Pierre-Vincent Koseleff
email: koseleff@math.jussieu.fr

Daniel Pecker
email: pecker@math.jussieu.fr

Pierre-Vincent Koseleff

Daniel Pecker Modulo

Conway Polynomials

P. -V Koseleff

MODULO 2 CONWAY POLYNOMIALS OF RATIONAL LINKS

teaching and research institutions in France or abroad, or from public or private research centers.

1. Introduction J. W. Alexander (1928), and independently K. Reidemeister, discovered the first effectively calculable knot invariant, now called the Alexander polynomial. It is still one of the most useful of all knot invariants. Rational knots (or links) are a very important and simple class of knots. They are the knots which have a representation such that the abcissa has only two local maxima and two local minima (see figure 1). For each rational number α β Schubert has constructed a ). This link is a knot if α is odd, and a two component link if α is even. Spectacularly, he proved that every rational knot (or link) is of this form (see [START_REF] Schubert | Knoten mit zwei Brücken[END_REF][START_REF] Kawauchi | A survey of knot theory[END_REF][START_REF] Kauffman | On the classification of rational knots[END_REF]). He also proved that the links S( α β ) and S( α ′ β ′ ) are isotopic if and only if α = α ′ and β ′ ≡ β ±1 (mod 2) (see [START_REF] Schubert | Knoten mit zwei Brücken[END_REF][START_REF] Kawauchi | A survey of knot theory[END_REF][START_REF] Kauffman | On the classification of rational knots[END_REF]).

J. H. Conway discovered a simple method to compute the Alexander polynomial of a knot (see [START_REF] Conway | An enumeration of knots and links, and some of their algebraic properties[END_REF][START_REF] Kauffman | On knots[END_REF]). By a simple change of variables, he deduced the Alexander polynomial from a simpler one, now named the Conway polynomial. His method provides an easy algorithm (theorem 8) to deduce the Conway polynomial of a rational knot (or link) from its Schubert fraction α β .

In this note, we shall study the converse problem: given a polynomial, is it the Conway polynomial of a rational knot?

We obtain the following result (where the Fibonacci polynomials are defined as in [START_REF] Webb | Divisibility of Fibonacci polynomials[END_REF] by:

f 0 = 0, f 1 = 1, f n+2 (z) = zf n+1 (z) + f n (z), (1)
Theorem 1. Let P (z) ∈ Z[z] be the Conway polynomial of a rational knot (or link). There exists a Fibonacci polynomial f (z) such that P (z) ≡ f (z) (mod 2).

Then, we deduce a simple proof of a beautiful criterion due to Murasugi ( [START_REF] Murasugi | On peridic knots[END_REF][START_REF] Burde | Das Alexanderpolynom der Knoten mit zwei Brücken[END_REF])

Corollary 2 (Murasugi (1971)). Let P (t) = a 0 -a 1 (t+t -1 )+a 2 (t 2 +t -2 )-• • •+(-1) n a n (t n + t -n
) be the Alexander polynomial of a rational knot. There exists an integer k ≤ n such that a 0 , a 1 , . . . , a k are odd, and a k+1 , . . . , a n are even.

Since the Conway polynomial of the torus link T(2, n) is the Fibonacci polynomial f n (z) (see [START_REF] Koseleff | On Fibonacci knots[END_REF]), we see that these results are in fact characterizations of the modulo 2 Conway and Alexander polynomials of rational knots.

The simplicity of our computations provide an easy algorithm for the modulo 2 Conway (and Alexander) polynomials of rational knots. Theorem 3. Let K be a rational link (or knot) of Schubert fraction

α β = [2b 1 , 2b 2 , . . . , 2b m ].
Let us define the sequence e i by e 0 = 0, e i = (e i-1

+ 1 + b i ) mod 2, i = 1, . . . , m.
Then the modulo 2 Conway polynomial of K is the Fibonacci polynomial f D (z), where

D = m i=1 (-1) e i × (b i mod 2) -e m .
This algorithm may be used to easily see that some rational knots cannot be Lissajous knots.

The Conway and Alexander polynomials

The Conway polynomial of a knot (or link) K is denoted by

∇ K (z) = c 0 + c 1 z + • • • + c n z n .
The Conway polynomials of knots are characterized by the fact that they are even polynomials such that ∇ K (0) = 1. The Alexander polynomial ∆ K (t) is deduced from the Conway polynomial:

∆ K (t) = ∇ K t 1/2 -t -1/2 .
The Alexander polynomial of a knot is a Laurent polynomial. The following result allows us to recover the Conway polynomial of a knot from its Alexander polynomial.

Lemma 4. If z = t 1/2 -t -1/2
, and n ∈ Z is an integer, we have the identity

f n+1 (z) + f n-1 (z) = (t 1/2 ) n + (-t -1/2 ) n ,
where f k (z) are the Fibonacci polynomials.

Proof. Let A = z 1 1 0 be the (polynomial) Fibonacci matrix. If z = t 1/2t -1/2 , the eigenvalues of A are t 1/2 and -t -1/2 , and consequently Tr A n = (t 1/2 ) n + (-t -1/2 ) n . On the other hand, we have

A n = f n+1 (z) f n (z) f n (z) f n-1 (z)
, and then Tr

A n = f n+1 (z) + f n-1 (z).
From this lemma, we immediately deduce:

Corollary 5. Let the Laurent polynomial P (t) be defined by

P (t) = a 0 -a 1 (t + t -1 ) + a 2 (t 2 + t -2 ) -• • • + (-1) n a n (t n + t -n ).
We have

P (t) = n k=0 (-1) k (a k -a k+1 )f 2k+1 (z),
where z = t 1/2t -1/2 , and a n+1 = 0.

The following formula is an immediate consequence of this corollary:

f 2k+1 t 1/2 -t -1/2 = (t k + t -k ) -(t k-1 + t 1-k ) + • • • + (-1) k . ( 2
) Example 6. Let K be a knot of Alexander polynomial ∆ K (t) = 1 -(t 2 + t -2 ) + (t 3 + t -3 ),
(for example the 8 19 knot). Using the lemma, we get its Conway polynomial

∇ K (z) = 1 -f 5 (z) + f 3 (z) + f 7 (z) + f 5 (z) = 1 -f 3 (z) + f 7 (z).
We shall use the classical notation for continued fractions:

[q 1 , . . . , q m ] = q 1 + 1

q 2 + 1 q 3 + 1 . . . + 1 q m . (3)
Euclidean algorithm with even quotients provides the following continued fractions: The only knot theoretic result that we need is the following remarkable theorem.

Theorem 8. [4] Let K = S( α β
) be a rational knot (or link), and let

α β = [2b 1 , 2b 2 , . . . , 2b m ].
The Conway polynomial of K is

∇ K (z) = [ 1 0 ] -b 1 z 1 1 0 b 2 z 1 1 0 • • • (-1) m b m z 1 1 0 1 0 . ( 4
) Since it is known that S( α β ) = S( α β -α
), we can use this theorem to compute the Conway polynomial of any rational knot (or link).

Let us consider some examples.

Example 9 (The torus links). The torus link T(2, m) is the rational link (or knot) of Schubert fraction m = m 1 . It is proved in [START_REF] Kauffman | On knots[END_REF] and [START_REF] Koseleff | On Fibonacci knots[END_REF] that the Conway polynomial of T(2, m) is

f m (z). If m = 2k + 1 (i.e. T(2, m
) is a knot) we obtain the Alexander polynomial

∆(t) = f 2k+1 t 1/2 -t -1/2 = (t k + t -k ) -(t k-1 + t 1-k ) + • • • + (-1) k . ( 5
)
Example 10 ( The twist knots). The Twist knot T n is the rational knot of Schubert fraction n + 1 2 . We have the continued fractions

α β = [n, 2] = [n + 1, -2]. We immediately get its Conway polynomial ∇(z) = 1 -(-1) n ⌊ n+1 2 ⌋z 2 .
It is congruent to a Fibonacci polynomial modulo 2. The Fibonacci knots, introduced by J. C. Turner ( [START_REF] Turner | On a class of knots with Fibonacci invariant numbers[END_REF]), are interesting examples of rational knots. Their Alexander and Conway polynomials are studied in [START_REF] Koseleff | On Fibonacci knots[END_REF] (see also [START_REF] Koseleff | Chebyshev knots[END_REF]).

