V 1 or V 2 , then it is a fixed point of the two of them by the generality of X. Applying the Proposition to V 1 and V 2 with Ω ′ := Ω ∪ {p} one thus gets dim

V -(K S + R) • ξ + g -1 + Card (X 1 ∩ R) \ Ω ′ + Card (X 2 ∩ R) \ Ω ′ =Card (X∩R)\Ω -1
and the result follows.

Otherwise p is variable for both V 1 and V 2 ; in this case V necessarily has codimension at least 1 in V 1 × V 2 (this may be proved for instance as in (4.2) by applying the Proposition to the fibres of the projection V → V 1 ), and therefore (4.4.1) gives the required inequality. Equality may hold but in any event condition (4.1.b) will not be fulfilled (see Remark (4.1.2)). ✷ curves in |2C ′ | have genus 9. Moreover the Riemann-Roch formula, which works as on a K3 as well, tells us that |2C ′ | has dimension 9.

On the other hand plane octics have arithmetic genus 21, so an octic with 12 nodes has geometric genus 9, confirming the above computations carried out on S ′ . The linear system of plane octics has dimension 44. Since a node at a prescribed point is 3 conditions, the expected dimension of a linear system of octics with 12 nodes at prescribed points is 44 -36 = 8. We thus find once again that there is 1 extra dimension when the 12 nodes are at the complete intersection C ∩ R.

This may be verified directly using a resolution of the ideal I 2 Z , where I Z is the ideal of Z ⊂ P 2 . Let r and f be homogeneous equations of the curves R and C respectively. While for I Z there is the Koszul resolution, for its square we have the exact sequence

0 → O(-10) O(-11) -f 0 r -f 0 r -------→ O(-6) O(-7) O(-8) (r 2 ,rf,f 2 )
------→ I 2 Z → 0, which gives h 0 (O P 2 (8) I 2 Z ) = 10 as required.

(5.5) This carries over to all systems |kC ′ |, to the effect that the condition of having a k-uple point at all 3d points of Z = C ∩ R imposes one less condition on plane d-ics than if the 3d points of Z were in general position. We leave this to the reader.

-Superabundant log Severi varieties coming from double covers

In this subsection we observe that linear systems on double covers of the projective plane provide examples of superabundant logarithmic Severi varieties. This is taken from [START_REF] Ciliberto | Double covers and extensions[END_REF].

(5.6) We shall use the following elementary facts about double covers. Let d be a positive integer, and B be a degree 2d curve in P 2 . We consider π : S → P 2 the double cover branched over B. Let H be the line class on P 2 , and L be its pull-back to S. For all k ∈ N we have H 0 (S, kL) = π * H 0 (P 2 , kH) π * H 0 (P 2 , kH -1 2 B), which is the isotypic decomposition of H 0 (S, kL) as a representation of Z/2. The first summand corresponds to divisors that are double covers of degree k curves in P 2 , and the second to divisors that decompose as B (seen as the ramification divisor in S) plus the double cover of a degree kd curve in P 2 .

(5.7) Proposition. For k d, the general member C of |kL| is not a double cover of some hypersurface in P 2 , the restriction π| C is birational on its image, a degree 2k hypersurface C ♭ in P 2 everywhere tangent to B, with a node at every point of a complete intersection Z of type (k, kd).

  Let p(d) be the arithmetic genus of plane curves of degree d. The "open" Severi variety V d p(d) [0, 1], d-2 parametrizes smooth d-ics tangent to R at p ; it has codimension 2 in the linear system |dH|. The "open" Severi variety V d p(d)-1 [0, 1], d -2 parametrizes plane d-ics of cogenus 1, tangent to R at p ; it has codimension 3 in the linear system |dH| (hence

  the normalization map φ is an immersion, except possibly at the points r i,j ; (b) the points q i,j and r i,j of C are pairwise distinct ; (c ♭ ) none of the points s i,j := φ(r i,j ) belongs to Ω ; (d) for every curve G ⊂ S and finite set Γ ⊂ S such that (G ∪ Γ) ∩ Ω = ∅, if [C] is general with respect to G and Γ then C intersects G transversely and does not intersect Γ. (1.4.2) If -K S • C ideg φ * D| Ci 3 for every irreducible component C i of C, then (a ♭ ) the normalization map φ is an immersion ;

( 2 . 1 )

 21 Let φ : C → S be a non-constant morphism from a smooth projective curve C. A deformation of φ with fixed target over a pointed base (B, 0) is the data of a deformation C π -→ B of C over (B, 0) together with a morphism Φ : C → S ×B of B-schemes, such that the restriction of Φ over 0 equals φ.This defines the deformation functor Def φ/S of φ with fixed target S. It is prorepresented by a complete local C-algebra R φ , see[START_REF] Sernesi | Deformations of algebraic schemes[END_REF] Thm. 3.4.8].

( 2 . 2 )

 22 The deformations of φ with fixed target S are controlled by the normal sheaf of φ, i.e. the sheaf N φ of O C -modules defined by the exact sequence on C (2.2.1) 0 → T C dφ -→ φ * T S → N φ → 0 :

( 2 . 5 )

 25 From maps to curves. Consider the universal morphism Φ : U M → S × M ξ,bir g defined over M ξ,bir g (pretending it exists, compare (2.4)). Let M be the semi-normalization of the reduced scheme underlying M ξ,bir g , ŪM := U M × M ξ,bir g M , and Φ : ŪM → S × M the induced morphism of M -schemes. I claim that the scheme-theoretic image Φ( ŪM ) is flat over M .

  then actually equality holds in (3.2.1) if and only if deg M = deg L. Proof. Assumption (a) ensures that L is non-special, hence the right-hand-side equality in (3.2.1) by the Riemann-Roch formula. If M is non-special as well, then h 0 (L)h 0 (M ) = deg(L)deg(M ) again by the Riemann-Roch formula, which gives the result.

