Nicolas Bouleau

Francis Hirsch

Some Results on Lipschitzian Stochastic Differential Equations by Dirichlet Forms Methods

Since the impulse given by P. Malliavin, the stochastic calculus of variations has been mainly applied to stochastic differential equations with C°° coefficients, see Ocone [01] for a comprehensive exposition.

But it is also important for applications to get regularity results for solutions of SDE with less smooth coefficients and in particular under Lipschitz hypotheses which axe, in dimension greater than one, the most natural hypotheses of existence and uniqueness of solutions.

The celebrated integration by parts method cannot apparently be extended beyond the case of functionals in the domain VL of the Ornstein-Uhlenbeck operator (ID 2 ,2 with the notations of Watanabe [Wl]), so that the regularity of solutions of Lipschitzian SDE must come from specific technics. Especially well adapted are Dirichlet forms methods which allow to exploit intensively the fact that Lipschitz functions operate on D 2 ,i = V\/-L.

We give here an account of results already obtained in this direction by Dirichlet forms methods and we present in details a new example which gives rise to an extension of the stochastic calculus. The first part introduces the framework of the Dirichlet space related to the Ornstein-Uhlenbeck semigroup on the Wiener space and recalls the absolute continuity criterion (cf [B-Hl] [B-H2]) for functionals in ID24 or ^24 an( i some consequences on Lipschitz SDE.

The second part is devoted to the regularity of solutions of Lipschitz SDE with respect to initial data. It is shown that the solution is differentiable in a slightly weakened sense. That gives for example the following simple result: under these hypotheses, if the initial variable XQ has a density, then X t has a density for all t.

After recalling the definition of the capacity associated with the Ornstein-Uhlenbeck Dirichlet form, it is shown in the third part, that the solutions of Lipschitz SDE can be refined, by taking quasi-continuous versions for each i, into processes with continuous paths outside a polar set and unique up to a quasievanescent set. The main tool here is an extension of the Kolmogorov theorem on existence of continuous versions to the case where the measure is changed to a capacity.

This allows to study the solutions of Lipschitz SDE under measures which do not charge polar sets. In the last part, using Wiener chaos decompositions of positive distributions, we show that this property allows an extension of the stochastic calculus by constructing a finite energy measure singular with respect to the Wiener measure and for which the coordinates do not build a semimartingale. This answers a conjecture formulated in [B-H4].

I The structure of Dirichlet space on the Wiener space associated with the Ornstein -Uhlenbeck semigroup*

The Wiener space ft = {UJ e C(1R+, JR d )]u;(0) = 0} is equipped with the topology of uniform convergence on compact sets, with its Borelian <r-a!gebra and with the Wiener measure m which makes the coordinates a standard Brownian motion. T denotes the m-completed <j-algebra of a(B t ;t £ K+), and T t the /"-m-completed a-algebra of cr(B 8 ; s < t). We consider on L 2 (m) the Ornstein-Uhlenbeck semigroup P t a strongly continuous symmetric Markovian semigroup characterized by

P t [exp{J h(s).dB a -l\\hf}} = expie-'* J h(s).dB. -i||e-t/a fc|| a }

Vh€H = L 2 (R+,TR d ).

The self-adjoint operator generator of P t is denoted by L. It corresponds to P t (cf [B-Hl]) a Dirichlet form with domain

D = V(V-L)

given by ((«,«)) = || V=Iu||i, (m) .

That means that the space ID with the norm (||w||£ /2 ( m ) + ((w,u)) ) 1 ^2 is complete and that normal contractions operate: For all u € ID, for all measurable v such that VU\V(UJ)\ < \u(u>)\ and Vu>,t</|v(u;) -v(u>')\ < \u(u) -u(u/)|, one has v £ TD and((v,v))<((t*,ti)).

This Dirichlet form is local ([B-Hl] p239) and possesses a carre-du-champ operator, i.e. a symmetric bilinear continuous map T from D x ID into jL 1 (m) such that W,vG DnX°°(m), 2((tiv,u)) -((v,u 2 )) == I vT (u,u)dm.

This Dirichlet structure (ft,.? 7 ,m; ((.,.)), D) is related to the Sobolev spaces which are classically defined on the Wiener space in the following way:

Let ID Pv ,,p G (1, oo), s G 1R the closure of the linear space generated by polynomials in continuous linear forms on ft for the norm

\\F\\ p , a = \\(I-L)^F\\ p
(cf [Wl]). Then D = D 2)1 and \\F\\l, +((F,F)) = M fl .

Let

= / L(s)ds

Jo

where £ n is a complete orthonormal system of H = X 2 (IR + , IR"*), then for all u £ D the following limit exists in probability (cf [B-H2])

V { »(«)(w) = limrMuCw + ttn) ~ «(w)]
and one has

The derivation operator D (cf [Wl]) which can be defined by n and which is continuous from D = D 2 ,i into L 2 (ft, #) is related to the carre-duchamp operator T by

r(u,tt) =< Du,Du >#, Vu G ID 2 ,i.
This relation between the carre-du-champ operator and the derivations in the directions of Cameron-Martin vectors (f G ft s.t. £ £ H) allows, by an extension of the co-area formula of Federer [Fl], to obtain the following absolute continuity criterion: Proposition 1 . Let u = (w 1? ... ,u n ) G (JD 2 ,i) n , </ &e image by u of the measure det [T(u,u*)].rn is absolutely continuous with respect to the Lebesgue measure on IR n .

When n = 1, this result is true for any local Dirichlet space (cf [B-Hl]) and also for the local energy part in any Dirichlet space on a locally compact space (cf [Bl]).

In fact proposition 1 remains valid for u in (D^i) n defined by E>Ji = {u : fi -* IR; 3Q n € T, fi" T ^ Vn3t/ n 6 B 2>1 , u = u n onft n } and for u G ID^J, T(u, tx) depends only on tx.

An important application of the extension of Dirichlet forms methods to the case of the Wiener space is the study of stochastic differential equations. Let us specify the Lipschitz hypotheses which will be in force in the sequel:

Two Borelian functions cr, 6 are given

a : H+ x R n -• U nXd b : IR+ x IR n -> IR n
and there exists K :

IR+ -• 1R + such that VTGIR+, V<G[0,T], Vx,yGlR n \v(t,x)\V\b(t,x)\<K(T)(l + \x\) \a(t, x) -o(t, y)\ V \b(t, x) -b(t, y)\ < K{T){\x -y\).
One is concerned by the equation

dX t = *(t,X i ).dB i + b(t,X t )dt. ( 1
)
From the fact that contractions hence Lipschitz functions operate on the Dirichlet space, it follows (cf [B-Hl] [B-H2]) that the solution of ( 1) is such that the map t --> X t is continuous from 1R+ into (©2,1 ) n and by writing down a stochastic differential equation satisfied by the matrix T(X t ^X*) it is possible to bring out conditions under which X t has a density by application of proposition 1.

For example if Ak = {(*,y) : <K*?y) ls °^ rank k} and if Tk is the essential beginning of Ak for (Xt)t>o> one gets that for t such that m({t > Tk}) > 0 and for almost all subspace V of JR n of dimension fc, the projection of X t on V, knowing {t > Tjt}, has a density with respect to the Lebesgue measure on V.

II Regularity of solutions of Lipschitz SDE with respect to the initial data

Under these Lipschitz hypotheses, it is known (cf [K2]) that there exists a version (Xf (k>))(t,a:)€]Ft+xR n of the solution of (1) starting at x, such that for almost all UJ the map (£, x) -» Xf(u) is continuous and for all t > 0 x -» X?(u>) is an onto homeomorphism of IR n . If it is supposed further that a and b are C 1,0f with respect to a: then x -» Xf (a;) is an onto C 1 -diffeomorphism.