Proofs

Proof of theorem 1. Let

A = z 1 1 0 , J = 0 1 1 0 . We have AJ = 1 z 0 1 , hence (AJ) 2 = 1 2z 0 1
≡ Id (mod 2). We also have J 2 = Id, and then JAJ ≡ A -1 (mod 2). We thus see that the modulo 2 matrices A and J generate an infinite dihedral group G.

Consider the matrix M(b) = bz 1 1 0 , we have M(b) ≡ A (mod 2) if b is odd, and M(b) ≡ J (mod 2) if b is even. Consequently, the matrix product of theorem 8 is equivalent modulo 2 to an element of the dihedral group G, that is to A m or A m J.

Since A m = f m+1 * * * and A m J = f m * * * , we conclude that ∇ K ≡ f D (mod 2), recall that the Fibonacci polynomials verify f -n (z) = (-1) n+1 f n (z).
Proof of corollary 2. If K is a rational knot, its Conway polynomial is an even Fibonacci polynomial modulo 2, i.e. of the form f 2k+1 (z). Since

f 2k+1 t 1/2 -t -1/2 = (t k + t -k ) - (t k-1 + t 1-k ) + • • • + (-1) k . The
e 0 = 0, e i = (e i-1 + 1 + b i ) mod 2, d 0 = 0, d i = d i-1 + (-1) e i × (b i mod 2).
Using JA ≡ A -1 J (mod 2), we easily show by induction that

P = M(b 1 ) • • • M(b i ) ≡ A d i J e i (mod 2).
As in the proof of theorem 1, we deduce that ∆ K ≡ f D (mod 2) where Using a formula of Stoimenow, we see that the crossing number of K is 59 (see [START_REF] Stoimenow | Generating functions, Fibonacci numbers and rational knots[END_REF]). Using theorem 8 and a computer, we obtain the Conway polynomial of K by evaluating

D = |d m + 1 -e m |.
P = M(b 1 ) • • • M(b 10 ).
The Conway polynomial of K is:

∇ K (z) = 2880 z 10 + 4944 z 8 + 2304 z 6 + 158 z 4 -61 z 2 + 1.
∇ K may be expressed in terms of Fibonacci polynomials and we obtain:

∇ K (z) = 2880 f 11 -20976 f 9 + 68496 f 7 -128482 f 5 + 140969 f 3 -62886 f 1 ≡ f 3 (mod 2).
On the other hand we have the simpler computations:

P ≡ AJ 2 A 2 JAJ 3 ≡ A 3 JAJ (mod 2) ≡ A 2 (mod 2).
Our algorithm is a formalization of these last computations. It gives i 0 1 2 3 4 5 6 7 8 9 10

b i 1 -2 -10 1 -1 -6 -1 2 -6 -2 b i (mod 2)
1 0 0 1 1 0 1 0 0 0 e i 0 0 1 0 0 0 1 1 0 1 0

d i 0 1 1 1 2 3 3 2 2 2 2
We see that the Conway polynomial of our knot is not congruent to 1 modulo 2. Hence, by a theorem of V. F. R. Jones, J. Przytycki and C. Lamm ([6], [START_REF] Lamm | There are infinitely many Lissajous knots[END_REF]) it cannot be a Lissajous knot.

We easily obtain the number of knots with Conway polynomial congruent to 1 modulo 2 (compare [START_REF] Boocher | Sampling Lissajous and Fourier knots[END_REF]). 1. The number of rational knots, rational knots with Conway polynomial congruent to 1 modulo 2.

Conclusion

We have found an elementary proof of the Murasugi criterion for rational knots. Furthermore our characterization with Conway polynomials is also valid for links.

There are other classical results on the Alexander polynomials of rational knots. K. Murasugi (1958) showed that the signs of coefficients of Alexander polynomials of alternating knots are alternating, which means that all the a i are non-negative ( [START_REF] Murasugi | Knot Theory and its Applications[END_REF]). In 1979, R. Hartley showed that the coefficient of the Alexander polynomial of rational knots satisfy the condition a 0 = . . . = a k > a k+1 > . . . > . . . > a n > 0 ( [START_REF] Hartley | On two-bridged knot polynomials[END_REF]). Y. Nakanishi and M. Suketa (1993) obtained upper and lower bounds for |a i | in terms of |a n | ( [START_REF] Nakanishi | Alexander polynomials of two-bridge knots[END_REF], see also [START_REF] Burde | Das Alexanderpolynom der Knoten mit zwei Brücken[END_REF]).

All these results can easily be translated to similar properties of Conway polynomials. As an application, let us mention the recent study of Lissajous knots. V. F. R. Jones, J. Przytycki (1998)and C. [START_REF] Lamm | There are infinitely many Lissajous knots[END_REF] showed that the Alexander polynomial of a rational Lissajous knot must be congruent to 1 modulo 2 ( [START_REF] Jones | Lissajous knots and billiard knots[END_REF][START_REF] Lamm | There are infinitely many Lissajous knots[END_REF]). This property is the main tool used by A. Boocher, J. Daigle, J. Hoste and W. Zheng (2009) to prove that some rational knots cannot be Lissajous ( [START_REF] Boocher | Sampling Lissajous and Fourier knots[END_REF]). Our algorithm provides a faster method to compute the modulo 2 Alexander polynomials.

Figure 1 .

 1 Figure 1. The Stevedore knot 6 1 and the Fibonacci link F (3) 2 rational link denoted S( α β). This link is a knot if α is odd, and a two component link if

Proposition 7 .

 7 ([4, p. 207]) Any fraction α β , α ≡ β (mod 2), has a continued fraction expansion with even quotients: α β = [2b 1 , 2b 2 , . . . , 2b m ], where b i ∈ Z.

3 1 4 1 6 1 Figure 2 .

 12 Figure 2. Some Twist knots, the trefoil T 1 = 3 1 , the figure-eight knot T 2 = 4 1 , and the stevedore knot T 4 = 6 1 .

result follows from corollary 5 .

 5 Proof of theorem 3. The simplicity of the computations in a dihedral group immediately allows us to obtain the Conway and Alexander polynomials. It clearly suffices to obtain their degrees. Let α β = [2b 1 , . . . , 2b m ]. Let us define the sequences e i and d i by:

  1 , . . . , b 10 ] = [2, -4, -20, 2, -2, -12, -2, 4, -12, -4]

Table

	3 4 5 6 7 8 9 10 11 12 13 14 15 16 17	18	19	20	21	22
	rational 1 1 2 3 7 12 24 45 91 176 352 693 1387 2752 5504 10965 21931 43776 87552 174933
	∇(t) ≡ 1 0 0 1 1 2 4 8 13 26 51 97 185 365 705 1369 2675 5233 10211 20011 39221


Sergey Barannikov

Maxim Kontsevich

FROBENIUS MANIFOLDS AND FORMALITY OF LIE ALGEBRAS OF POLYVECTOR FIELDS

We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus = 0 Gromov-Witten invariants.

Introduction

Probably the best mathematically understood part of the Mirror Symmetry program is the theory of Gromov -Witten invariants (see [KM]). In this paper we will construct a Mirror partner for the genus = 0 sector of this theory. It may be considered as a generalization of the theory of variations of Hodge structures on Calabi-Yau manifolds.