( 3 . 10 )

 310 We eventually prove that under the Assumption(1.4.3) the curve C is nodal, which is Assertion (a) of Thm.(1.4).Since we already know that the curve C is immersed, it is enough to show that for all point p ∈ C, C has neither 3 or more local branches, nor 2 or more tangent local branches.To exclude the former possibility, suppose by contradiction that there exist a, b, c ∈ C pairwise distinct such that φ(a) = φ(b) = φ(c). The assumption(1.4.3) implies that the linear series ω C φ * ω -1 S (-D) separates any three points, so there exists a section σ ∈H 0 C, ω C φ * ω -1 S (-D) such that σ(a)= σ(b) = 0 and σ(c) = 0; it corresponds to a first-order deformation of φ leaving both φ(a) and φ(b) fixed while moving φ(c). By generality of [C], there is correspondingly an actual deformation of the curve C for which the 3 local branches under consideration are no longer intersecting in a common point, a contradiction to the generality of [C] in V (see, e.g., [5, Prop. 1.4]). We exclude the second possibility in a similar fashion. Suppose by contradiction that there exist a, b ∈ C distinct such that φ(a) = φ(b) and Im dφ a = Im dφ b . Again since ω C φ * ω -1 S (-D) separates any three points, there exists σ ∈ H 0 C, ω C φ * ω -1 S (-D) such that σ(a) = 0 and σ(b) ∈ Im dφ a . It corresponds to a first-order deformation of φ leaving φ(a) fixed while moving φ(b) in a direction transverse to the common tangent to the 2 local branches of C under consideration. This contradicts the generality of [C] as before.✷

( 4 . 1 . 2 )

 412 Remark. (a) Assumption (4.1.b) ensures that the normalization of X is unibranch over the points in X ∩ R. Example. Set (S, R) = (P 2 , L) whith L a line, and ξ = 3[H] with [H] the hyperplane class. The family of V of plane cubics with one node on L and otherwise smooth has dimension 7, which equals

  [START_REF] Arbarello | Su una congettura di Petri[END_REF][START_REF] Arbarello | Su una congettura di Petri[END_REF])(∅) of rational plane cubics with one variable tangency along L, which has dimension 7 as well.(b) In the equality case, if one replaces (4.1.b) with the weaker condition that

Card

3[H] 0 ( 3 , 1 )

 031 (p, p, p).

( 4 . 2 )

 42 We first prove the Proposition under the assumption that X is reduced and irreducible ; in this case φ is the normalization of X and C ∼ = X.If e := Card (X ∩ R) \ Ω = 0, then the statement is a slight variant of part (1.4.0) of the main Theorem (1.4) : we get the required inequality (4.1.a) exactly as in paragraph (3.4), with D = φ * R and D 0 = 0 in (3.4.1) and (3.4.2) respectively. The only difference with the setting of (3.4) is that here two distinct points of the support of φ * R ⊂ X may have the same image by φ in X ; this makes absolutely no difference in the argument. Now if (4.1.b) holds, then X is unibranch over the points of X ∩ R. This implies that V is contained in a certain log-Severi variety V ξ g (α, 0)(Ω). If moreover equality holds in (4.1.a), then V is dense in an irreducible component of this same log-Severi variety by (1.4.0).

Proof.( 5 . 8 ) 2 =

 582 The divisor C belongs to a unique pencil A ′ , B + D ′ , with A ′ and D ′ the double covers of curves A and D in P 2 of respective degrees k and kd. Thus C ♭ := π(C) belongs to the pencil 2A, B + 2D , from which it follows that C ♭ is double along Z := A ∩ D, and touches B doubly along A ∩ B, which accounts for the whole intersection scheme of C ♭ and B. The base locus of this pencil is the scheme defined by the ideal I 2 Z (I 2 A + I B ). The pull-back π * C ♭ ∈ |2kL| splits as C + i(C), with i the involution on V associated to π ; it has a double singularity along Z ′ := π -1 (Z) and π -1 (B ∩ A), with at each point one local sheet belonging to C and another to i(C). The unionZ ′ ∪ π -1 (B ∩ A) is the base locus of the pencil A ′ , B + D ′ .✷ We consider the image V B,k in |2kH| of the linear system |kL| on S. It has dimensionh 0 (S, kL) -1 = h 0 (P 2 , kH) + h 0 (P 2 , (kd)H) -1,and parametrizes curves of geometric genusg k,d = 1 2 (2k -1)(2k -2)k(kd)everywhere tangent to B, the number of contact points is thus 2kd. The family of curves V B,k is therefore contained in the log-Severi variety V 2k g k,d (0, [0, 2kd]) of the pair (P 2 , B), which has the expected dimension-(K P 2 + B) • 2kH + g k,d -1 + 2kd = k(k + 3d).By (1.4.0) a component of the Severi variety has the expected dimension if it has an irreducible member and-K P 2 • 2kH -2kd 1 ⇐⇒ 2k(3d) 1,the latter inequality holding if and only if d 2.It turns out that the dimension of our family V B,k exceeds the expected dimension of the log-Severi variety. Indeed a direct computation shows that dim(V B,k )expdim V 2kH g k,d (0, [0, 2kd]) = (d-1)(d-2) p g (S)(cf. [2, V.22 p.237] for the last equality).

If R is reducible one should specify the distribution of Ω on the various components of R. This distribution however doesn't change anything in the framework of this text, so we don't dwell on this.

a very curious phenomenon ; in the case when φ is birational between C and Γ, the presence of "cusps" on Γ comports the existence, at the infinitesimal level, of more than one smooth model of the curve Γ, if we may say so.

I use the (probably french) notation [[a, b]] = [a, b] ∩ Z for a, b ∈ R ∪ {±∞}.

(3.4) We now turn to the proof that the dimension of V equals its expected dimension under assumption (1.4.0).

Note that the points q i,j are necessarily pairwise distinct because they have distinct images p i,j ∈ R. Let us first assume in addition that the points q i,j and r i,j are all together pairwise distinct ; the case when this does not hold will be dealt with in (3.5).

We set (3.4.1) D :=

the divisor on C of "infinitesimal tangency conditions with R" (compare (3.3)), and

is the order of vanishing of the differential dφ at the point r i,j , i.e. D 0 is the ramification divisor of φ "in the points r i,j ". We then decompose the difference of these two divisors as

where D 1 and D ′ 1 are non-negative divisors on C with disjoint supports ; note that D ′ 1 may be nonzero only at the points r i,j , and that it is so if and only if l i,j > i.

It follows from Cor. (2.12) that

Let Z 0 be the non-negative divisor on C such that the ramification divisor of φ is

Now by assumption (1.4.0) the line bundle

which, together with (3.3.3) implies that V has the expected dimension if indeed the points q i,j and r i,j are all together pairwise distinct.

(3.5) Now if it is not true that the points q i,j and r i,j are all together pairwise distinct, then V is actually a component of some Severi variety

Note that we have proved the additional fact that the tangent space at [C] of V identifies with

-Examples

-Logarithmic K3 surfaces

(5.1) Let us first consider the case of "absolute" K3 surfaces : let S be a K3 surface, and R = 0. In this case Theorem (1.4) is not quite accurate, a prominent problem being that the expected dimension given in (1.4.0) is not the actual dimension. Suppose S is equipped with a polarisation L of genus p (i.e., L 2 = 2p -2). The expected dimension of V L g is g -1 whereas its actual dimension is g (if 0 g p and S is very general, say). Technically, the deformations of [C] ∈ V L g are governed by the invertible sheaf L| C ∼ = ω C , hence the obstruction space is H 1 (C, ω C ) which is 1-dimensional, although the equigeneric deformations of C are in fact unobstructed. In some sense, the reason behind this is that there exist non-algebraic K3 surfaces. I refer to [5, §4.2] for a detailed account.