Under the only Lipschitz hypotheses, X* is of course not C l with respect to x in general, but it is possible to show that the Jacobian - §^(X*(a;)) exists in a weakened sense and satisfies a SDE which can be written explicitly.

For this, consider the space Q, = H n x Q, equipped with the probability m = h(x) dxxm where m is the Wiener measure on 0, and h a strictly positive continuous function such that fh(x)dx = 1, / \x\ 2 h(x)dx < +oo. The cr-algebras generated by applications J5 5 , s < t and completed for m are denoted by T t .

(fi, in) gets a natural Dirichlet form associated with the derivations in directions given by the canonical basis of IR n . In other words the tool is here the form du dv

«-.•))-/»££)*

with domain and operators d/dxi suitably defined.

We denote by (Xt)t>o [resp. (I?t)t>o] the class of the process (X?) t >o [resp. of the Brownian motion (B t )t>o\ enlarged up to m-evanescent sets.

Proposition 2 . Under the Lipschitz hypotheses, a) for m-almost all u, V* > 0, Xfa) G {Hi oc (lR n )) n b) there exists a process (Mt(x,u)) t >o, (ft)-adapted, with continuous paths and values in (j?£ n (IR), such that for m-almost all u>, Vi > 0, { §^(X*(u;)) = M t (x,uj) dx -a.e.] c) let a 1 and V be fixed Borelian versions of the derivatives - §^cr(t,x) and T^b(t,x), then M is the unique (Tt)-adapted continuous solution, defined up to an m-evanescent set, of the SDE

i dM t = [a'(t, X t ).M t ] dB t + [b'(t, X t ).M t ] dt 1 Mo = /
It follows from this proposition and from a variant of proposition 1 applied to the Dirichlet structure on Q explained above that the equation (1) with initial value a random variable independent of (Bt) possessing a density , has a solution which admits a density for all t > 0. This was known, apparently, in dimension greater than one, only under C 1,Qf hypotheses.

In dimension 1, there is an explicit solution : if we write as before o\, V for fixed Borelian versions of the derivatives of a and b with respect to x, the process Y' = expjf (£ot(s,X')dBi -±£te(s y Xn?ds) + £v{3,Xf)da} is such that for m-almost all u, Va, 0 6 H, V* > 0, Xf(«) -X?(u>) = /' Y?(u) dx. Ja III Regularity, up to a polar set, of the solutions and their flows*

The Dirichlet form on the Wiener space associated with the Ornstein-Uhlenbeck operator makes it possible to look at properties of the Brownian motion satisfied up to a zero capacity set (cf [F3], [K1], [S2]).

We study here, from this point of view, properties of solutions of Lipschitz SDE. A work in the same spirit was done independently by J. Ren (cf [Rl]) for equations with C°°-coefficients and with thin sets associated with C p , 5 -capacities (cf [Ml]).

We denote by C the capacity associated with the Ornstein-Uhlenbeck Dirichlet form. It is defined by

C(G) = m/{||tx||2 a ; u £ D, xz > 1 m -a.e. on G} if G is an open set,

and by

C(G) = inf{C(G) G open and G D A}

ifAef.

If C(A) = 0, A is said to be a polar set. / : fi -* H is said to be quasi-continuous with respect to the capacity C if Ve > 0,3ft c open with C(Q € ) < e such that / restricted to the complementary Q c e of fi c is continuous.

Two processes (UA)A€A> (^A)ACA defined on Q, are said to be C-indistinguishable if there exists a polar set A such that Va; ^ A, VA £ A, u\(u) = v\(u>).

Under the Lipschitz hypotheses, we know that the solution X%(UJ) of equation (1) starting at x is such that for fixed i,x, Xf £ IDJj. It follows that this random variable admits a quasi-continuous version defined up to a polar set . The following extension of the Kolmogorov theorem gives conditions under which it is possible to put these quasi-continuous versions together to get a continuous process outside a polar set.

The family (v x ) is unique up to C-indistinguishability and the following uniformity properties hold:

There exist open sets (fi c )oo with compact complementary fi£ such that a) Ve > 0, C(fi c ) < e and the map (x,u;)

G IR r x ft£ -• v«(w) G IR w continuous b) V#, 0 < A < a,(l -E-=i l/a,-)/2p * = 1,..., r 3tf > 0, Ve > 0, Vi? > 0, 3?7 > 0, (a; G Ql \x\ V |y| < IR, |* -y\ < r,) =» |ti r («,) -t;»| < If EU I*, -V.-| ft -
This criterion allows to show that under the Lipschitz hypotheses and for a given fixed initial condition x G H n , the solution X? of equation ( 1) can be made more accurate into a process (X t )t>o unique up to C-indistinguishability such that i) t -» X t is continuous, ii) for all t X t is quasi-continuous and X t = X* m -a.s.. This result has been extended, by using a Banach valued space TD 2l i by D. Feyel and A. de la Pradelle [F2] to the case of Ito processes of the form X t = / a s .dB s + I p 8 .ds

Jo Jo

with a, /3 G L 2 (TR+,TD) and adapted.

The previous criterion of Kolmogorov type, allows also to obtain a quasicontinuous version Xf of Xf which is for w outside a polar set, continuous in (t,x) and an onto homeomorphism with respect to x; but for this C 1,or -hypotheses in x are needed for a and b (cf [B-H4]).

With C 2 ' a -hypotheses, the differentiability with respect to x of the flow is obtained with a quasi-continuous regular Jacobian matrix -^Xf(uj) continuously depending on (t,x) for u; outside a polar set (see [B-H4] theorems V.l and V.2 for more precise results).

The family (u t ) is a distribution martingale in the sense of Yan [Yl] and the i/ t 's are probabilities on fi.

Following the notations of [Yl] we compute now the adapted projection of the distribution Du where D is the gradient operator. We have ( D ,nt)=g ^yv. ((w M )*-,2 ^ai vo)

and by using the formulae of Shigekawa [Si] hl n (h**) = 7 n+1 (fc®<" +1 >) + nWhlfl^ih^-V)

we obtain

(^M " ^I-SP'W" (8)
where || . || is the norm of £ 2 ([0,1]) and

(^ol[o,*]) = / h 0 (s)dB sl */0
and with a = inf{t : / h 2 0 (s)ds = l}. Jo If we write (8) in the following form

(Du)«(t) = CM-*

the formula of Ito-Ustunel [Ul] gives u = rn + 6(((.)v.)

From now on, we suppose a < 1 and h 0 with bounded variation. (/i 0 l[o,t]) possesses then a version which is an (J^)-adapted process continuous in (t,u>) on [0,1] x O. In the sequel £ is supposed to be defined from this version. (The following construction is also possible without supposing the variation of h 0 to be bounded. A version of (/iol [o,t]) should be chosen continuous in t and quasi-continuous in a;, what is always possible).

That leads to the following lemma:

Lemma 4 . The process rt ,ds

M t = B t -f (s

Jo (where the integral is, for t > a, a semi-convergent integral) is an (J^)-Brownian motion under v.

Proof. Using the fact that the measure v restricted to the <7-field J^ is the measure v u we break up the interval [0,1] in [0,a), {a}, (a, 1]. a) First if t < a, the distribution v t is a random variable in L 2 , in other words the measure v t has a density in L 2 with respect to m.

Indeed setting $ = JjJ hl(s) ds.