One of the puzzles in Mirror symmetry was to find an interpretation of the mysterious objects involved in the famous predictions of the numbers of rational curves. Such an interpretation should, in particular, give the meaning to the "extended" moduli space H * (M, Λ * T M )[2], thought as generalized deformations of complex structure. This moduli space should be equipped with the analog of the 3-tensor C ijk (t) ("Yukawa coupling") arising from a generalization of the variation of Hodge structure on H * (M ). To find such structure is essential for the extension of the predictions of Mirror Symmetry in the dimensions n > 3.

Background philosophy.

The Mirror Symmetry conjecture, as it was formulated in [K1], states that the derived category of coherent sheaves on a Calabi-Yau manifold M is equivalent to the derived category constructed from (conjectured) Fukaya category associated with the dual Calabi-Yau manifold M . The conjecture implies existence of the structure of Frobenius manifold on the moduli space of A ∞ -deformations of the derived category of coherent sheaves on M . This structure coincides conjecturally with the Frobenius structure on formal neighborhood of zero in H * ( M , C) constructed via Gromov-Witten classes of the dual Calaby-Yau manifold M .

S. B. was partially supported by the Fellowship for Graduate Study of the University of California at Berkeley

Typeset by A M S-T E X 1 0.2 Formulation of the results.

Consider the differential graded Lie algebra

t = k t k , t k = q+p-1=k Γ(M, Λ q T * M ⊗ Λ p T M ) (0.1)
endowed with the differential ∂ and the bracket coming from the cup-product on ∂forms and standard Schouten-Nijenhuys bracket on holomorphic polyvector fields. Deformation theory associates a formal (super)moduli space M t to the Lie algebra t. It can be described informally as universal moduli space of solutions to the Maurer-Cartan equation over Z-graded Artin algebras modulo gauge equivalence. The formal moduli space M t can be identified with a formal neighborhood of zero in graded vector space

H * (M, Λ * T M )[2]
* (M, Λ * T M )[2]. 0.3 Connection with A ∞ -deformations of D b Coh(M ).
The Formality theorem (see [K2]) identifies the germ of the moduli space of A ∞deformations of the derived category of coherent sheaves on M with the moduli space M t . The tangent bundle of this moduli space after the shift by [-2] has natural structure of the graded commutative associative algebra. The multiplication arises from the Yoneda product on Ext-groups. The identification of moduli spaces provided by the Formality theorem respects the algebra structure on the tangent bundles of the moduli spaces. This implies, in particular, that the usual predictions of numerical Mirror Symmetry can be deduced from the homological Mirror Symmetry conjecture proposed in [K1]. We hope to elaborate on this elsewhere.

Frobenius manifolds

Remind the definition of formal Frobenius (super) manifold as given in [D], [M], [KM]. Let H be a finite-dimensional Z 2 -graded vector space over C. 2 It is convenient to choose some set of coordinates x H = {x a } which defines the basis {∂ a := ∂/∂x a } of vector fields. One of the given coordinates is distinguished and is denoted by

x 0 . Let A c ab ∈ C[[x H
]] be a formal power series representing 3-tensor field, g ab be a nondegenerate symmetric pairing on H. To simplify notations in superscripts we replace deg (x a ) by ā.

One can use the A c ab in order to define a structure of

C[[x H ]]-algebra on H ⊗ C[[x H ]], the (super)space of all continuous derivations of C[[x H ]], with multiplica- tion denoted by •: ∂ a • ∂ b := c A c ab ∂ c
1 For a graded object t denote t[n] the tensor product of t with the trivial object concentrated in degree -n.

2 One can use an arbitrary field of characteristic zero instead of C everywhere equivalently, the pairing g ab is invariant with respect to the multiplication • defined by

A c ab . (3) (Identity) ∀a, b A b 0a = δ b a equivalently ∂ 0 is an identity of the algebra H ⊗ C[[x H ]] (4) (Potential) ∀a, b, c, d ∂ d A c ab = (-1)
ā d∂ a A c db , which implies, assuming (1a) and (2), that the series A abc are the third derivatives of a single power series Φ

∈ H ⊗ C[[x H ]] A abc = ∂ a ∂ b ∂ c Φ

Moduli space via deformation functor

The material presented in this section is standard (see [K2] and references therein).

Let us remind the definition of the functor Def g associated with a differential graded Lie algebra g . It acts from the category of Artin algebras to the category of sets. Let A be an Artin algebra with the maximal ideal denoted by m. Define the set

Def g (A) := {dγ + [γ, γ] 2 = 0|γ ∈ (g ⊗ m) 1 }/Γ 0 A
where the quotient is taken modulo action of the group Γ 0 A corresponding to the nilpotent Lie algebra (g ⊗ m) 0 . The action of the group can be described via the infinitesimal action of its Lie algebra:

α ∈ g ⊗ m → γ = dα + [γ, α]
Sometimes functor Def g is represented by some topological algebra O Mg (projective limit of Artin algebras) in the sense that the functor Def g is equivalent to the functor Hom continuous (O Mg , • ). For example, H 0 (g) = 0 is a sufficient condition for this. If Def g is representable then one can associate formal moduli space to g by defining the "algebra of functions" on the formal moduli space to be the algebra O Mg .

We will need the Z-graded extension of the functor Def g . The definition of Def Z g is obtained from the definition of Def g via inserting Z-graded Artin algebras instead of the usual ones everywhere. A sufficient and probably necessary condition for the functor Def Z g to be representable is that g must be quasi-isomorphic to an abelian graded Lie algebra. We will see in §2.1 that this is the case for g = t. Hence one can associate formal (graded) moduli space3 M t to the Lie algebra g.

2.1 Extended moduli space of complex structure.

Let M be a connected compact complex manifold of dimension n, with vanishing 1-st Chern class c 1 (T M ) = 0 ∈ Pic (M ). We assume that there exists a Kähler metric on M , although we will not fix it. By Yau's theorem there exists a Calabi-Yau metric on M .

It follows from the condition c 1 (T M ) = 0 there exists an everywhere nonvanishing holomorphic volume form Ω ∈ Γ(X, Λ n T * M ). It is defined up to a multiplication by a constant. Let us fix a choice of Ω. It induces isomorphism of cohomology groups:

H q (M, Λ p T M ) ≃ H q (M, Ω n-p ); γ → γ ⊢ Ω. Define differential ∆ of degree -1 on t by the formula : (∆γ) ⊢ Ω = ∂(γ ⊢ Ω)
The operator ∆ satisfies the following identity (Tian-Todorov lemma) :

[γ 1 , γ 2 ] = (-1) degγ1+1 (∆(γ 1 ∧ γ 2 ) -(∆γ 1 ) ∧ γ 2 -(-1) degγ1+1 γ 1 ∧ ∆γ 2 ) (2.1) where degγ = p + q -1 for γ ∈ Γ(M, Λ q T * M ⊗ Λ q T M ).
Denote by H the total homology space of ∆ acting on t[1]. Let {∆ a } denote a graded basis in the vector space ⊕ p,q H q (M, Λ p T M ), ∆ 0 = 1 ∈ H 0 (M, Λ 0 T M ) . Let us redefine the degree of ∆ a as follows Proof. It follows from (2.1) that the maps

|∆ a | := p + q -2 for ∆ a ∈ H q (M, Λ p T M ) Then {∆ a }
(t, ∂) ← (Ker ∆, ∂) → (H[-1], d = 0) (2.2) are morphisms of differential graded Lie algebras. Then the ∂ ∂-lemma, which says that Ker ∂ ∩ Ker ∆ ∩ (Im ∆ ⊕ Im ∂) = Im ∆ • ∂, (2.3)
shows that these morphisms are quasi-isomorphisms (this argument is standard in the theory of minimal models, see [DGMS]). Hence (see e.g. theorem in §4.4 of [K2]) the deformation functors associated with the three differential graded Lie algebras are canonically equivalent. The deformation functor associated with trivial algebra

(H[-1], d = 0) is represented by the algebra C[[t H ]].
Corollary 2.2. The moduli space M t associated to t is smooth. The dimension of M t is equal to p,q dim H q (M, Λ p T M ) of the dimension of the space of first order deformations associated with t.