(5.2) In this subsection we shall describe some analogous phenomena for K3 pairs, by looking at a typical example. From now on we let S = P 2 , and R be a smooth cubic ; note that in this case one has K S + R = 0. Let C be a smooth curve of degree d 4 on S, and set Z = C ∩ R (for simplicity we shall assume that C and R intersect transversely). Then the blow-up S ′ of P 2 at Z is a smooth surface having a unique anticanonical divisor, namely the proper transform of R. The linear system |C ′ | of the proper transform C ′ of C gives a birational model of S ′ in P g , whose hyperplane sections are the canonical models of the degree d plane curves passing through Z (we let g = p(d) be the genus of smooth plane d-ic). In many aspects the surface S ′ behaves like a K3 surface.

(5.3) We may view the linear system |C ′ | as the Severi variety V d g (3d, 0)(Z) of (P 2 , R). Then its expected dimension given in (1.4.0) is g -1, whereas its actual dimension is g. In this case the discrepancy is readily explained : the points in Z are not general points on R (and therefore, according to our Definition (1.3) V d g (3d, 0)(Z) is not a Severi variety). To realize |C ′ | as a genuine Severi variety, we may replace V d g (3d, 0)(Z) by V d g (3d-1, 1)(Z -p) for an arbitrary point p ∈ Z. The latter has both expected and actual dimension equal to g, and all its members automatically pass through p as well.

Let us illustrate this in the concrete case d = 4 (then, g = 3). The linear system of plane quartics has dimension 14. If we take a set Ω of 12 general points on R, then the Severi variety V 4 3 (12, 0)(Ω) is empty : Ω imposes 12 independent conditions on quartics, the linear system of quartics through Ω is 2-dimensional, and all its members are made of R plus a line. On the other hand if we take Z the complete intersection of R with a smooth quartic C, then one sees using the restriction exact sequence 0 → O P 2 (1) → O P 2 (4) → O R (4) → 0 that the linear system of quartics through Ω is 3-dimensional, generated by C and the net of reducible quartics containing R. If we take a set Ω of 11 general points on R, it imposes 11 independent conditions on quartics, and the linear system of quartics through Ω is 3-dimensional with a 12-th base point on R.

(5.4) We may consider the linear system |2C ′ | in a similar fashion, although there is no completely convincing way of realizing it as a genuine Severi variety. Seen on P 2 , it is the system of plane (2d)-ics with a node at each of the 3d points of Z = C ∩ R.

Again we shall work this out in the case d = 4. One has (2C ′ ) 2 = 4 • (C ′ ) 2 = 16 = 2 • 9 -2, so the adjunction formula on S ′ , which is essentially the same as on a K3 surface, tells us that


Ghada Bouillass

Joris Masafont

Maëlle Baronnet

Nathan Bertho

Philomene Blot

Jeremy Costanzo

Thomas Dubard

Pierre Frodé

La Forêt

Exploring complementarity of Life Cycle Thinking and Landscape & Urban Planning towards

published or not. The documents may come   L'archive ouverte pluridisciplinaire

• European regulations are pushing further urban access restrictions and promoting a shift to more sustainable solutions.

• Introducing sustainability into communities living conditions and territorial policies requires the consideration of:

• living environment characteristics,

• stakeholders' needs,

• sustainable technologies with high environmental, social and economic performance.

• To face this challenges, the complementarity of two consistent management approaches have been studied:

• Life Cycle Assessment (LCA)

• Landscape and Urban Planning (LUP)

INTRODUCTION & OBJECTIVE MATERIALS AND METHODS

REAL CASE STUDY

• Study of the complementarity of both LCA and LUP throughout a real case study in the historical city of Nice, South of France.

• -Field visits: urban configuration for logistics 2. LCA application -Diesel-power cargo vans vs electricity tricycles: from 285 to 28 gCO 2 eq/km in EU -EU mix vs PV: from 400 to 22gCO 2 eq/kWh

Characterization step

Crossing LCA and LUP through urban metabolism requirements

•

  ghada.bouillass@mines-paristech.fr joris.mosafont@mines-paristech.fr www.mines-paristech.fr www.oie.mines-paristech.fr CONTACT AFFILIATION AUTHORS City logistics is one of the most polluting segments of the transport sector.

→

  Optimization of delivery flows through a Clarke and Wright algorithm for 4 ULP and 1000 deliveries' points → ULP integration in the urban metabolism → Dimensioning of ULP and possible other usages of the platform (such as solar energy-based chargers for public electric vehicules) meeting stakeholders' needs and improving their living conditions. Clarke and Wright Algorithm performed for 4 ULP and 1000 delivery points Distribution flows between the ULP and the delivery points in Nice city Design of the ULP and its integration in the city metabolism Tricycle technology for city logistics « Triporteurs de l'Ouest » Global annual average horizontal irradiance (kWh/m²) Center Observation Impacts & Energy (O.I.E) → Environmental living conditions such as air pollution, noise and urban congestion → Stakeholders' needs and concerns (shopkeepers, workers, transport companies, public authorities, etc.) → Social hotspots → Constraints for scheduling and delivery distribution including technology capacity, time, topography and energy required → Economic performance of electric tricycles & charging infrastructures → Potential of renewable energy resources in electric mobility applications → Real-world local solar data, to be explored through Geographical Information Systems for energy use optimization. STEPS FOR THE SUSTAINBLE CO-DESIGN OF CITY LOGISTIC SERVICES Identification step 1. LUP application -Territorial reading grid based on: i) Physical, socio-cultural & economic resources ii) Political context

ACKNOWLEDGMENTS

Design and optimization step : from the electric tricycle technology to Urban Logistic Platforms.

• Opportunities:

Remarkable solar deposit encouraging the use of photovoltaics (PV).

Transport policy shift towards electric mobility


Keywords: Protein conformation, Protein structural variation, Structural alphabet, Molecular dynamics

Understanding the structural plasticity of proteins is key to understanding the intricacies of their functions and mechanistic basis. In the current study, we analyzed the available multiple crystal structures of the same protein for the structural differences. For this purpose we used an abstraction of protein structures referred as Protein Blocks (PBs) that was previously established. We also characterized the nature of the structural variations for a few proteins using molecular dynamics simulations. In both the cases, the structural variations were summarized in the form of substitution matrices of PBs. We show that certain conformational states are preferably replaced by other specific conformational states.

Interestingly, these structural variations are highly similar to those previously observed across structures of homologous proteins (r 2 =0.923) or across the ensemble of conformations from NMR data (r 2 =0.919). Thus our study quantitatively shows that overall trends of structural changes in a given protein are nearly identical to the trends of structural differences that occur in the topologically equivalent positions in homologous proteins. Specific case studies are used to illustrate the nature of these structural variations.