It follows then from (8) that if we set n t = ^ we have • Writing B t = M t + /o C* ^> we see *^a' * n or der that the coordinates (2? t ) fail to be a semimartingale, it is sufficient to choose h 0 in such a way that the continuous process Jg C* ds fails to have a finite variation in the neighbourhood of a under i/.

n t = 1 + / (5^,
As m and i/ are mutually singular, we must express ( s in terms of the Brownian motion M t under v.

Lemma 5 . For t < a, it holds h Q (s)

' -^f ^

Therefore there exists a Brownian motion (W t ) such that Y t = W^t) and the integral to be studied can be written

I =ti\ho(t)\\W m \dt

Since by ( 12) it holds £'(£ _1 ( 5)) = $j$ one S ets with ( 13)

'-r^w-( M)

Hence it is enough to find a function u > 0 satisfying (11) and such that in ( 14) one gets +oo. For this we use the following version of a lemma of Jeulin [Jl]:

Lemma 6 . Let R t be a positive measurable real process on a probability space (0,P) such that 1) the law v of R t does not depend on t 2) K{0}) = o

3) f xdu(x) < +oo then for any positive Radon measure \i on

IR+ 0 f~dfi(t) <+oc =» •f?R t dii(t)eL 1 (W) ii) f~ d/i(t) = +oo =» / 0 °° R t dfjL(t) = +oo Fa.s. Proof. The point i) is clear because ]EiJ t = J x du(x) < +oo.
For the second point let n € IN and

J n = {f£° R t dfi(t) < n}. Suppose F( J n ) > 0, then E[lj.ft] = JJ° du E[l Jn l w >.}] = J 0 ~ duE[(l Jn -1 { *,<*}) + ] >f~du(lP(J n )-v([0,u}))+
and by the hypothesis 2) \im u -+ o v([0,u]) = 0, hence the last integral is equal to a n > 0. By integration There are several functions satifying ( 11) and ( 15), for example u(s) = l/(-+ log(s + 1)), which gives g2a 2(a-t)

V') = at WO-Let us summarize the preceding discussion. Let ho associated with u by ( 12) and ( 13) and let v be the distribution on the Wiener space associated with ho by (3) and (4). v is a distribution of Watanabe in JD 2 ,_i and is also a positive measure which does not charge polar sets.

For t < a, on the a-field J^ the measures m and v are equivalent, (B s ) s < t is an (^)-Brownian motion under m and an (^)-semimartingale under v.

For t > a, the measures m and v are mutually singular on the <7-field ^, (B 3 ) s <t is not an (^))-semimartingale under i/, nevertheless the process

h Q (v)
B °~ f r* J ^ f h o^)dB u dv

Jo [jy h£(u)du Jo

is an (^)-Brownian motion under i/.

It is possible to build examples of measures which do not charge polar sets and for which the singularity which is here at the point a, appears along a whole interval. Such measures are solutions in sense of distributions of Watanabe of the stochastic differential equation which defines the exponential of Doleans. This will be published elsewhere.

Proposition 3 .

 3 Let (u x ) xe ^r be a family of elements of D and p, <*i, ...,a r stricly positive real numbers. Suppose the following conditions hold <> Vx,y€K r \u x -u y \ p €JD o 3L : IR+ -»IR+, V.R > 0, Vx, y G HT 1*1 V |y| < IR =• || |u. -tt,|»||l 4 < £(*) EU |*.--y,f Tfeen ifeere exists a family (v x ) x€ j^-such that i) x -> v x (u>) ts continuous ii) for all x v x is a quasi-continuous version of u x .

  Jn) > On / dl*(t) Jo nl what gives ii) by contraposition. It follows by taking R t = ^p that / = +oo as soon as r°° u(s)x/s f /irN /o (JW^ = +C ° (15)

M a + B t -B a for t > a.

  > 0 and in fact v t and m are equivalent, v t being a probability it holds JEn t = 1 and the classical Girsanov theorem (cf[LI]) applies and gives the result. b) The study of the limit of M t for t f a is obvious under */, and M t is an J^-

	d£ 5	for * < a	(9)
	and because P[/J £* <£s < +oo] = 1 for t < a we get
	n t = exp[/ ( s dB s -~ Jo	Z Jo	( 2	3 ds)
	hence n Brownian motion under v on [0,a].
	c) At last it follows easily from formula (7) that under v a-fields J^ and cr(B 3 -
	B a ,Q, < s < 1) axe independent and that
	< v, GF >=< i/, G	><m 1 F>
	if G is ^-measurable and if F is <r(B s -B a ,a < 5 < Immeasurable, what gives
	the result by
	M t =

t

IV Stochastic calculus under a probability which does not charge polar sets

We keep in the sequel the preceding globally Lipschitz hypotheses and look at the solution of X t = x + f <T(S, X,).dB, + f b(s, X s ) ds (2) Jo Jo which is continuous in t, quasi-continuous in u> and unique up to C-indistinguishability.

This process is well defined under any probability measure on the Wiener space which does not charge polar sets. A. The first case is when the right hand side of (2) also makes sense under such a measure v.

To be precise with the changes of measure we introduce the a-fields J^ = &{Bs-> s < t) without any completion.

It can be shown (cf [B-H4]) that there exists an (^)-adapted solution , X t , of (2) such that, for fixed i, X t is quasi-continuous in a;, and for quasi every u>, t -• X t (w) is continuous. Then if v is a probability measure on £1 which does not charge polar sets and such that the process (B t ) is an (^)-semimartingale under i/, the process X t is the solution of the same SDE under i/, that is to say X t satisfies i/-a.e.

Vi X t = x+ f t a(s,X s ) l <dB 9 + f'bis.X^ds Jo Jo where / a(s, X s ) v -dB a denotes the stochastic integral under v.

For a one dimensional Brownian motion (d = 1), the law of the Brownian bridge 1E[ . \Bi = a] is an example of such a measure v which is singular with respect to the Wiener measure (cf [B-H4]). For d > 1 the same result is obtained by taking the conditional law of the Brownian motion given that B\ belongs to an (n -l)-dimensional hyperplan with the Gauss measure on it.

B. The case which gives rise to a true extension of the classical stochastic calculus is when under v (B t ) fails to be a semimartingale so that the right hand side has no direct meaning by itself.

We construct now a family of such measures on the Wiener space in the case d = 1 for simplicity.

The idea is to consider a conditional law of the form IE[. | f$ h 0 (s)dB s = 0] for /io€£ 2 ([0,l]),/o h 2 0 {s)ds = l. For using computations by decomposition on the Wiener chaos, we define this object as the positive measure which coincides on ID fl C(Q) with the distribution on the Wiener space

where S(0 = exp{£z,dB a -±£e a ds}.

Let h 0 ,hi,..., /i n ,... be a complete orthonormal system of L 2 ([0,1]), it follows from (6) that if g is a polynomial one has < v,g{h 0 ,hi,...,ti n ) >=< m,flf(0,fei,...,fc n ) > ( 7)

and v is then a positive distribution of Watanabe hence v is a measure (cf [N-U]).

For t G [0,1] we consider also the distrbutions v t defined by U ut (0 =<

M, = B l + [\ hM JlJ!gL rv aM^

To show that it is possible to choose h 0 in such a way that Jo v -a.s.

• (10) we perform some transformations: Let u be a function from [0, oo) into (0, oo) such that


Catherine Teyssier

Muriel Le Romancer

Stéphanie Sentis

Stéphan Jalaguier

Laura Corbo

Vincent Cavaillès
email: v.cavailles@valdorel.fnclcc.fr

Protein arginine methylation in estrogen signaling and estrogen-related cancers

Estrogen signaling pathways regulate cellular processes such as proliferation and differentiation, and if deregulated, are involved in several human pathologies. Posttranslational modifications (PTMs) play important roles in estrogen signaling pathways. This review focuses on recent findings pertinent to arginine methylation of non-histone proteins and their implications in estrogen signaling. We describe protein arginine methyltransferases and demethylases, the role of methylarginine proteins in estrogen action and cross-talk with other PTMs such as phosphorylation and lysine methylation. The relationships between various PTMs form a specific code which might play an important role in hormone signaling.