Remark. The Formality theorem proven in [K2] implies that the differential graded Lie algebra controlling the A ∞ -deformations of D b Coh(M ) is quasi-isomorphic to t. Here we have proved that t is quasi-isomorphic to an abelian graded Lie algebra. Therefore, the two differential graded Lie algebras are formal, i.e. quasi-isomorphic to their cohomology Lie algebras endowed with zero differential.

Corollary 2.3. There exists a solution to the Maurer-Cartan equation

∂γ(t) + [γ(t), γ(t)] 2 = 0 (2.4)
in formal power series with values in t

γ(t) = a γa t a + 1 2! a1,a2 γa1a2 t a1 t a2 + . . . ∈ (t ⊗ C[[t H ]]) 1
such that the cohomology classes [γ a ] form a basis of cohomology of the complex (t, ∂)

Remark. The deformations of the complex structure are controlled by the differential graded Lie algebra

t (0) := k t k (0) , t k (0) = Γ(M, Λ k T * M ⊗ T M )
The meaning of this is that the completion of the algebra of functions on the moduli space of complex structures on M represents Def t (0) (or Def Z t (0) restricted to the category of Artin algebras concentrated in degree 0). The natural embeddings t (0) ֒→ t induces embedding of the corresponding moduli spaces. In terms of the solutions to Maurer-Cartan equation the deformations of complex structure are singled out by the condition γ

(t) ∈ Γ(M, Λ 1 T * M ⊗ Λ 1 T M ).
Remark. Similar thickening of the moduli space of complex structures were considered by Z. Ran in [R].

Algebra structure on the tangent sheaf of the moduli space

Let R denotes a Z-graded Artin algebra over C, γ R ∈ (t ⊗ R) 1 denotes a solution to the Maurer-Cartan equation (2.4).

The linear extension of the wedge product gives a structure of graded commutative algebra on t

⊗ R[-1]. Let γ R be a solution to the Maurer-Cartan equation in (t ⊗ R) 1 . It defines a differential ∂γ R = ∂ + {γ R , •} on t ⊗ R[1]. Denote the cohomology of ∂γ R by T γ R .
The space of first order variations of γ R modulo gauge equivalence is identified with T γ R . Geometrically one can think of γ R as a morphism from the formal variety corresponding to algebra R to the formal moduli space. An element of T γ R corresponds to a section of the preimage of the tangent sheaf. Note that ∂γ R acts as a differentiation of the (super)commutative R-algebra

t ⊗ R[-1]. Therefore T γ R [-2]
inherits the structure of (super)commutative associative algebra over R. This structure is functorial with respect to the morphisms φ * :

T γ R 1 → T γ R 2 induced by homomorphisms φ : R 1 → R 2 . Let γ(t) ∈ (t ⊗C[[t H ]]
) 1 be a solution to the Maurer-Cartan satisfying the condition of corollary 2.3. It follows from this condition that the C[[t H ]]-module T γ(t) is freely generated by the classes of partial derivatives [∂ a γ(t)]. Therefore we have Proposition 3.1. There exists formal power series

A c ab (t) ∈ t ⊗C[[t H ]] satisfying ∂ a γ ∧ ∂ b γ = c A c ab ∂ c γ mod ∂γ(t)
The series A c ab (t) are the structure constants of the commutative associative

C[[t H ]]-algebra structure on H ⊗ C[[t H ]][-2].
Remark. Note that on the tangent space at zero this algebra structure is given by the obvious multiplication on ⊕ p,q H q (M, Λ p T M ). This is "Mirror dual" to the ordinary multiplication on ⊕ p,q H q (M, Ω p M ).

Integral

Introduce linear functional on t

γ := M (γ ⊢ Ω) ∧ Ω Claim 4.1.
It satisfies the following identyties:

∂γ 1 ∧ γ 2 = (-1) degγ1 γ 1 ∧ ∂γ 2 ∆γ 1 ∧ γ 2 = (-1) degγ1+1 γ 1 ∧ ∆γ 2 (4.1) for γ i ∈ Γ(M, Λ qi T * M ⊗ Λ pi T M ), i = 1, 2 where deg γ i = p i + q i -1 .

Metric on T M

There exists a natural metric (i.e. a nondegenerate (super)symmetric O Mt -linear pairing) on the sheaf T Mt . In terms of a solution to the Maurer-Cartan equation γ R ∈ (t ⊗ R) 1 it means that there exists an R-linear graded symmetric pairing on T γ R , which is functorial with respect to R. Here T γ R denotes the cohomology of ∂γ R defined in §3. The pairing is defined by the formula

h 1 , h 2 := h 1 ∧ h 2 for h 1 , h 2 ∈ T γ R
where we assumed for simplicity that γ R ∈ Ker ∆ ⊗ R. It follows from (2.2) (see also lemma 6.1) that such a choice of γ R in the given class of gauge equivalence is always possible.

Claim 5.1. The pairing is compatible with the algebra structure.

Flat coordinates on moduli space.

Another ingredient in the definition of Frobenius structure is the choice of affine structure on the moduli space associated with t. The lemma 2.1 identifies M t with the moduli space associated with trivial algebra (H[-1], d = 0). The latter moduli space is the affine space H. The affine coordinates {t a } on H give coordinates on M t . This choice of coordinates on the moduli space corresponds to a specific choice of a universal solution to the Maurer-Cartan equation over

(2) (Flat coordinates) γa ∈ Ker ∆, γa1...a k ∈ Im ∆ for k ≥ 2 (3) (Flat identity) ∂ 0 γ(t) = 1, where ∂ 0 is the coordinate vector field corre- sponding to [1] ∈ H[-1]
Proof. The theorem of §4.4 in [K2] shows that there exists L ∞ morphism f homotopy inverse to the natural morphism (Ker ∆, ∂)

→ H[-1] (for the definition of L ∞ -morphism see §4.3 in [K2]). Put ∆(t) = a (∆ a [-1])t a
, where ∆ a [-1] denotes the element ∆ a having degree shifted by one. Then

γ(t) = n 1 n! f n (∆(t) ∧ • • • ∧ ∆(t))
satisfies the conditions (1) -(2). To fulfill the condition (3) γ(t) must be improved slightly. Define the differential graded Lie algebra Ker as follows

(1)

Ker i = Ker ∆ ⊂ t i for i ≥ 0 (2) Ker -1 = Im ∆ ⊂ t -1
Note that the algebra Ker ∆ is the sum of the algebra Ker and trivial algebra of constants C⊗1[-1]. Let f be a homotopy inverse to the natural quasi-isomorphism

Ker → H[-1] ≥0 . Put ∆(t) = a =0 (∆ a [-1])t a . Then γ(t) = 1t 0 + n 1 n! fn ( ∆(t) ∧ • • • ∧ ∆(t))
satisfies all the conditions.

Remark. Any two formal power series satisfying conditions of lemma 6.1 are equivalent under the natural action of the group associated with the Lie algebra ( Ker∆ ⊗C[[t H ]]) 0 .

Remark. It is possible to write down an explicit formula for the components of the morphism f in terms of Green functions of the Laplace operator acting on differential forms on M .

Remark. After the identification of the moduli space M t with H, provided by lemma 2.1, the complex structure moduli space corresponds to the subspace H 1 (M, Λ 1 T M ). In the case of classical moduli space of the complex structures on M the analogous lemma was proved in [T]. The coordinates arising on the classical moduli space of complex structures coincide with so called "special" coordinates of [BCOV].