Introduction

Proteins exhibit a higher degree of structural conservation compared to sequence conservation [START_REF] Chothia | The Relation between the Divergence of Sequence and Structure in Proteins[END_REF][START_REF] Bashford | Determinants of a Protein Fold. Unique Features of the Globin Amino Acid Sequences[END_REF][START_REF] Bowie | Deciphering the Message in Protein Sequences: Tolerance to Amino Acid Substitutions[END_REF][START_REF] Gan | Analysis of Protein Sequence/Structure Similarity Relationships[END_REF] as they require maintaining certain key residues in a specific orientation to fulfill their functional constraints [START_REF] Nagl | Evolutionary Constraint Networks in Ligand-Binding Domains: An Information-Theoretic Approach[END_REF][START_REF] Lecomte | Structural Divergence and Distant Relationships in Proteins: Evolution of the Globins[END_REF][START_REF] Dessailly | Relating Destabilizing Regions to Known Functional Sites in Proteins[END_REF][START_REF] Najmanovich | Detection of 3D Atomic Similarities and Their Use in the Discrimination of Small Molecule Protein-Binding Sites[END_REF][START_REF] Saldaño | Evolutionary Conserved Positions Define Protein Conformational Diversity[END_REF] . Proteins are also known to be dynamic entities. Their intrinsic dynamic nature is encoded in the amino acid sequence itself as suggested by the high dynamics associated with certain intrinsically disordered proteins displaying compositional bias [START_REF] Hansen | Intrinsic Protein Disorder, Amino Acid Composition, and Histone Terminal Domains[END_REF] . These conformational changes occur also due to natural events such as binding of a ligand and association with other proteins or nucleic acids and also due to thermodynamic fluctuations in proteins. Extrinsic factors like molecular packing in the crystal lattice, pH, temperature, pressure and crystallization conditions are also known to affect the protein conformation. There are several studies that document the dynamic nature of proteins [START_REF] Gerstein | Structural Mechanisms for Domain Movements in Proteins[END_REF][START_REF] Narunsky | ConTemplate Suggests Possible Alternative Conformations for a Query Protein of Known Structure[END_REF][START_REF] Hrabe | PDBFlex: exploring flexibility in protein structures[END_REF][START_REF] Monzon | CoDNaS 2.0: a comprehensive database of protein conformational diversity in the native state[END_REF][START_REF] Chang | CCProf: exploring conformational change profile of proteins[END_REF] contributed by one or more of these factors. The user friendly ConTemplate [START_REF] Narunsky | ConTemplate Suggests Possible Alternative Conformations for a Query Protein of Known Structure[END_REF] webserver takes as input a protein structure (or model) and proposes possible alternative conformations for the protein. The Protein Data Bank [START_REF] Rose | The RCSB Protein Data Bank: Views of Structural Biology for Basic and Applied Research and Education[END_REF] Flexibility (PDBFlex) [START_REF] Hrabe | PDBFlex: exploring flexibility in protein structures[END_REF] , Conformational Diversity of the Native State (CoDNaS) [START_REF] Monzon | CoDNaS 2.0: a comprehensive database of protein conformational diversity in the native state[END_REF] and Conformational Change Profile (CCProf) [START_REF] Chang | CCProf: exploring conformational change profile of proteins[END_REF] and studies document the structural variations in proteins sharing >=95% sequence similarity. For the current study we focus only on proteins sharing exact (100%) sequence identity. The observed structural changes in this dataset may be caused by diverse factors. Considered collectively, multiple structural entries for a given protein represent stable (crystallizable) conformers and they represent the extent of structural diversity possible in a protein irrespective of the factor causing the structural change.

In this work, inspired by the availability of CoDNaS resource, we studied the local structural changes observed over a large collection of alternate crystal structures of the same protein using Protein Blocks (PBs), a structural alphabet. Protein Blocks are a 16 state classification of pentapeptide fragments that can abstract any part of a protein backbone structure [START_REF] De Brevern | Bayesian Probabilistic Approach for Predicting Backbone[END_REF][START_REF] De Brevern | New Assessment of a Structural Alphabet[END_REF] . The versatility of PBs to abstract protein structures have been exploited for diverse applications: to predict protein structure [START_REF] De Brevern | Local backbone structure prediction of proteins[END_REF][START_REF] Suresh | SVM-PB-Pred: SVM based protein block prediction method using sequence profiles and secondary structures[END_REF][START_REF] Vetrivel | Knowledge-based prediction of protein backbone conformation using a structural alphabet[END_REF] , in fold recognition [START_REF] Ghouzam | Improving protein fold recognition with hybrid profiles combining sequence and structure evolution[END_REF][START_REF] Mahajan | Use of a structural alphabet to find compatible folds for amino acid sequences[END_REF] , for structural motif identification [START_REF] Dudev | Discovering structural motifs using a structural alphabet: application to magnesium-binding sites[END_REF][START_REF] Schneider | Bioinformatic analysis of the protein/DNA interface[END_REF][START_REF] Wu | A structural-alphabet-based strategy for finding structural motifs across protein families[END_REF] , molecular dynamics trajectory analysis [START_REF] Barnoud | PBxplore: a tool to analyze local protein structure and deformability with Protein Blocks[END_REF][START_REF] Vetrivel | PBmapclust: Mapping and Clustering the Protein Conformational Space Using a Structural Alphabet[END_REF] etc. [START_REF] Joseph | A Short Survey on Protein Blocks[END_REF] Furthermore, PBs have been shown to be convenient in identifying local structural variations in proteins 29,[START_REF] Mahajan | Correlation between Local Structural Dynamics of Proteins Inferred from NMR Ensembles and Evolutionary Dynamics of Homologues of Known Structure[END_REF] . It is also effective in differentiating between structurally variable and conserved regions in protein structures [START_REF] Tyagi | Protein Structure Mining Using a Structural Alphabet[END_REF][START_REF] Joseph | Improvement of Protein Structure Comparison Using a Structural Alphabet[END_REF] . The relationship between this PB based metric and the conventionally used root mean squared deviation (RMSD) have been extensively presented and discussed in other studies 29,[START_REF] Tyagi | Protein Structure Mining Using a Structural Alphabet[END_REF] . Previously, non PB based studies have been undertaken to analyze structural variations in homologous proteins using normal mode analysis [START_REF] Leo-Macias | Core deformations in protein families: A physical perspective[END_REF] and Gaussian network approach [START_REF] Maguid | Exploring the common dynamics of homologous proteins. Application to the globin family[END_REF][START_REF] Maguid | Evolutionary conservation of protein backbone flexibility[END_REF] and also structural variations inherent to proteins [START_REF] Monzon | Conformational diversity analysis reveals three functional mechanisms in proteins[END_REF] . The novelty of our current work lies in analyzing the structural variations occurring within a set proteins and to compare them with those occurring between their respective homologues. PBs have been used to characterize the nature of the structural variations observed across structures of homologous proteins [START_REF] Joseph | Improvement of Protein Structure Comparison Using a Structural Alphabet[END_REF]29 and conformers from NMR ensembles [START_REF] Mahajan | Correlation between Local Structural Dynamics of Proteins Inferred from NMR Ensembles and Evolutionary Dynamics of Homologues of Known Structure[END_REF] . From these analyses, two PB substitution matrices were derived. One PB substitution matrix summarizes the local structure variations occurring between structurally equivalent regions across homologous structures and another between alternate NMR models. The comparison of the two matrices showed a high degree of similarity in the nature of structural variations at play in these two contexts.