In addition, deregulations of arginine methylation or of enzymes responsible for these modifications could be key events in estrogen-dependent cancers such as breast cancer.

Arginine methylation was first described as a post-translational modification (PTM) of histones in the late 1960s. However, only recently the responsible enzymes (Box 1) and wide variety of substrates of this modification have been identified. While the addition of each methyl group does not modify the charge of the residue, it does increase its bulkiness and hydrophobicity. Interactions of a methylated protein with its binding partners can therefore be affected by this modification and impact the physiological functions of the substrate protein.

Methylarginine substrates include transcription factors, nucleic acid-binding factors, signal transducers, splicing factors and histones. Because of the large number of substrates, protein arginine methyltransferases (PRMTs) and arginine methylation regulate various cellular processes such as cell differentiation, DNA repair, RNA processing, signal transduction, cellular localization, and apoptosis [START_REF] Bedford | Protein arginine methylation in mammals: who, what, and why[END_REF][START_REF] Bedford | Arginine methylation an emerging regulator of protein function[END_REF][START_REF] Aletta | Protein arginine methylation in health and disease[END_REF]. In this review, we describe the recent studies implicating arginine methylation in estrogen transcriptional regulation and likewise, in estrogen-related diseases.

Role of PRMTs and methylarginine proteins in estrogen signaling

Estrogen action -Genomic and non-genomic effects

Nuclear estrogen receptors act mainly as ligand-activated transcription factors [START_REF] O'malley | The Year in Basic Science: nuclear receptors and coregulators[END_REF]. The binding of estrogens such as 17 -estradiol (E2) induces a protein conformation change in the receptor that allows recruitment of coactivator complexes with chromatin-remodeling or histone-modifying activities [START_REF] Lonard | The expanding cosmos of nuclear receptor coactivators[END_REF] (Box 2). Steroid-regulated promoters recruit the ligand-bound receptors and regulatory proteins in an ordered, cyclical manner with multiple rounds of coactivator assembly and disassembly [START_REF] Rosenfeld | Sensors and signals: a coactivator/corepressor/epigenetic code for integrating signal-dependent programs of transcriptional response[END_REF]. Histone acetylases and some methyltransferases lead to a more open chromatin and increase gene transcription. P300, an acetyltransferase, and PRMT1 and CARM1, two well-characterized PRMTs, cooperate synergistically to regulate hormone target genes [START_REF] Lee | Role of protein methylation in regulation of transcription[END_REF]. The activated estrogen receptor can also bind corepressors, such as RIP140 [START_REF] Augereau | The nuclear receptor transcriptional coregulator RIP140[END_REF], which recruit enzymes with histone deacetylase activity (HDAC) to repress transcription. Anti-estrogen compounds like tamoxifen prevent steroid action by inducing a conformation in the ligand binding pocket of the receptor that fails to bind coactivators and allows the recruitment of corepressor proteins (such as NCoR and SMRT) together with HDACs [START_REF] Jones | N-CoR-HDAC corepressor complexes: roles in transcriptional regulation by nuclear hormone receptors[END_REF].

In addition to transcriptional regulation, ERα also mediates events through its association with signaling molecules outside the nuclei and independent of its direct influence on the genome [START_REF] Bjornstrom | Mechanisms of estrogen receptor signaling: convergence of genomic and nongenomic actions on target genes[END_REF]. For example, estradiol triggers cell proliferation and cell survival through activation of MAPK kinases and Akt pathways [START_REF] Migliaccio | Crosstalk between EGFR and extranuclear steroid receptors[END_REF]. These non-genomic actions of estrogens occur rapidly and independently of protein synthesis. At the molecular level, ERα palmitoylation anchors a pool of ERα at the plasma membrane [START_REF] Acconcia | Palmitoylation-dependent estrogen receptor alpha membrane localization: regulation by 17beta-estradiol[END_REF] where it interacts with Src, PI3K and other scaffold proteins as MNAR (modulator of non-genomic activity of ER). This complex can therefore activate the downstream pathways [START_REF] Greger | Phosphorylation of MNAR promotes estrogen activation of phosphatidylinositol 3-kinase[END_REF]. Recently, novel non-genomic action of estrogens in breast cancer cells has been described, involving the association of membrane ERα with HDAC6. This association induces tubulin deacetylation, potentially contributing to estrogeninduced cell migration [START_REF] Azuma | Association of estrogen receptor alpha and histone deacetylase 6 causes rapid deacetylation of tubulin in breast cancer cells[END_REF].

Methylarginine proteins involved in estrogen action

Arginine methylation affects estrogen-mediated transcription by modifying both histone and non-histone proteins. Since histone methylation has been widely described in a various number of reviews [START_REF] Litt | Histone arginine methylations: their roles in chromatin dynamics and transcriptional regulation[END_REF][START_REF] Wysocka | Histone arginine methylation and its dynamic regulation[END_REF][START_REF] Pal | Interplay between chromatin remodelers and protein arginine methyltransferases[END_REF], this section focuses on methylation of non-histone proteins and their role in estrogen action (Table 1 and Figure 1).

Estrogen receptor

Because PRMT1 [START_REF] Koh | Synergistic enhancement of nuclear receptor function by p160 coactivators and two coactivators with protein methyltransferase activities[END_REF] and CARM1 (PRMT4) [START_REF] Chen | Regulation of transcription by a protein methyltransferase[END_REF] are ER coregulators (Box 2), ER could be a target for arginine methylation. Concordant with this, a recent study described ER as a methylarginine substrate [START_REF] Le Romancer | Regulation of estrogen rapid signaling through arginine methylation by PRMT1[END_REF]. In this study, Le Romancer et al. used in vitro methylation assays and showed that PRMT1, but not CARM1, methylated ER within the DNA binding domain. Mutation of arginine 260 into alanine (R260A) specifically abolished the modification by PRMT1. An antibody specific to methylated R260 confirmed ER methylation in living cells. Perhaps more interestingly, estradiol treatments of MCF7 cells drastically increased ER methylation within 5 minutes of treatment. A decrease of the methylated form was observed within less than one hour, suggesting enzymatic removal of the methyl group. Indeed, this disappearance was not due to ER degradation by the proteasome. Moreover, immunohistochemical experiments performed on human breast tumors showed that the methylated form of ER was exclusively localized in the cytoplasm of breast epithelial cells. Since rapid effects have been described for non-genomic estrogen actions, the role of methylated ER in those pathways was investigated. Interestingly, methylated ER was essential for E2-induced assembly of ER with Src, the p85 subunit of PI3K and the focal adhesion kinase (FAK), a Src substrate involved in the migration process. E2 activation of Akt was not observed if ER R260 was mutated to an alanine. Collectively, these results show that ER methylation is a prerequisite for its association with certain molecules involved in growth factor signaling. Since formation of the ER /Src/PI3K/FAK complex activates Akt and corresponding downstream pathways, ER methylation is likely involved in regulating cell proliferation and cell survival.