Notation. Denote γ(t) = a γa t a + 1 2! a1,a2 γa1a2 t a1 t a2 + . . . ∈ (t ⊗ C[[t H ]]) 1
a solution to the Maurer-Cartan equation satisfying conditions of lemma 6.1.

The parameters of a miniversal solution to the Maurer-Cartan equation over C[[t H ]] serve as coordinates on the moduli space. The specific choice of coordinates corresponding to the solution to the Maurer-Cartan equation satisfying conditions (1)-( 2) of lemma 6.1 corresponds to choice of coordinates on moduli space that are flat with respect to the natural (holomorphic) metric g ab .

Claim 6.2. The power series

∂ a γ(t), ∂ b γ(t) ∈ C[[t H ]
] has only constant term in the power series expansion at t = 0.

Proof. x, y = 0 for x ∈ Ker ∆, y ∈ Im ∆.

Notation. Denote g

ab := ∂ a γ(t), ∂ b γ(t) .
Thus we have constructed all the ingredients of the Frobenius structure on M t : the tensors A c ab (t), g ab and the coordinates {t a } that are flat with respect to g ab . Remark. The 3-tensor A c ab (t) on M t does not depend on the choice of Ω. The 2-tensor g ab is multiplied by λ 2 when Ω is replaced by λΩ Claim 6.3. The structure constants satisfy A b 0a = δ b a . Proof. It follows from the condition (3) imposed on γ(t)

We have checked that the tensors A c ab (t), g ab have the properties (1)-(3) from the definition of the Frobenius structure.

It remains to us to check the property (4).

Flat connection and periods

Let γ(t) ∈ (t ⊗ C[[t H ]]
) 1 be a solution to the Maurer-Cartan equation satisfying conditions (1)-( 2) of lemma 6.1. Then the formula Ω(t) := e γ(t) ⊢ Ω defines a closed form of mixed degree depending formally on t ∈ H. For t ∈ H -1,1 γ(t) ∈ Γ(M, T ⊗ T * ) represents a deformation of complex structure. Then Ω(t) is a holomorphic n-form in the complex structure corresponding to t ∈ H -1,1 , where

n = dim C M .
Let {p a } denote the set of sections of T * H that form a framing dual to {∂ a }.

Define a (formal) connection on T *

H by the covariant derivatives:

∇ ∂a (p c ) := b A c ab p b (7.1)
Strictly speaking this covariant derivatives are formal power series sections of T * H . Let us put

Π ai = ∂ ∂t a Γi Ω(t) (7.2)
where {Γ i } form a basis in H * (M, C). In particular

Π 0i = Γi Ω(t)
if γ(t) satisfies the condition (3) of lemma 6.1.

Lemma 7.1. The periods Π i = a Π ai p a are flat sections of ∇ Proof. Let ∂ τ = a τ a ∂ a be an even constant vector field, i.e. τ a are even constants for even ∂ a and odd for odd ∂ a . It is enough to prove that

∂ τ ∂ τ Γi Ω(t) = c A c τ τ ∂ c Γi Ω(t)
where A c τ τ are the algebra structure constants defined via

a τ a ∂ a • a τ a ∂ a = c A c τ τ ∂ c (see §1). Note that the operators ∆ and ∂γ(t) acting on t ⊗C[[t H ]] satisfy a version of ∂ ∂-lemma : Ker ∆ ∩ Ker ∂γ(t) ∩ (Im ∆ ⊕ Im ∂γ(t) ) = Im ∆ • ∂γ(t) (7.3) Equivalently, there exists decomposition of t ⊗C[[t H ]] = X 0 ⊕ X 1 ⊕ X 2 ⊕ X 3 ⊕ Y
into direct sum of graded vector spaces, such that the only nonzero components of ∂γ(t) , ∆ are isomorphisms ∂γ(t) :

X 0 → X 1 , X 2 → X 3 ; ∆ : X 0 → X 2 , X 1 → X 3 .
Differentiating twice the Maurer-Cartan equation (2.4) with respect to ∂ τ and using (2.1) one obtaines

∆(∂ τ γ(t) ∧ ∂ τ γ(t) - c A c τ τ ∂ c γ(t)) = -∂γ(t) ∂ τ ∂ τ γ(t) (7.4)
It follows from ∂ ∂-lemma for ∆, ∂γ(t) and the equation (7.4) that there exist formal power series α τ (t

) ∈ t ⊗C[[t H ]] such that ∂ τ γ(t) ∧ ∂ τ γ(t) - c A c τ τ ∂ c γ(t) = ∂γ(t) α τ (t), ∂ τ ∂ τ γ(t) = ∆(α τ (t)) Therefore ∂ τ ∂ τ Ω(t) = c A c τ τ ∂ c Ω(t) + d(α τ e γ(t) ⊢ Ω)
It follows from the condition (1) imposed on γ(t) that Π i form a (formal) framing of T * H . Corollary 7.2. The connection ∇ is flat Claim 7.3. The structure constants A c ab satisfy the potentiality condition (4) in flat coordinates.

Proof. If one puts symbolically ∇ = ∇ 0 + A then the flatness of ∇ implies that

∇ 0 A + 1 2 [A, A] = 0
Notice that associativity and commutativity of the algebra defined by A c ab imply that [A, A] = 0. Therefore ∇ 0 (A) = 0.

We have completed the proof of the fact that A c ab (t) and g ab define the Frobenius structure on M t in the flat coordinates {t a }.

Remark. In fact one can write an explicit formula for the potential of the Frobenius structure. Let us put γ(t) = a γa t a + ∆α(t), α(t)

∈ (t ⊗t 2 H C[[t H ]]) 0 . Put Φ = - 1 2 ∂α ∧ ∆α + 1 6 γ ∧ γ ∧ γ Then one checks easily that A abc = ∂ a ∂ b ∂ c Φ (see Appendix). In the case dim C M = 3, γ ∈ t -1,1 = Γ(M, T * M ⊗ T M
) this formula gives the critical value of so called Kodaira-Spencer Lagrangian of [BCOV].

Remark. Define differential Batalin-Vilkovisky algebra as Z 2 -graded commutative associative algebra A equipped with odd differentiation ∂, ∂2 = 0 and odd differential operator ∆ of order ≤ 2 such that ∆ 2 = 0, ∆ ∂ + ∂∆ = 0, ∆(1) = 0. One can use the formula (2.1) to define the structure of Z 2 -graded Lie algebra on ΠA. Assume that the operators ∂, ∆ satisfy ∂ ∂-lemma (2.3). Assume in addition that A is equipped with a linear functional : A → C satisfying (4.1) such that the metric defined as in §5 is nodegenerate. Then the same construction as above produces the Frobenius structure on the Z 2 -graded moduli space M ΠA . One can define the tensor product of two such Batalin-Vilkovisky algebras. Operator ∆ on A 1 ⊗ A 2 is given by ∆

1 ⊗ 1 + 1 ⊗ ∆ 2 . Also, ∂ on A 1 ⊗ A 2 is ∂1 ⊗ 1 + 1 ⊗ ∂2 .
It is naturally to expect that the Frobenius manifold corresponding to A 1 ⊗ A 2 is equal to the tensor product of Frobenius manifolds corresponding to A 1 , A 2 , defined in [KM] in terms of the corresponding algebras over operad {H * (M 0,n+1 )}.

Scaling transformations

The vector field E = a -1 2 |∆ a |t a ∂ a generates the scaling symmetry on H. Proposition 8.1.

Lie E A abc = ( 1 2 (|∆ a | + |∆ b | + |∆ c |) + 3 -dim C M )A abc Proof. A abc = M ∂ a γ ∧ ∂ b γ ∧ ∂ c γ.
Note that γ = 0 implies that γ ∈ t 2n-1 . The proposition follows from the grading condition on γ(t).