In this work, we extended this analysis by investigating the nature of the structural variations observed between conformers of the same protein derived from multiple crystallographic instances and from conformations explored during molecular dynamics simulations. Inherent to the nature of these datasets, these structural variations are not drastic but the structural variations that do occur in the protein backbone, are statistically very similar to those observed across homologous structures or across models from NMR experiments.We summarized these structural variations in the form of two additional substitution matrices and investigated how they relate to the matrices established previously [29][START_REF] Mahajan | Correlation between Local Structural Dynamics of Proteins Inferred from NMR Ensembles and Evolutionary Dynamics of Homologues of Known Structure[END_REF][START_REF] Tyagi | Protein Structure Mining Using a Structural Alphabet[END_REF][START_REF] Joseph | Improvement of Protein Structure Comparison Using a Structural Alphabet[END_REF] .

Materials and Methods

Datasets

To analyze the structural variations across multiple entries of the same protein in the PDB [START_REF] Rose | The RCSB Protein Data Bank: Views of Structural Biology for Basic and Applied Research and Education[END_REF] , all protein chains were considered independently. If a PDB entry has 2 identical protein chains (e.g a homodimer), some structural variations could be observed between them, hence justifying they are considered separately. All the 250,000 protein chains available in the PDB at the time of the study were clustered at 100% sequence identity using the BLASTClust algorithm [START_REF] Altschul | Basic Local Alignment Search Tool[END_REF] . This resulted in 56,829 clusters of identical protein chains with the largest cluster being the one with the capsid protein of HIV-1 with 2,580 chains. These are viral capsid proteins in multimeric forms that were solved by different research groups. Out of these 56,829 clusters, singletons were not kept since a minimum of two structures in a cluster is needed in order to study their structural variability. Further our study focuses only on structures solved by X-ray crystallography with a resolution of 2.5 Å or better. Protein chains with chain breaks were also removed. After this filtering process we were left with a total of 254,979 protein chains distributed in 23,631 clusters. From this dataset, a total of 7.8 million pairwise comparisons were performed. For example, in a cluster with n=30 protein chains, we made a total of 435 (i.e. (n*(n-1))/2) pairwise comparisons .

Protein Blocks

Protein Blocks are a library of 16 backbone conformational states (structural alphabet) denoted by the letters a to p. The orientation of the backbone atoms in any five consecutive residues in a protein structure can be defined by a series of 8 torsion angles (φ,ψ). PBs are defined by 16 sets of these 8 torsion angles. This standard library of torsion angles were determined by unsupervised clustering of the torsion angles from a non-redundant set of proteins [START_REF] De Brevern | Bayesian Probabilistic Approach for Predicting Backbone[END_REF] . Parsing a protein structure through a sliding window of 5 residues enables abstraction of the backbone 3D structure into 1D string of PBs.

The PBE-T (Protein Block Transform) tool available on the PBE (Protein Blocks Expert) web-server (http://www.boprotscience.fr/pbe/?page_id=10) can be used to encode protein structures into PB sequences [START_REF] Tyagi | Protein Block Expert (PBE): A Web-Based Protein Structure Analysis Server Using a Structural Alphabet[END_REF] . There are many advantages to approximating protein backbone conformation using PBs and a number of applications have been developed based on PBs 39 .

Molecular dynamics

A collection of 130 structures of unrelated proteins were subjected to molecular dynamics simulation using the GROMACS [START_REF] Abraham | GROMACS: High Performance Molecular Simulations through Multi-Level Parallelism from Laptops to Supercomputers[END_REF] program for a duration of 10 ns each. These proteins shared less than 30% sequence identity with one another, were single chain proteins with no chain breaks and no ligands bound. Further structural criteria imposed were a X-ray crystallographic resolution better than 1.75 Å, R-factor <0.18 and free R-factor <0.24. The simulations were carried out in NPT conditions using the Amber99SB-ILDN force field with TIP3P water model at 300K temperature and 1 atm pressure. The leapfrog integrator was used with the timestep set as 2 fs. Periodic conditions were applied. Global charge was neutralized to 0.15 mol/L with Na + and Cl -ions.

Generating the matrices

Structures of 254,979 protein chains in our data set were represented as sequences of PBs using an in-house Perl script.

Multiple structural entries of the same protein represented as PB sequences were compared in an all-vs-all fashion. PB sequences that constituted our dataset are provided in supplementary material S4. To ensure only structurally equivalent regions of both the proteins were being compared, the missing residues at the N-terminal and C-terminal were adjusted by introducing appropriate number of gaps to the shorter sequence. The PB variations from these 7.8 million pairwise comparisons were cumulated and used to populate a 16 X 16 variation matrix. Henceforth this matrix is referred to as the altconfMAT because it summarizes the alternate conformations of crystallographic structures.

An equivalent 16 x 16 matrix was constructed that summaries the PB change between snapshots during the MD simulation.

Out of the 10 ns simulation, the first 5 ns were not taken into consideration for the analysis to allow for the system to stabilize. Coordinates of the protein structure were written out after every 250 ps. Hence we get 19 sets of coordinates for each of the 130 simulations. These coordinates were converted into PB sequences exactly in the same manner as described before. Similarly, the corresponding PB sequences that constituted this dataset are provided in supplementary material S4.

The PB variations among these PB sequences were used to populate the second PB variation matrix, henceforth called the dynamicsMAT.

Since all the 16 PBs do not occur in the same frequency in protein structures, the matrices so obtained needs to be normalized. As described previously 29,[START_REF] Mahajan | Correlation between Local Structural Dynamics of Proteins Inferred from NMR Ensembles and Evolutionary Dynamics of Homologues of Known Structure[END_REF] , the normalization was carried out as per the procedure described by Johnson and Overington [START_REF] Johnson | A Structural Basis for Sequence Comparisons. An Evaluation of Scoring Methodologies[END_REF] (Eqn. 1). S x,y is the normalized score for variation of PB x to y, N x,y is the raw count of PB x varying to PB y and M is the total number of PBs i.e. 16.