Transcriptional coregulators

Transcriptional coregulators play critical roles in controlling ER-mediated transcription. They function through protein-protein interactions, by facilitating or inhibiting recruitment of other coregulators or specific components of the transcription machinery (Box 2). Methylation of these cofactors may influence complex formation, enzymatic activity, subcellular localization, and stability, leading to a subtle regulation of ER-mediated transcription (Figure 1). SRC-3: p/CIP/AIB1/SRC-3 is a member of the p160 coactivator family involved in CARM1 and PRMT1 recruitment to ER target genes [START_REF] Lee | Role of protein methylation in regulation of transcription[END_REF]. SRC-3 (steroid receptor coactivator-3) protein levels are amplified in breast cancer and associated with poor prognosis [START_REF] Lahusen | The role and regulation of the nuclear receptor co-activator AIB1 in breast cancer[END_REF][START_REF] Harigopal | Estrogen receptor co-activator (AIB1) protein expression by automated quantitative analysis (AQUA) in a breast cancer tissue microarray and association with patient outcome[END_REF]. In fact, studies classified SRC-3 as an authentic oncogene [START_REF] Torres-Arzayus | High tumor incidence and activation of the PI3K/AKT pathway in transgenic mice define AIB1 as an oncogene[END_REF][START_REF] Kuang | AIB1/SRC-3 deficiency affects insulin-like growth factor I signaling pathway and suppresses v-Ha-ras-induced breast cancer initiation and progression in mice[END_REF]. Two independent groups found that CARM1 modifies SRC-3 [START_REF] Feng | Signaling within a coactivator complex: methylation of SRC-3/AIB1 is a molecular switch for complex disassembly[END_REF][START_REF] Naeem | The activity and stability of the transcriptional coactivator p/CIP/SRC-3 are regulated by CARM1-dependent methylation[END_REF], methylating it at the C-terminal region which contains the p300 and CARM1-binding sites. SRC-3 methylation, which is induced by estradiol, leads to the dissociation of CBP and CARM1 from SRC-3. Moreover, SRC-3 arginine methylation reduces its stability and causes an increase in its turnover. These results show that arginine methylation is implicated in the regulation of coregulator stability in response to estradiol. In addition to this role, arginine methylation also regulates the balance between coactivator complex assembly and disassembly. Studies suggest that repetitive association and dissociation of steroid receptors and coactivators from their target promoters may be required to maintain an activated state of transcription [START_REF] Rosenfeld | Sensors and signals: a coactivator/corepressor/epigenetic code for integrating signal-dependent programs of transcriptional response[END_REF]. Altogether, these results highlight coactivator methylation as an important regulatory mechanism in hormonal signaling.

CBP/p300: Recently, Lee et al. reported that p300 is methylated by CARM1 at R2142 which is located within the C-terminal GRIP1 binding domain. Interestingly, methylation of R2142 inhibits the interaction between p300 and GRIP1 whereas PADI 4 removes this methylation mark, thereby enhancing the p300-GRIP1 interaction. These methylation and demethylation events alter the conformation and activity of the coactivator complex and regulate estrogen receptor-mediated transcription [START_REF] Lee | Regulation of coactivator complex assembly and function by protein arginine methylation and demethylimination[END_REF]. This provides another example of arginine methylation regulating coactivator complex assembly, conformation and function.

PGC-1 : PGC-1α (peroxisome proliferator-activated receptor gamma coactivator 1 alpha) serves as a coactivator for several nuclear receptors, including ER , as well as other transcription factors, such as nuclear respiratory factor 1 (NRF-1). Expression of PGC-1α is induced by a variety of physiological stimuli that regulate metabolic activity, such as exposure to cold, exercise, and fasting. PGC-1α regulates metabolic processes by affecting genes involved in mitochondrial biogenesis, respiration and gluconeogenesis. Teyssier et al.

showed that the C-terminal region of PGC-1α is methylated by PRMT1 at one or more sites within a glutamate and arginine rich region. PGC-1α coactivator activity and the ability of PGC-1α to induce expression of target genes are both compromised by mutation of modified arginine residues of PGC-1α or as a result of reduced PRMT1 levels by siRNA. Because inhibition of PRMT1 leads to inhibition of the expression of some PGC-1α target genes involved in mitochondrial biogenesis, it is tempting to speculate a role for PRMT1 in this regulatory pathway [START_REF] Teyssier | Activation of nuclear receptor coactivator PGC-1alpha by arginine methylation[END_REF]. However, the binding partners of methylated PGC-1α remain to be determined. RIP140: RIP140 (Receptor Interacting Protein of 140 kDa) is a well-known ER hormonedependent binding corepressor [START_REF] Augereau | The nuclear receptor transcriptional coregulator RIP140[END_REF]. In vitro and in vivo arginine methylation of RIP140 by PRMT1 has been described in 3T3-L1 adipocytes [START_REF] Mostaqul Huq | Suppression of receptor interacting protein 140 repressive activity by protein arginine methylation[END_REF], and liquid chromatography-tandem mass spectroscopy identified that arginine methylation occurs on R240, R650, and R948 as a mono-methyl mark, suppressing RIP140 repressive activity by two mechanisms. First, methylation of R240, located in the HDAC3 interaction domain of RIP140, impairs its interaction with HDAC3, reducing its repressive function. Second, arginine methylation of the three sites increases RIP140's interaction with CRM1, a component of the export machinery; this leads to RIP140 export to the cytoplasm, thereby reducing its nuclear repressive function.

So far, RIP140 arginine methylation links to adipocyte differentiation in a physiological context. RIP140 expression enhances fat accumulation in differentiated adipocytes cells by inhibiting lipolysis enzyme expression, whereas a constitutive hypermethylated mutant (where arginine is replaced by phenylalanine) failed to exert an effect on fat accumulation because of reduced repressor activity [START_REF] Mostaqul Huq | Suppression of receptor interacting protein 140 repressive activity by protein arginine methylation[END_REF]. Because RIP140 repressive activity is inhibited by arginine methylation and because RIP140 is a key factor in estrogen signaling, it will be of interest to verify whether RIP140 arginine methylation plays a role in estrogen genomic and non-genomic pathways.

Arginine methylation cross-talks with other modifications A code for protein PTMs

This concept has been initially proposed for histone tails which are heavily modified by methylation, in addition to other modifications like acetylation, phosphorylation, and ubiquitination. These epigenetic marks (defined as the "histone code") extend the genetic message beyond DNA sequences. They can be recognized or "read" by non-histone proteins containing for instance bromo-or chromodomains and participate in chromatin remodelling and transcriptional regulation [START_REF] Kouzarides | Chromatin modifications and their function[END_REF][START_REF] Taverna | How chromatin-binding modules interpret histone modifications: lessons from professional pocket pickers[END_REF]. An important point deals with cross-talks that exist between the different PTMs targeting histone tails, i.e. the modification of one residue influencing the modification of neighbouring amino acids. Finally, recent publications extended these observations to non-histone proteins, with nuclear receptor and coregulator PTM coding emerging as a major level of regulation [START_REF] O'malley | Cracking the coregulator codes[END_REF].

Lysine methylation of RIP140

RIP140 has been described as a substrate for several PTMs, including phosphorylation, acetylation and arginine methylation [START_REF] Mostaqul Huq | Post-translational modifications of nuclear corepressor RIP140: a therapeutic target for metabolic diseases[END_REF]. PTMs affect its subcellular distribution, protein-protein interaction, and biological activity in adipocyte differentiation. Huq et al. found recently that endogenous RIP140 is also modified by lysine methylation in differentiated 3T3-L1 cells [START_REF] Huq | Lysine methylation of nuclear co-repressor receptor interacting protein 140[END_REF]. Using mass spectrometry, they found three lysine residues (K591, K653, and K757) as potential methylation sites. The loss of lysine methylation by mutation of the target sites enhances arginine methylation, suggesting a communication between lysine and arginine methylation. This study unraveled a potential code of modifications between lysine and arginine methylation, which regulates the functionality of a non-histone protein.