Corollary 8.2. Lie

E A c ab = ( 1 2 (|∆ a | + |∆ b | -|∆ c |) + 1)A c ab
Note that the vector field E is conformal with respect to the metric g ab . Therefore the proposition 8. 1 shows that E is the Euler vector field of the Frobenius structure on H (see [M]). Such a vector field is defined uniquely up to a multiplication by a constant. The simplest invariant of Frobenius manifolds is the spectrum of the operator [E, •] acting on infinitesimal generators of translations and the weight of the tensor A c ab under the Lie E -action. Usually the normalization of E is chosen so that [E, ∂ 0 ] = 1.

In our case the spectrum of

[E, •] is equal to d {1 -d/2} with multiplicity q-p=d-n dimH q (M, Ω p )
Note that this spectrum and the weight of A c ab coincide identically with the corresponding quantities of the Frobenius structure arising from the Gromov-Witten invariants of the dual Calabi-Yau manifold M .

9.Further developments.

Conjecturally the constructed Frobenius manifold is related to the Gromov-Witten invariants of M in the following way. One can rephrase the present construction in purely algebraic terms using Čech instead of Dolbeault realization of the simplicial graded Lie algebra Λ * T M . The only additional "antiholomorphic" ingredient that is used is the choice of a filtration on H * (M, C) complementary to the Hodge filtration. The Frobenius structure, arising from the limiting weight filtration corresponding to a point with maximal unipotent monodromy on moduli space of complex structures on M, coincides conjecturally with Frobenius structure on H * ( M , C), obtained from the Gromov-Witten invariants.

Our construction of Frobenius manifold is a particular case of a more general construction. Other cases of this construction include the Frobenius manifold structure on the moduli space of singularities of analytic functions found by K. Saito, the Frobenius manifold structure on the moduli space of "exponents of algebraic functions". The latter case is a Mirror Symmetry partner to the structure arising from Gromov-Witten invariants on Fano varieties. All these cases seem to be related with yet undiscovered generalization of theory of Hodge structures. We hope to return to this in the next paper.

Appendix

Let γ(t) = a γa t a + ∆α(t), α(t) ∈ (t ⊗ t

(∂ τ ∂α) ∧ (∂ 2 τ τ ∆α) = 1 2 ∂ τ (γ ∧ γ) ∧ ∂ 2 τ τ γ (∂ 2 τ τ ∂α) ∧ (∂ τ ∆α) = 1 2 (∂ 2 τ τ γ) ∧ ∂ τ (γ ∧ γ)

  One can use g ab to define the symmetric C[[x H ]]-pairing on H ⊗ C[[x H ]]: ∂ a , ∂ b := g ab These data define the structure of formal Frobenius manifold on H iff the following c ab are structure constants of a supercommutative associative C[[x H ]]-algebra (2) (Invariance) Put A abc = e A e ab g ec ∀a, b, c A abc = (-1) ā( b+c) A bca ,

  2 H C[[t H ]]) 1 be a solution to Maurer-Cartan equation satisfying conditions (1)-(2) of lemma 6Proposition. A abc = ∂ a ∂ b ∂ c Φ Proof. . Let ∂ τ = a τ a ∂ a be an even constant vector field in H. It is enough to prove that (∂ τ γ ∧ ∂ τ γ ∧ ∂ τ γ) = ∂ 3 τ τ τ Φ. Let us differentiate the terms in Φ ∂ 3 τ τ τ (γ ∧ γ ∧ γ) = 18∂ 2 τ τ γ ∧ ∂ τ γ ∧ γ + 3∂ 3 τ τ τ γ ∧ γ ∧ γ + 6∂ τ γ ∧ ∂ τ γ ∧ ∂ τ γ ∂ 3 τ τ τ ( ∂α ∧ ∆α) = ∂α ∧ (∂ 3 τ τ τ ∆α) + (∂ 3 τ τ τ ∂α) ∧ ∆α + 3(∂ τ ∂α) ∧ (∂ 2 τ τ ∆α)+ + 3(∂ 2 τ τ ∂α) ∧ (∂ τ ∆α) ∂α) ∧ ∆α = (-1) deg ∂α+1 (∂ 3 τ τ τ ∆ ∂α) ∧ α = (∂ 3 τ τ τ ∆ ∂α) ∧ α = = -(∂ 3 τ τ τ ∂∆α) ∧ α = -(-1) deg∆α (∂ 3 τ τ τ ∆α) ∧ ∂α = (∂ 3 τ τ τ ∆α) ∧ ∂α = = (-1) (deg∆α+1)(deg ∂α+1) ∂α ∧ (∂ 3 τ τ τ ∆α) = ∂α ∧ (∂ 3 τ τ τ ∆α)Hence, the first two terms in (*) give the same contribution.We have∂α ∧ (∂ 3 τ τ τ ∆α) = (-1) deg ∂α+1 ∆ ∂α ∧ (∂ 3 τ τ τ α) = ∆ ∂α ∧ (∂ 3 τ τ τ α) = = -∂∆α ∧ (∂ 3 τ τ τ α) = -∂γ ∧ (∂ 3 τ τ τ α) ∧ γ) ∧ (∂ 3 τ τ τ α) = (-1) (deg(γ∧γ)+1) 1 2 (γ ∧ γ) ∧ (∂ 3 τ τ τ ∆α)

  1 . The tangent sheaf to M t after the shift by[-2] has natural structure of graded commutative associative algebra over O Mt . In this note we show that this algebra structure gives rise to the structure of (formal) Frobenius manifold on H

* (M, Λ * T M )[2]. More specifically, using a universal solution to the analog of Maurer-Cartan equation in t, we construct generalized holomorphic volume element for generalized deformations of complex structure. The integrals of this element, which can be thought of as generalized periods, produce the Frobenius manifold structure on H

  form a graded basis in H. Denote by {t a }, t a ∈ H * , deg t a = -|∆ a | the basis dual to {∆ a }. Denote by C[[t H ]] the algebra of formal power series on Z-graded vector space H.

	Lemma 2.1. The functor Def Z

t associated with t is canonically equivalent to the functor represented by the algebra C[[t H ]].

  C[[t H ]].

	Lemma 6.1. There exists a solution to the Maurer-Cartan equation in formal
	power series with values in t
	∂γ(t) +	[γ(t), γ(t)] 2	= 0, γ(t) =	a	γa t a +	1 2! a1,a2	γa1a2 t a1 t a2 + . . . ∈ (t ⊗C[[t H ]]) 1
	such that
	(1) (Universality) the cohomology classes [γ

a ] form a basis of cohomology of the complex (t, ∂)

We will omit the superscript Z where it does not seem to lead to a confusion.

University of California at Berkeley, Berkeley CA 94720, USA Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France E-mail address: barannik@ihes.fr , maxim@ihes.fr


Ilaria Bellantuono
email: i.bellantuono@shef.ac.uk

Abdullah Aldahmash

Moustapha Kassem

Aging of marrow stromal (skeletal) stem cells and their contribution to age-related bone loss

Keywords: mesenchymal stem cells, ageing, marrow stromal cells, senescence, telomeres Notch

Marrow stromal cells (MSC) are thought to be stem cells with osteogenic potential and therefore responsible for the repair and maintenance of the skeleton. Age related bone loss is one of the most prevalent diseases in the elder population. It is controversial whether MSC undergo a process of aging in vivo, leading to decreased ability to form and maintain bone homeostasis with age. In this review we summarize evidence of MSC involvement in age related bone loss and suggest new emerging targets for intervention.