Eqn. 1

In the dynamicsMAT dataset there were no PB n ↔ PB d changes observed, hence in order to avoid undefined values during the normalization process, the value was changed from 0 to a nominal value of 0.0001. This value was chosen to be lower than the lowest observed value in the matrix.

Results and Discussion

Inferences from qualitative analysis of structural plasticity

Previously, backbone structural differences, as inferred from Protein Block (PB) variations observed among models in NMR ensembles of large number of NMR structures of proteins were compared to the structural differences observed in topologically equivalent regions of homologous protein structures [START_REF] Mahajan | Correlation between Local Structural Dynamics of Proteins Inferred from NMR Ensembles and Evolutionary Dynamics of Homologues of Known Structure[END_REF] . The cumulated normalized frequencies of PB variations in NMR ensembles and homologous protein structures were represented as matrices referred to as NMRMAT and homologueMAT respectively. The Pearson correlation coefficient between the two matrices (0.92) suggested an excellent correlation [START_REF] Mahajan | Correlation between Local Structural Dynamics of Proteins Inferred from NMR Ensembles and Evolutionary Dynamics of Homologues of Known Structure[END_REF] . We were interested to see how the newly established PB variation matrices, altconfMAT and dynamicsMAT (see section 2.4), related to the previously established NMRMAT and the homologueMAT matrices.

The correlations between each of these matrices are summarized in Table 1. All the correlation coefficient values are quite high with lowest value of about 0.87. This means backbone conformational variations observed in the same proteins in different crystal structures, variations observed during molecular dynamics simulations, variations among the models in NMR ensembles and differences observed in topologically equivalent positions in homologous protein structures are all well correlated with one another. The best correlation is between NMRMAT and homologueMAT matrices. All the other matrices show similar correlation values with the lowest correlation coefficient between NMRMAT and dynamicsMAT matrices (r 2 =0.87). A finding from our work is that, overall, the backbone structural variations due to dynamics and structural differences between homologues are highly similar. Furthermore, to ascertain that these correlations are not an artifact of the non-variant PBs occuring in the diagonals of these matrices, we recalculated the correlations excluding the diagonals. These results show a slight reduction in the correlation but very similar trends hence indicating that the bias brought about by the non-variant PBs is minimal.

Inferences from quantitative analysis of structural plasticity

All the 4 matrices are symmetric across the diagonal. Diagonal elements represent extent of retention of PBs and the off diagonal values correspond to PB substitutions. The proportion of the sum of the raw counts of off-diagonal elements to the sum of all the raw counts of the matrix reflects the amount of structural variability captured by that particular matrix. For the altconfMAT this value is 4.88% and it is 7.04% for the dynamicsMAT matrix. The sum of the values of a row in the normalized matrix is a measure of how favorable or unfavorable the particular PB is to structural variations. The more negative this value is, the more resistant that particular PB is to structural variations (see supplementary material S1 for individual PB values for all the four matrices). PBs m and d often correspond to the central regions of α-helices and β-sheets respectively (92% and 75% of -helix and -sheet respectively [START_REF] De Brevern | New Assessment of a Structural Alphabet[END_REF] ). The extent to which PBs m and d are refractory to structural variation are very comparable in the altconfMAT and dynamicsMAT. However, this is not the case for the previously established matrices-the NMRMAT and the homologueMAT. In both these cases, PB m is significantly better retained than PB d.

Once it is established that the central regions of the regular Secondary Structural Elements (SSE), α-helix and β-sheet, are mainly structurally invariant, we now focus on the loop regions and the interface of loop and SSE. In the altconfMAT matrix, this amounts to 78.2% of the structural variations out of which 64.3% are from the capping regions of the SSEs. In the case of dynamicsMAT, 65.9% of the variations are from the capping regions and this value increases to 79.4% if the loop regions are included (see Supplementary materials S2 for a more detailed break-up of which regions contribute to the variations). It must be noticed that the distribution of PBs in the central regions of SSEs, capping regions of SSEs and the loop regions are not absolute. For example, although the PBs g, h, i and j represent the loop regions we can occasionally find other PBs in the loop regions as well.

The correlation coefficients listed in Table 1 are overall values for all the 16 PBs. They can hide differences in correlation at the individual PB level. Hence we decided to calculate the individual PB wise correlation between these four matrices. 1 are represented by the horizontal red lines. Points above this line contribute positively to the correlation and the ones below negatively.

Case study of structural variations in alternate conformations of the same protein

Implications of different functional states or conformational changes potentially induced by ligand binding on the backbone conformational variations in a given protein were analyzed. The cluster we get from BLASTClust filtered at 100% sequence identity for human tyrosine kinase consists of three structures (PDBID: 2SRC 42 , 1Y57 [START_REF] Sw | The Crystal Structure of a c-Src Complex in an Active Conformation Suggests Possible Steps in c-Src Activation[END_REF] , 1FMK [START_REF] Xu | Three-dimensional structure of the tyrosine kinase c-Src[END_REF] ). Their superimposition featured in Figure 3 clearly shows that there are large structural variations that are induced upon binding to MPZ ligand. Shown in blue is the unbound form (PDBID 1FMK) [START_REF] Xu | Three-dimensional structure of the tyrosine kinase c-Src[END_REF] , in red is the conformation (PDBID 2SRC) when it binds to phosphoaminophosphonic acid adenylate ester (ANP) [START_REF] Xu | Crystal Structures of c-Src Reveal Features of Its Autoinhibitory Mechanism[END_REF] and in green is bound form to MPZ ligand (PDBID 1Y57) [START_REF] Sw | The Crystal Structure of a c-Src Complex in an Active Conformation Suggests Possible Steps in c-Src Activation[END_REF] These are further compared using protein blocks below. Figure 4 shows the PB alignment between the structures of human tyrosine kinase C-SRC bound to ANP ligand (PDBID 2SRC) and to MPZ ligand (PDBID 1Y57). The regions marked in red are the regions that show structural changes based on PBs. A total of 75 out of the 452 PBs variations are observed. Rigid body superposition of the two structures using the ProFit 45 software (http://www.bioinf.org.uk/software/profit/) results in a high RMSD of 21.3 Å (Figure 3c). When the same two structures are aligned flexibly using the FATCAT program [START_REF] Ye | FATCAT: A Web Server for Flexible Structure Comparison and Structure Similarity Searching[END_REF] the RMSD reduces to 2.7 Å (Figure 3d). This was achieved by the FATCAT algorithm [START_REF] Ye | FATCAT: A Web Server for Flexible Structure Comparison and Structure Similarity Searching[END_REF] by introducing a twist located at residue 259. Indeed, in this region PB sequence between the two structures showed a major difference with a twist at residue Leucine 255, here PBs behia are aligned with PBs addfk (Figure 4). When the same structure is compared with the ligand unbound form (PDBID 1FMK) 68 PBs show variation and most importantly the region of the twist shows only one PB change as opposed to 5 PB changes in the previous case (Figure 5). Shown is alignment of chain A for both structures. The region responsible for the high RMSD value calculated after rigid superimposition is circled in red. This region displays drastic PB variations that reflect a major twist in the backbone at the level of leucine 255.