Phosphorylation regulates arginine methylation

The proximity of SRC-3 methylation and phosphorylation sites suggests potential cross-talk between methylation and phosphorylation of SRC-3. Indeed, Naeem et al. showed that phosphorylation of SRC-3 decreased its methylation by approximately fivefold, indicating that prior phosphorylation antagonizes methylation at least in vitro [START_REF] Naeem | The activity and stability of the transcriptional coactivator p/CIP/SRC-3 are regulated by CARM1-dependent methylation[END_REF]. By contrast, RIP140 phosphorylation by PKCε triggers its arginine methylation by inducing subsequent recruitment of the chaperone 14-3-3 necessary for PRMT1 recruitment and therefore RIP140 methylation [START_REF] Gupta | PKCepsilon stimulated arginine methylation of RIP140 for its nuclear-cytoplasmic export in adipocyte differentiation[END_REF]. The combination of these PTMs stimulates RIP140 nuclear export and decreases its repressive activity. Altogether, these studies show that arginine methylation of a non-histone protein can be influenced by phosphorylation, enhancing the arguments in favor of a code of modifications extended to coregulators.

Regulation of PRMT activity: effects on estrogen signaling

Protein-protein interactions

Recent studies demonstrate that the methyltransferase activity of PRMTs can be modulated by protein-protein interactions. PRMT1 was initially identified as an interactor of the antiproliferative proteins BTG1 (B-cell translocation gene 1) and TIS21/BTG2, stimulating its activity towards selected substrates [START_REF] Berthet | Interaction of PRMT1 with BTG/TOB proteins in cell signalling: molecular analysis and functional aspects[END_REF][START_REF] Lin | The mammalian immediate-early TIS21 protein and the leukemia-associated BTG1 protein interact with a protein-arginine N-methyltransferase[END_REF]. PRMT1 activity is also modulated in a substratedependent manner by the BTG protein partner, hCAF1 (CCR4 associated factor 1) [START_REF] Rouault | Interaction of BTG1 and p53-regulated BTG2 gene products with mCaf1, the murine homolog of a component of the yeast CCR4 transcriptional regulatory complex[END_REF], which specifically inhibits Sam68 and histone H4 Arg3 methylation [START_REF] Robin-Lespinasse | hCAF1, a new regulator of PRMT1-dependent arginine methylation[END_REF]. Notably histone H4, when methylated by PRMT1 at Arg3, becomes a better substrate for p300, whereas acetylation of H4 by p300 inhibits its methylation by PRMT1 [START_REF] Wang | Methylation of histone H4 at arginine 3 facilitating transcriptional activation by nuclear hormone receptor[END_REF][START_REF] Huang | Methylation of histone H4 by arginine methyltransferase PRMT1 is essential in vivo for many subsequent histone modifications[END_REF]. This cross-talk has been described to contribute to the complex "histone code" in hormone signaling. Because both hCAF1 [START_REF] Prevot | Relationships of the antiproliferative proteins BTG1 and BTG2 with CAF1, the human homolog of a component of the yeast CCR4 transcriptional complex: involvement in estrogen receptor alpha signaling pathway[END_REF] and PRMT1 [START_REF] Koh | Synergistic enhancement of nuclear receptor function by p160 coactivators and two coactivators with protein methyltransferase activities[END_REF] have been described as transcriptional regulators of the nuclear receptor response, these results suggest a putative mechanism for hCAF1 in estrogenstimulated transcription through participation in PTM coding.

CARM1 activity is also regulated by protein-protein interactions. Indeed, CARM1 is a component of a nucleosomal methylation activator complex (NUMAC) and interacts with BRG1 (brahma-related gene 1), among others [START_REF] Xu | A methylation-mediator complex in hormone signaling[END_REF]. Once CARM1 interacts with BRG1, it can then methylate histones. Moreover, CARM1 and BRG1 are both recruited to ER-target genes and cooperatively activate ER-dependent transcription [START_REF] Xu | A methylation-mediator complex in hormone signaling[END_REF]. Therefore, modulation of PRMT1 and CARM1 activities by protein-protein interaction can be considered an important component of regulation in estrogen signaling pathway.

Phosphorylation of CARM1

The methyltransferase activity of CARM1 is negatively regulated by phosphorylation. Two groups describe phosphorylation of CARM1 at two different serine residues (S229, S217) during mitosis [START_REF] Higashimoto | Phosphorylation-mediated inactivation of coactivatorassociated arginine methyltransferase 1[END_REF][START_REF] Feng | CARM1 regulates estrogen-stimulated breast cancer growth through up-regulation of E2F1[END_REF]. Both phosphorylations abolish CARM1's ability to bind the methyl donor adenosyl-methionine and subsequently inhibit CARM1 methyltransferase activity. In both cases, CARM1 transactivation of estrogen receptor-dependent transcription is reduced.

Moreover, phosphorylation at S217 promotes CARM1 cytoplasmic localization, which occurs mainly during mitosis, suggesting that the CARM1 methyltransferase activity is turned off during mitosis when gene transcription is silent and turned on in G1 phase when gene transcription becomes active. Deregulation of this precise switch of CARM1 activity may affect progression of the cell cycle in breast cancer cells. Indeed, it was shown that CARM1 is involved in estrogen-induced cell cycle progression of MCF-7 breast cancer cells [46].

Arginine methylation and estrogen-dependent cancers

Prmt knockout mice have been developed, providing interesting findings on the relevance of arginine methylation in vivo [START_REF] Bedford | Protein arginine methylation in mammals: who, what, and why[END_REF]. For instance, Prmt1 and Carm1 gene disruptions result in an embryonically lethal phenotype and neonatal death respectively, confirming a fundamental role for these enzymes in cellular metabolism [START_REF] Pawlak | Arginine N-methyltransferase 1 is required for early postimplantation mouse development, but cells deficient in the enzyme are viable[END_REF][START_REF] Yadav | Specific protein methylation defects and gene expression perturbations in coactivator-associated arginine methyltransferase 1-deficient mice[END_REF]. However, until now, these genetically modified animals have not provided information regarding roles for PRMT in estrogendependent physiological processes. By contrast, recent studies using microarray and quantitative PCR-based approaches described the aberrant expression of arginine methylation enzymes in estrogen-dependent cancers (Table 2).

PRMT expression in estrogen-related cancers

Several recent papers analyzed PRMT expression in estrogen-dependent cancers. Prmt1 and Fbxo11 expression is up-regulated in high tumor grade breast carcinomas [START_REF] Miller | An expression signature for p53 status in human breast cancer predicts mutation status, transcriptional effects, and patient survival[END_REF][START_REF] Van 't Veer | Gene expression profiling predicts clinical outcome of breast cancer[END_REF] and Prmt2, 5 and 10 expression is down-regulated in breast carcinomas [START_REF] Richardson | X chromosomal abnormalities in basal-like human breast cancer[END_REF][START_REF] Finak | Stromal gene expression predicts clinical outcome in breast cancer[END_REF]. In ovarian adenocarcinomas, Prmt1, 2 and 5 expression is down-regulated compared to normal tissues [START_REF] Ivshina | Genetic reclassification of histologic grade delineates new clinical subtypes of breast cancer[END_REF][START_REF] Hendrix | Fibroblast growth factor 9 has oncogenic activity and is a downstream target of Wnt signaling in ovarian endometrioid adenocarcinomas[END_REF].