A C C E P T E D M A N U S C R I P T

Introduction

Human aging is associated with bone loss leading to bone fragility and increased risk for fractures, a disease known as osteoporosis. Osteoporosis is one of the most prevalent and serious diseases affecting the elderly population and constitutes a major public health problem. The cellular and molecular causes of age-related bone loss are currently intensive topics of investigation with the aim of identifying new approaches to prevent and treat osteoporotic bone loss. The aim of this review is to summarise the recent literature that relates senescence of stromal (also known as mesenchymal stem cells and abbreviated MSC) to aged-related decrease in bone formation.

The concept of bone remodelling in relation to MSC

Bone as a tissue, is composed of bone matrix and bone cells. Two main types of mature bone cells have been identified. Osteoblasts (bone forming cells) and osteoclasts (bone resorbing cells) [START_REF] Frost | Why should many skeletal scientists and clinicians learn the Utah paradigm of skeletal physiology?[END_REF]. The main function of those cells is to mediate a bone replacement mechanism called "bone remodelling" aiming at maintenance of the integrity of the skeleton by removing old bone of high mineral density and high prevalence of fatigue micro-fractures and replacing it with young bone of low mineral density and better mechanical properties [START_REF] Frost | Why should many skeletal scientists and clinicians learn the Utah paradigm of skeletal physiology?[END_REF][START_REF] Parfitt | Bone forming cells in clinical conditions[END_REF].Bone remodelling comprises a specific sequence of cellular events. It starts with recruitment of osteoclast precursors, their fusion to form mature osteoclasts followed by their migration to bone surfaces. The osteoclasts remove an amount of bone and create a resorption lacuna.

Bone resorption is followed temporally in the same anatomical location by bone formation.

Key to the initiation of the process of bone formation is the recruitment of osteoblasts from

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stem cells and precursor cells in the bone marrow and their migration to the bottom of the resorption lacuna. Osteoblasts deposit matrix that mineralizes and forms lamellar bone.

Usually the amount of bone removed by the osteoclasts is equal to the amount of bone formed by the osteoblasts and a stable bone mass is maintained. However, this is not the case during aging. Increased bone resorption and/or decreased bone formation can lead to bone loss.

Based on a number of histomorphometric studies performed on iliac crest biopsies, a decrease in bone formation seems to be the principal pathophysiological mechanism responsible for age-related decreased bone mass [START_REF] Parfitt | Bone forming cells in clinical conditions[END_REF].

Mesenchymal stem cells and their contribution to bone formation

Bone formation is dependent on the number and the activity of osteoblasts recruited at bone formation sites during bone remodelling. It is assumed that osteoblasts are differentiated from osteoprogenitor and stem cells present in the bone marrow. Based on the pioneer work of Friedenstein and co-workers [START_REF] Friedenstein | Origin of bone marrow stromal mechanocytes in radiochimeras and heterotopic transplants[END_REF], it has been recognised that the non-hematopoietic compartment of bone marrow (known as bone marrow stroma) contains a group of fibroblastlike stem cells with osteogenic differentiation potential. In vivo transplantation of MSC in syngeneic or immunodeficient mice in ectopic sites resulted in the formation of a mixture of tissues including bone, cartilage and hematopoietic-supporting stroma [START_REF] Friedenstein | Origin of bone marrow stromal mechanocytes in radiochimeras and heterotopic transplants[END_REF][START_REF] Friedenstein | Osteogenesis in transplants of bone marrow cells[END_REF]. However, the identity of these cells, and whether they hold true properties of stem cells is still matter of debate (see below).

Biological characteristics of MSC

Are MSC stem cells?

A Inefficient migration has been thought as the cause of lack of engraftment [START_REF] Bentzon | Tissue distribution and engraftment of human mesenchymal stem cells immortalized by human telomerase reverse transcriptase gene[END_REF][START_REF] Wynn | A small proportion of mesenchymal stem cells strongly express functionally active CXCR4 receptor capable of promoting migration to bone marrow[END_REF]. However, intra-femoral injection of a subset of murine marrow stromal cells showed limited engraftment at the site of injection 4-6 weeks post-transplantation with few cells expressing osteoblastic markers [START_REF] Short | Prospective isolation of mesenchymal progenitor cells from mouse compact bone[END_REF], suggesting that the quality of the cells injected may be the main cause of low engraftment .

More encouraging are the studies testing the preservation of long term self-renewal by implantation of MSC in heterotopic sites [START_REF] Friedenstein | Osteogenesis in transplants of bone marrow cells[END_REF][START_REF] Stenderup | Aged human bone marrow stromal cells maintaining bone forming capacity in vivo evaluated using an improved method of visualization[END_REF]. In a recent study by Sacchetti et al. [START_REF] Sacchetti | Self-renewing osteoprogenitors in bone marrow sinusoids can organize a hematopoietic microenvironment[END_REF],

implantation of a subpopulation of MSC, which are CD45 low /CD146 bright , regenerated bone and stroma supporting hematopoiesis in ectopic bone formation assay. Most importantly highly clonogenic CFU-F were re-isolated from the implants, suggesting that some of the implanted MSC underwent self-renewal in vivo. In the study by Quarto et al. MSC isolated by plastic adherence and transduced with an erythropoietin cDNA, were transplanted (along with hydroxy-apatite particles) under the skin of nude mice. The cells were able to form bone and

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ACCEPTED MANUSCRIPT

bone marrow stroma and the animals maintained increased hematocrit levels for 12-14 weeks [START_REF] Daga | Enhanced engraftment of EPO-transduced human bone marrow stromal cells transplanted in a 3D matrix in nonconditioned NOD/SCID mice[END_REF]. More interestingly MSC were re-isolated from the implants and used in secondary transplantation showing sustained increased hematocrit levels similar to those obtained in primary transplantation. These studies provide a proof of principle that a stem cell with osteogenic potential exists among MSC.

Identification of MSC

The paucity of studies demonstrating stem cell properties of MSC in vivo highlights the difficulties in the isolation and identification of the true MSC with self-renewing and multipotential differentiation abilities. The absence of suitable markers to establish a hierarchical relationship between stem and progenitor cells has left scientists with a variety of protocols for the isolation of MSC, which vary from laboratory to laboratory and rely mostly on physical-chemical conditions of the in vitro culture. The most popular method for isolating MSC from bone marrow is through plastic adherence [14, 15, [14]. The resulting MSC cultures are heterogeneous and contain a mix of progenitors at different stages of commitment [START_REF] Post | Demonstration of the presence of independent pre-osteoblastic and pre-adipocytic cell populations in bone marrowderived mesenchymal stem cells[END_REF][START_REF] Kuznetsov | Single-colony derived strains of human marrow stromal fibroblasts form bone after transplantation in vivo[END_REF].

In vitro, MSC are identified retrospectively by their clonogenic growth capacity (known as colony forming unit-fibroblast CFU-F) and their ability to differentiate into a broad spectrum of fully differentiated connective tissues, including cartilage, bone, adipose tissue and myelosupportive stroma in vitro [START_REF] Owen | Marrow stromal stem cells[END_REF]. However, remarkable differences have been observed

among the different CFU-Fs in terms of cell morphology, rate of replication, expression of markers for osteoblastic, chondroblastic, adipogenic phenotypes and the number of differentiated progeny they can give rise to, with some CFU-F capable of multipotent

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differentiation, others only capable of forming bone and yet others capable of giving origin to myelo-supportive stroma [START_REF] Post | Demonstration of the presence of independent pre-osteoblastic and pre-adipocytic cell populations in bone marrowderived mesenchymal stem cells[END_REF][START_REF] Kuznetsov | Single-colony derived strains of human marrow stromal fibroblasts form bone after transplantation in vivo[END_REF][START_REF] Banfi | Proliferation kinetics and differentiation potential of ex vivo expanded human bone marrow stromal cells: Implications for their use in cell therapy[END_REF][START_REF] Castro-Malaspina | Characterization of human bone marrow fibroblast colony-forming cells (CFU-F) and their progeny[END_REF][START_REF] Digirolamo | Propagation and senescence of human marrow stromal cells in culture: a simple colony-forming assay identifies samples with the greatest potential to propagate and differentiate[END_REF][START_REF] Muraglia | Clonal mesenchymal progenitors from human bone marrow differentiate in vitro according to a hierarchical model[END_REF].