Figure 5. PB alignment between two structures of human tyrosine kinase C-SRC in the unbound form (PDB ID 1FMK) and bound with ANP ligand (PDB ID 2SRC).

This is an example of structural variation captured by altconfMAT matrix. Shown is alignment of chain A for both structures. The region featured in Figure 10 is again circled here. This region displays minor PB variations that reflect very little change in the backbone conformation in this region.

Figure 6 shows the structure of a mutant amidase from Geobacillus pallidus (PDBID 4LF0) [START_REF] Weber | The mechanism of the amidases: mutating the glutamate adjacent to the catalytic triad inactivates the enzyme due to substrate mispositioning[END_REF] at the end of 5 and 10 ns of a molecular dynamics simulation. The regions marked in red are the regions that show structural changes between these two states. In this case 25 out of 340 PBs showed variations. In both the cases, the PBs corresponding to the capping regions of secondary structural elements and loop regions are highly represented. The logical succession to this work would be to characterize the structural changes during a molecular dynamics simulation at different time steps.

Furthermore, when we consider the above examples of tyrosine kinase and amidase and compare them to their respective 95% sequence homologues, we see similar PB substitutions between them. The frequently occurring PB l ↔ b variation in alternate structures of tyrosine kinase (8 instances) are also seen between their homologues (5 instances). There are also three instances of PB l ↔ b variation each

for alternate structures vs. homologues of amidase, hence indicating a correlation in structural variations. Detailed representation of these alignments are included in supplementary material S3.

Case study of structural variations between two homologous proteins having alternate conformations

Implications of backbone conformational variations in a given protein on the perceived structural differences between homologous proteins were analyzed in this case study. In this analysis we used PB-ALIGN [START_REF] Tyagi | Protein Structure Mining Using a Structural Alphabet[END_REF] to compare two homologous proteins each having multiple alternate conformations. The main results are featured in the form of network (Figure 7). The cluster in red corresponds to alternate structures of human sulfotransferase and the ones in green are various conformational states of the homologue from mouse. All the members of a single group share 100% sequence identity but inter group members share 59% sequence identity. The edges in red and green represent intra-group alignments and the blue edges are inter-group alignments. The alignment scores were normalized by length. In all cases the edge length corresponds to the normalized alignment scores, higher alignment scores correspond to shorter edges. Global distribution of the normalized alignment scores are featured in (Figure 8). On the left is the distribution of the normalized alignment scores for the mouse sulfotransferase structures group. On the right is the distribution of the normalized alignment scores for the human sulfotransferase structures group. In red are the inter group normalized alignment scores. In green are intra group alignment scores. Intragroup scores for the mouse sulfotransferases are higher than any of its intergroup scores. Intragroup scores for the human sulfotransferases show some overlap with some of its intergroup scores. This overlap is indicated in blue in the histogram on the left. RMSD (0.06 Å) seem to be minor but it is noteworthy that this variability is seen in homologues sharing a high sequence identity of almost 60%. We can expect this variability effect to be more pronounced in homologues sharing much lower sequence identity and also when the sizes of the groups are larger. More importantly, as shown in Figure 7, taking one conformation of the human sulfotransferase as a reference (PDB ID 1LS6 chain A), it can be seen that a mouse sulfotransferase (PDB ID 2ZYV chain X) conformation is strikingly more structurally similar to it than another conformation of the same human sulfotransferase (PDB ID 2D06 chain A).

Conclusions

We showed that the backbone structural differences, represented as Protein Blocks (PBs) observed in the data set of multiple crystal structures of a given protein are remarkably similar to the structural differences between the homologous protein structures. In other words, overall trends observed in structural changes in a given protein are nearly identical to the trends of structural differences that occur in the topologically equivalent positions in homologous proteins. Different PBs in topologically equivalent positions in two homologous proteins can be characterized by high exchange frequency. Therefore, if one considers the fact that the easily exchangeable PBs within a protein and between the homologous proteins are nearly the same, it could be inferred that backbone structures of homologous proteins are more similar than previously thought.

This also means that upon superposition of two homologous proteins, two residues which are not found to be topologically equivalent, maybe topologically equivalent if different crystal structures of the homologues are superimposed.

Our analysis of comparison two homologous protein structures by considering all multiple crystal structures available for the two homologues shows that the extent of structural variability within a protein can be similar to the extent of structural differences between the homologues. Indeed we have shown that structural similarity measure among multiple crystal forms of a given protein can be less than the structural similarity measure between the protein under study and its homologue.

Therefore if one considers conformational changes intrinsic and extrinsic in two homologous proteins, it appears that homologous protein structures more closely resemble each other than what one might infer based only on comparing one of the conformers of each of the two homologues. Therefore, a more complete understanding of how similar the structures of two homologues is arrived at by considering multiple crystal structures of same two proteins.

Molecular dynamics simulation and models from NMR ensembles seem to capture different aspects of protein structural dynamics, as indicated by the lowest correlation between the dynamicsMAT and NMRMAT, see Table 1.

Table 1 :

 1 Correlation among the three PB variation matrices (NMRMAT 30 , altconfMAT, dynamicsMAT) and the PB transition matrix (homologueMAT 29 ). The values in the brackets are the correlations calculated by excluding the nonthe PBs are organized in such a way that specific PBs have a higher predisposition to be present in certain secondary structure elements (SSE). PBs a, b and c correspond to the N-terminal capping regions of β-strands, PB d corresponds to the geometry of β-strand central regions, PBs e and f to C-terminal caps of β-strands. The counterparts for the helices are PBs k and l for the N-terminal caps, PB m for the central region and PBs n, o and p for the C-terminal caps. PBs g, h, i and j represent the loop regions. Therefore, as expected, PBs representing the ends of the spectrum namely PBs m and d show the least structural variation. This is represented by Figure 1's rows (or columns) of PB m and PB d having the highest shades of red in the heat map. This pattern is consistent across all the four matrices. Apart from the PBs m and d, the PB f and PB k are highly represented in the top five invariant PBs. They represent the C-terminal of β-strands and Nterminal of helices respectively. The PBs g, j, b, h and i are highly represented in the top 5 varying PBs across all the four matrices. PB b is the exception in this list as it typically corresponds to the N-terminal cap of β-strands whereas the other PBs correspond to the loop regions.

Figure 1 .

 1 Figure 1. Tri colored heat maps of the four substitution matrices. (a) altconfMAT. (b) dynamicsMAT. (c) homologueMAT. (d) NMRMAT.