Prmt1 isoforms exist as a result of alternative mRNA splicing, and amino acid sequence comparison indicates that they are all enzymatically active, but with different N-terminal hydrophobic regions. Goulet et al. found that the expression profile of Prmt1 splicing variants is altered in breast cancer [START_REF] Goulet | Alternative splicing yields protein arginine methyltransferase 1 isoforms with distinct activity, substrate specificity, and subcellular localization[END_REF]. Furthermore, this study showed that increased Prmt1 expression was detected in human breast tumor samples compared with adjacent normal breast tissue, confirming the studies mentioned previously. Strikingly, increased argininemethylated protein levels were also observed in breast cancer cell lines. Therefore, an altered Prmt1 isoform expression profile correlates with a differential pattern of arginine methylation in breast cancer cell lines, suggesting that misregulation of arginine methylation could contribute to the propagation of breast cancer.

Increased PADI4 expression in breast tumors

Immunohistochemistry detected significant PADI4 expression in various malignancies including breast carcinomas, endometrial carcinomas and uterine adenocarcinomas, with no detectable PADI4 expression in benign and healthy tissues [START_REF] Chang | Expression of peptidylarginine deiminase type 4 (PAD4) in various tumors[END_REF]. Quantitative PCR and western blot analyses also showed higher PADI4 mRNA and protein levels in malignant tissues compared to benign and non-tumor tissues [START_REF] Chang | Increased PADI4 expression in blood and tissues of patients with malignant tumors[END_REF]. Interestingly, in MCF7 breast cancer cells, PADI4 mRNA expression gradually increased with time after estradiol stimulation through both classical and non-classical ER-mediated pathways [START_REF] Dong | Estrogen-enhanced peptidylarginine deiminase type IV gene (PADI4) expression in MCF-7 cells is mediated by estrogen receptor-alpha-promoted transfactors activator protein-1, nuclear factor-Y, and Sp1[END_REF]. Altogether, these results suggest increased PADI4 expression in breast cancer tissues, probably in response to estradiol.

CARM1/E2F1 breast cancer growth induced by estrogens

ER controls the expression of cell cycle genes which in turn mediate breast cancer proliferation. Frietze et al. showed that CARM1 is essential for estrogen-induced cell cycle progression in MCF-7 breast cancer cells. Upon silencing of CARM1 by siRNA, the E2mediated stimulation of MCF-7 cell cycle progression was strongly reduced. This silencing resulted in decreased expression of E2F1 and E2F1-target genes (cyclin E1, cyclin A, cdc25A), providing a direct link between CARM1 and cell cycle regulation and identifying CARM1 as a potential new target in the treatment of estrogen-dependent breast cancer [46]. Aberrant expression of CARM1 has also been linked to human breast cancer [START_REF] El Messaoudi | Coactivator-associated arginine methyltransferase 1 (CARM1) is a positive regulator of the Cyclin E1 gene[END_REF], with elevated CARM1 levels found in aggressive breast tumors that also express high levels of the oncogenic coactivator AIB1 (amplified in breast cancer 1). Compiled high levels of CARM1 and AIB1 could work in synergy to enhance target gene expression and thereby cell proliferation.

ER methylation in breast cancer

In MCF-7 cells, E2-induced ER methylation is transitory suggesting the involvement of a not yet identified arginine demethylase whose expression or activity could be deregulated in breast cancer [START_REF] Le Romancer | Regulation of estrogen rapid signaling through arginine methylation by PRMT1[END_REF]. The evaluation of methylated ER with a specific antibody showed that ER is hypermethylated in 50% of human breast tumors [START_REF] Le Romancer | Regulation of estrogen rapid signaling through arginine methylation by PRMT1[END_REF]. Because ER methylation is necessary for estrogen-induced cellular kinase pathways, this deregulation in breast cancer could lead to sustained activation of those kinases which in turn would activate estrogen signaling. This bidirectional crosstalk appears to play a critical role in breast cancer development by maintaining the activation of signaling pathways and survival of breast cancer cells even in the presence of tamoxifen [START_REF] Tokunaga | The association between Akt activation and resistance to hormone therapy in metastatic breast cancer[END_REF]. It is then tempting to speculate that the deregulation of ER methylation may be involved in breast tumorigenesis and resistance to hormonal therapy. Further analyses are needed to consider methylated ER a new prognostic marker.

Concluding remarks

Arginine methylation impacts various levels of regulation in estrogen signaling, and thereby appears to be a key regulatory event in genomic and non-genomic estrogen actions. In genomic events, arginine methylation of transcriptional coregulators influences coregulators complex formation, activity, subcellular localization, and stability. Methylation of histones participates in chromatin remodeling in concert with other PTMs according to the histone code. In non-genomic events, estrogen receptor methylation is necessary for the formation of a transduction signaling complex and may participate in downstream events. Although, recent findings strongly enhanced our knowledge of the role of arginine methylation in estrogen signaling, a complete understanding will only be realized through answering fundamental questions (Box 3). For instance, since expression of other PRMTs beside PRMT1 and CARM1 is deregulated in estrogen-dependent cancers (Table 2), could other PRMTs be also involved in genomic and non-genomic estrogen regulatory mechanisms? Are other substrates implicated in estrogen signaling? Deregulation of arginine methylation, methyltransferases and demethylases are described in estrogen-related cancers, strengthening the notion of a connection between arginine methylation patterns and cancer progression. Moreover, hypermethylation of ER in human breast cancers may indicate that methylarginine proteins represent novel interesting prognostic biomarkers. Moreover, inhibiting methyltransferase activity, in particular that of PRMT1 and CARM1, by selective molecules appears as a potential therapeutic tool.

Recently, using an approach based on a protein virtual screen, Spannhoff et al. identified specific PRMT1 inhibitors. These compounds operate as a brake on steroid hormone actions, suggesting their potential for future drug development in cancer therapy [START_REF] Spannhoff | Cancer treatment of the future: inhibitors of histone methyltransferases[END_REF]. There is little doubt that detailed insights into the function and regulation of arginine methylation will unravel the pathogenesis of various diseases, in particular hormone-dependent cancers, and eventually contribute to the discovery of novel biological markers or therapeutic targets.

Box 1. Arginine methylation enzymes

The enzymes responsible for arginine methylation are called protein arginine methyltransferases (PRMTs). So far, eleven members have been identified in the PRMT family [START_REF] Lee | Minireview: protein arginine methylation of nonhistone proteins in transcriptional regulation[END_REF][START_REF] Krause | Protein arginine methyltransferases: evolution and assessment of their pharmacological and therapeutic potential[END_REF]. PRMT-encoding genes are well-conserved through evolution [START_REF] Krause | Protein arginine methyltransferases: evolution and assessment of their pharmacological and therapeutic potential[END_REF], sharing common structural and functional domains (Figure I). Although circumstantial evidence over the past 40 years depicted arginine methylation as an irreversible PTM, some data suggest that this modification could be reversed. For instance, histone H4 is transiently and cyclically methylated on arginine 3 [START_REF] Metivier | Estrogen receptor-alpha directs ordered, cyclical, and combinatorial recruitment of cofactors on a natural target promoter[END_REF] and estrogen receptor α methylation also appears transient [START_REF] Le Romancer | Regulation of estrogen rapid signaling through arginine methylation by PRMT1[END_REF].