Figure 2

 2 Figure2is a plot of these PB-wise correlations. In each case the red line corresponds to the overall correlation, the points above contribute positively to the correlation and the ones below negatively. The NMRMAT vs dynamicsMAT PB wise correlation shows the maximum dispersion.

Figure 2 .

 2 Figure 2. Individual PB wise correlation between the four matrices. (a) homologueMAT vs NMRMAT (b) homologueMAT vs altconfMAT, (c) NMRMAT vs altconfMAT, (d) homologueMAT vs dynamicsMAT, (e) NMRMAT vs dynamicsMAT, (f) altconfMAT vs dynamicsMAT. The global correlation values in Table1are represented by the horizontal red lines. Points above this line contribute positively to the correlation and the ones below negatively.

Figure 3 .

 3 Figure 3. Comparison between 3 structures of human tyrosine kinase C-SRC by rigid body & flexible superimposition.Shown in blue is the unbound form (PDBID 1FMK)[START_REF] Xu | Three-dimensional structure of the tyrosine kinase c-Src[END_REF] , in red is the conformation (PDBID 2SRC) when it binds to phosphoaminophosphonic acid adenylate ester (ANP)[START_REF] Xu | Crystal Structures of c-Src Reveal Features of Its Autoinhibitory Mechanism[END_REF] and in green is bound form to MPZ ligand (PDBID 1Y57)[START_REF] Sw | The Crystal Structure of a c-Src Complex in an Active Conformation Suggests Possible Steps in c-Src Activation[END_REF] . Ligands are not shown for clarity. (a) Shown is the rigid body superimposition of the 3 structures. (b) Shown is the rigid body pairwise superimposition between 2SRC and 1FMK with a calculated RMSD of 0.8 Å. (c) Shown is the rigid body pairwise superimposition between 2SRC and 1Y57 with a calculated RMSD of 21.0 Å. (d) Shown is the pairwise flexible superimposition between 2SRC and 1Y57 after introducing a twist at position 259 using FATCAT that resulted into a calculated RMSD of 2.7 Å.

  Figure 3. Comparison between 3 structures of human tyrosine kinase C-SRC by rigid body & flexible superimposition.Shown in blue is the unbound form (PDBID 1FMK)[START_REF] Xu | Three-dimensional structure of the tyrosine kinase c-Src[END_REF] , in red is the conformation (PDBID 2SRC) when it binds to phosphoaminophosphonic acid adenylate ester (ANP)[START_REF] Xu | Crystal Structures of c-Src Reveal Features of Its Autoinhibitory Mechanism[END_REF] and in green is bound form to MPZ ligand (PDBID 1Y57)[START_REF] Sw | The Crystal Structure of a c-Src Complex in an Active Conformation Suggests Possible Steps in c-Src Activation[END_REF] . Ligands are not shown for clarity. (a) Shown is the rigid body superimposition of the 3 structures. (b) Shown is the rigid body pairwise superimposition between 2SRC and 1FMK with a calculated RMSD of 0.8 Å. (c) Shown is the rigid body pairwise superimposition between 2SRC and 1Y57 with a calculated RMSD of 21.0 Å. (d) Shown is the pairwise flexible superimposition between 2SRC and 1Y57 after introducing a twist at position 259 using FATCAT that resulted into a calculated RMSD of 2.7 Å.

Figure 4 .

 4 Figure 4. PB alignment between two structures of human tyrosine kinase C-SRC bound to ANP ligand (PDB ID 2SRC) and MPZ ligand (PDB ID 1Y57) captured by altconfMAT matrix.Shown is alignment of chain A for both structures. The region responsible for the high RMSD value calculated after rigid superimposition is circled in red. This region displays drastic PB variations that reflect a major twist in the backbone at the level of leucine 255.

Figure 6 .

 6 Figure 6. Example of the structural variation captured by dynamicsMAT matrix after 5 and 10 ns MD simulations. Shown is the PB variations observed in an amidase (PDB ID 4LF0) after 5 and 10 ns MD simulations. On the left is the cartoon representation of the x-ray structure and on the right are its corresponding PB sequences after 5 and 10 ns MD simulation. The regions marked in red represent structurally varying regions between the two structures extracted at these simulation times.

Figure 7 .

 7 Figure 7. Network showing structural (dis)similarity between mouse and human sulfotransferases each having multiple alternate conformations. The red nodes are the group of human sulfotransferase structures while the green nodes represent the group of mouse structures. Intragroup alignments are colored in red and green respectively. Intergroup alignments are colored in blue.Distance between nodes are inversely proportional to the normalized alignment scores. Intragroup distances for the mouse sulfotransferases are lower than any intergroup distance. Some of the intragroup distances for the human sulfotransferases are greater than some its intergroup distances.

Figure 8 .

 8 Figure 8. Distribution of PB-ALIGN normalized alignment scores.On the left is the distribution of the normalized alignment scores for the mouse sulfotransferase structures group. On the right is the distribution of the normalized alignment scores for the human sulfotransferase structures group. In red are the inter group normalized alignment scores. In green are intra group alignment scores. Intragroup scores for the mouse sulfotransferases are higher than any of its intergroup scores. Intragroup scores for the human sulfotransferases show some overlap with some of its intergroup scores. This overlap is indicated in blue in the histogram on the left.

  The closest inter group structures (PDB ID 1LS6 chain A and PDBID 2ZYV chain X) have a normalized alignment score of 1.83 and the farthest pair (PDB ID 2D06 chain A and PDB ID 2ZVP chain X) have 1.65. Their corresponding all atom superposition RMSDs are 0.40 Å and 0.46 Å, respectively. The difference in normalized alignment score (0.177) and

Acknowledgement

The authors are thankful to Dr. Swapnil Mahajan and Lionel Hoffmann for valuable discussions on PB substitution matrix and help with MD simulations respectively.

Funding

This work was supported by the Région Réunion and the Fond Social Européen, in the form of a PhD. grant [grant no.20131528] to IV. This work was in part supported by Conseil Regional des Pays de la Loire in the framework of GRIOTE project. AdB and FC acknowledge grants from the Ministry of Research (France), National Institute for Blood Transfusion (INTS, France), National Institute for Health and Medical Research (INSERM, France) and labex GR-Ex. The labex GR-Ex, reference ANR-11-LABX-0051 is funded by the program "Investissements d'avenir" of the French National Research Agency, reference ANR-11-IDEX-0005-02. AdB acknowledge supports by University Paris Diderot, Sorbonne, Paris Cité (France), FC acknowledge supports by Université de La Réunion, Faculty of Sciences and Technology. NS and AdB acknowledge to Indo-French Centre for the Promotion of Advanced Research / CEFIPRA for collaborative grant (number 5302-2). Research in NS laboratory is also supported by Department of Biotechnology, Government of India. NS is a J.C. Bose National Fellow.