While enzymes capable of removing or preventing such methylation have been identified, their roles are still controversial. Peptidylarginine deiminase 4 (PADI4) has been described to convert monomethylated arginine to citrulline by deimination [START_REF] Lee | Regulation of coactivator complex assembly and function by protein arginine methylation and demethylimination[END_REF][START_REF] Cuthbert | Histone deimination antagonizes arginine methylation[END_REF][START_REF] Wang | Human PAD4 regulates histone arginine methylation levels via demethylimination[END_REF]. However, a full reversion would then need an enzyme to convert citrullines to arginine residues. More importantly, PADI enzymes do not deiminate methylated arginine residues in vitro [START_REF] Hidaka | Methylation of the guanidino group of arginine residues prevents citrullination by peptidylarginine deiminase IV[END_REF][START_REF] Raijmakers | Methylation of arginine residues interferes with citrullination by peptidylarginine deiminases in vitro[END_REF], and it seems rather that histone citrunillation simply interferes with methylation of arginine residues. The first histone demethylase removing asymmetrical dimethylation at arginine 2 of histone H3 and symmetrical dimethylation at arginine 3 of histone H4 is the Jumonji domaincontaining 6 protein (JMJD6) [START_REF] Chang | JMJD6 is a histone arginine demethylase[END_REF]. However, no publication has confirmed these results and, very recently, the Bottger's group demonstrated that JMJD6 is a lysine hydroxylase involved in RNA splicing indicating that demethylation of methylarginine residues is not its major activity [START_REF] Webby | Jmjd6 catalyses lysyl-hydroxylation of U2AF65, a protein associated with RNA splicing[END_REF]. Altogether, this suggests that additional arginine demethylases remain to be identified.

Box 2. Estrogen receptor and coregulator complexes

Estrogens or anti-estrogens induce ER ligand binding domain (LBD) conformational change, DNA binding to specific response elements in promoter regions of target genes and recruitment of coregulator complexes [START_REF] O'malley | The Year in Basic Science: nuclear receptors and coregulators[END_REF]. Agonist-induced LBD conformational change allows recruitment of coactivator complexes composed of p160 coactivators or PGC-1 and of histone-modifying enzymes (p300, CARM1 and PRMT1), whereas antagonists permit recruitment of corepressors complexes containing SMRT, NCoR and HDAC enzymes.

Recruitment of coactivator complexes helps to pull down chromatin remodeling ATPase complexes (e.g. BRG1, SWI/SNF1), which participate in chromatin remodeling. This event facilitates the recruitment of the Mediator complex and thereby of the transcription machinery (which contains among others the RNA Polymerase II and the TATA Binding Protein, TBP) to the initiation start point [START_REF] Rosenfeld | Sensors and signals: a coactivator/corepressor/epigenetic code for integrating signal-dependent programs of transcriptional response[END_REF]. Enzymes associated with coactivator complexes are acetyltransferases (p300, SRC-1, SRC-3) and methyltransferases (CARM1, PRMT1). They modify histone tails inducing the open state of chromatin and gene expression. On the contrary, HDACs deacetylate histones leading to the closed conformation of chromatin and gene repression. In the presence of estradiol, ER can also bind corepressors, such as RIP140 which interacts with HDACs, leading to gene repression (Figure I).

Box 3. Outstanding questions -Arginine methylome and methylarginine target proteins

A fundamental issue is to define the entire arginine methylome in estrogen signaling, i.e. to set up proteomic approaches with high-performance mass spectrometry methods in order to describe all the methylated arginine residues in proteins involved in the estrogen pathway. In addition, although several methyl lysine-binding proteins have been identified [START_REF] Kim | Tudor, MBT and chromo domains gauge the degree of lysine methylation[END_REF], effectors for arginine-methylated proteins remain to be found. Actually, only three mammalian proteins have been demonstrated to bind methylarginine motifs through their Tudor domains [START_REF] Cote | Tudor domains bind symmetrical dimethylated arginines[END_REF][START_REF] Cheng | The arginine methyltransferase CARM1 regulates the coupling of transcription and mRNA processing[END_REF].

Defining the proteins which recognize methylated arginines of estrogen receptors and coregulators will certainly enhance our understanding of the downstream cascades dependent on arginine methylation in estrogen signaling.

-Regulation of arginine methylation

Studies addressing the integration of signal transduction and arginine methylation pathways are also needed. Specifically, how are arginine methylation and PRMT activities regulated by PTMs such as phosphorylation, acetylation, ubiquitination, sumoylation and trans-or automethylation? How are PRMT activity and expression regulated by estrogens or other stimuli? Finally, we also clearly need a better characterization of the enzymes that demethylate arginine residues.

-In vivo function of arginine methylation

Most of the published studies discuss the importance of arginine methylation in a cellular context. It will be of great interest to elucidate further the in vivo function of the modifications in estrogen target tissues. To achieve this goal, conditional knock-out approaches targeting specifically breast, ovarian or uterus tissues together with knock-in strategies in mice with unmethylatable mutants will be of a great help.

Figure legends

Figures inside Boxes: Box 1. Figure I-The PRMT family

The nine members of the mammalian PRMT family are shown. Recently, two related members, FBXO11 (also called PRMT9) and FBXO10, have been added to the family [START_REF] Krause | Protein arginine methyltransferases: evolution and assessment of their pharmacological and therapeutic potential[END_REF][START_REF] Cook | FBXO11/PRMT9, a new protein arginine methyltransferase, symmetrically dimethylates arginine residues[END_REF]. The common methyltransferase domain consists of a series of short conserved motifs (blue bars) that are important for binding the methyl donor and for catalysis. This catalytic core region is formed by a Rossman fold and two helices (black boxes). The less conserved -barrel structure (gray boxes) folds against the catalytic region to form the protein substrate binding cleft [START_REF] Troffer-Charlier | Functional insights from structures of coactivatorassociated arginine methyltransferase 1 domains[END_REF]. Additional specific motifs, such as the SH3 domain (SH3), the zinc finger (ZnF), the myristylation motif (Myr), the FBox motif and the tetratricopeptide repeat motif (TPR) are represented in green boxes. CARM1 uniquely contains a substantial C-terminal region which contains an autonomous transcriptional activation domain [START_REF] Teyssier | Requirement for multiple domains of the protein arginine methyltransferase CARM1 in its transcriptional coactivator function[END_REF].

Box 2 .

 2 Figure I-Gene regulation mechanism by estrogen receptor and its coregulators Estrogen receptor (ER) binds to specific regions called estrogen response element (ERE) represented by a pink box on the DNA target gene drawn as a thick black line wrapped around histones symbolized by yellow plots. The initiation start point is represented by a black arrow. The green and red arrows represent the impacts of coactivator and corepressor complexes on chromatin state and gene transcription.

Figure 1 -

 1 Figure 1 -Arginine methylation and estrogen signaling

  Figure

Table 1 . Methylarginine proteins involved in estrogen pathway

 1

	Enzyme	Substrates Arginine (R)	Impact	Refs
	PRMT1	Histone H4 R3	Gene activation	[40]
		ER	R260	Interaction with Src and PI3K	[20]
		PGC-1	R665, 667, 669	Stimulation of coactivator activity [28]
		RIP140	R240, 650, 948	Inhibition of repressive activity	[29]
	CARM1	Histone H3 R2, 17, 26	Gene activation	[19, 74]
		p300/CBP	R2142	Complex assembly regulation	[27]
			R1171	Complex assembly regulation
		SRC-3			[25, 26]
			R1178, 1184, 1195	Activity and stability regulation

Table

Table 2 . Arginine methylation enzymes in estrogen-dependent cancers

 2 Abbreviations: OC: ovarian cancer; BC: breast cancer; ER: estrogen receptor All studies were found in the Oncomine database (www.oncomine.org) but b .

	Enzyme (Locus)	Expression levels

a

Table

Acknowledgments

This work was supported by a grant from the "Association pour la Recherche sur le Cancer" (#3169) to L.C. and V.C. and the Ligue Contre le Cancer (L.C.). C.T. is funded by the "Ligue Contre le Cancer"