Unfortunately, there are no commonly accepted surface antigens, which can be used to prospectively isolate MSC from bone marrow. To-date the markers used are not specific to stemness characteristics and are common to several mature cell types. MSC are recognised as negative for CD45 (non-hematopoietic) CD34 (non-endothelial) and positive for CD90, CD105 (SH2), CD73 (SH3), CD63, Stro-1 [START_REF] Haynesworth | Cell surface antigens on human marrowderived mesenchymal cells are detected by monoclonal antibodies[END_REF][START_REF] Simmons | Identification of stromal cell precursors in human bone marrow by a novel monoclonal antibody, STRO-1[END_REF]. There have been several attempts to identify novel markers for prospective isolation of MSC with stem cell properties (Table1).

Stro1 is among the first and one of the most investigated marker. Stro-1 antibody identifies a glycoprotein component on all the clonogenic MSC [START_REF] Simmons | Identification of stromal cell precursors in human bone marrow by a novel monoclonal antibody, STRO-1[END_REF]. However expression of Stro1 is not unique to MSC as it is also expressed in erythroid precursors. Transplantation of Stro1+ cells in vivo did not show improved engraftment potential [START_REF] Chapel | Mesenchymal stem cells home to injured tissues when co-infused with hematopoietic cells to treat a radiation-induced multi-organ failure syndrome[END_REF]. Moreover, there has been no evidence that Stro1+ cells have the ability of in vivo self-renewal. It remains unclear whether

Stro1 identifies a population of cells, which is more primitive compared to cells isolated by plastic adherence, and whether the more recently isolated CD45 low /CD146 bright subpopulation of MSC [START_REF] Sacchetti | Self-renewing osteoprogenitors in bone marrow sinusoids can organize a hematopoietic microenvironment[END_REF] (see above) performs better in serial transplantation. A turning point in the establishment of a hierarchical relationship within the BMSC may be represented by a recent study by Nishikawa's group [START_REF] Takashima | Neuroepithelial cells supply an initial transient wave of MSC differentiation[END_REF]. The authors examined the developmental origin of MSC, using murine ES cell as model. A very early progenitor cell, the Sox1 + neuroepithelial cell was identified as the primary cell of origin of MSC. Sox1 + cells subsequently became cells with properties of MSC progenitors through the sequential acquisition of PDGFRα and PDGFRβ expression. It is unknown if any of the cells emerging from the Sox1 + PDGFRα

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and PDGFRβ populations were CD45 low /CD146 high or whether any of these intermediate cell populations are present in the post-natal human bone marrow.

Age-related changes in MSC

As mentioned above, stem cells are required for the continuous supply of mature, functionally competent cells for normal tissue turnover and regeneration. An age-related decrease in the number or functional abilities of stem cells is thought to lead to tissue deterioration. In the following we will review evidence that MSC exhibit changes in their biological characteristics during aging that are relevant for their bone forming abilities.

Age-related changes in size of the MSC pool

Considering the lack of appropriate markers and assays to identify MSC it is not surprising that no agreement has been reached as to whether MSC decrease in numbers with age and whether this has a role in the decrease bone mass observed during aging. Several investigators have utilized the in vitro clonogenic ability of CFU-F to determine the number of MSC progenitors. Others have measured the clonogenic ability of their osteoblastic-committed populations, the CFU-O, which is alkaline phophatase positive and is obtained following exposure of bone marrow cells to osteogenic supplements. In rats a decrease in MSC number is reported but this is not consistent in murine MSC (Table 2). The reason for this discrepancy may be related to the experimental method utilized. In the murine studies, the number of CFU-F has been corrected for the number of cells present in the femur. In contrast in the rat and human studies, CFU-F or CFU-O has been calculated as frequency of bone marrow mononuclear cells. Moreover, age-related effects on MSC may be strain dependent. This is well known from studies performed on haematopoietic stem cells (HSC), where the

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repopulation ability of HSC is compromised with age in Balb/c but not in C57Bl/6 [START_REF] Ertl | Effects of dietary restriction on hematopoietic stem-cell aging are genetically regulated[END_REF].

Similarly, femoral bone density decrease with age in Balb/c but not in C57Bl/6 [START_REF] Beamer | Genetic variability in adult bone density among inbred strains of mice[END_REF] suggesting that Balb/c mice can be more sensitive to age related changes. Unfortunately most of the studies to assess MSC numbers have been carried out in C57Bl/6 strain. Interestingly, the only study partly carried out using Balb/c showed a decrease in the number of CFU-F and CFU-O [START_REF] Martinez | Human bone marrow mesenchymal stromal cells express the neural ganglioside GD2: a novel surface marker for the identification of MSCs[END_REF].

Discrepancies exist also in the human studies that examined the effect of age on the number of CFU-F or CFU-O [START_REF] Haynesworth | Diminution of the number of mesenchymal stem cells as a cause for skeletal aging[END_REF][START_REF] Egrise | The number of fibroblastic colonies formed from bone marrow is decreased and the in vitro proliferation rate of trabecular bone cells increased in aged rats[END_REF][START_REF] Quarto | Bone progenitor cell deficits and the ageassociated decline in bone repair capacity[END_REF][START_REF] Mets | Variations in the stromal cell population of human bone marrow during aging[END_REF][START_REF] Oreffo | Skeletal progenitor cells and ageing human populations[END_REF][START_REF] Baxter | Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion[END_REF][33][START_REF] Stolzing | Age-related changes in human bone marrow-derived mesenchymal stem cells: consequences for cell therapies[END_REF]. It is plausible that these inconsistencies in the results are due to differences in the methods and source of bone marrow samples utilized. Some studies used bone marrow obtained during surgical procedures for treatment of osteoarthritis or other operations, which may not represent the bone marrow of a healthy aging population. Also, bone marrow was obtained from different sites (spine, proximal knee, iliac crest) and with different techniques. Some authors used bone marrow aspirates and other used "flushing" technique to remove bone marrow cells from the bone fragments obtained during orthopaedic procedures. Methods of collection of bone marrow aspirates can influence widely the number of MSC present depending on the extent of blood contamination. Moreover, the site of harvest may determine the type of MSC obtained. Sacchetti et al [START_REF] Sacchetti | Self-renewing osteoprogenitors in bone marrow sinusoids can organize a hematopoietic microenvironment[END_REF] showed that the more primitive CD45 low /CD146 high MSC is contained in the bone marrow obtained from the iliac crest and not from cells from the periosteum or trabecular bone. Another important factor, which can influence the outcome of the studies, is the way a CFU-F or a CFU-O is defined. Some investigators defined a CFU as a group of cells with more than 16 cells whilst other more then 50 cells. It is of interest that only the studies where colonies with more than 50 cells were 9

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counted showed a decrease with age (Table 2) and it raises the question whether these two colony types represent different cell populations with different biology.

In an attempt to overcome some of these experimental limitations, Stenderup et al [33] examined the number of CFU-F and CFU-O present in Stro-1 + cells isolated from bone marrow mononuclear cells in 38 normal healthy volunteers (23 young and 15 old individuals) and 13 patients with osteoporosis. No age-related effects on the total number of CFU-O, CFU-F or CFU colony size was observed [33]. In contrast, the number of CD45 low /D7fib + /LNGF + cells was reported to decrease with age and was positively correlated to the number of CFU-F [START_REF] Stolzing | Age-related changes in human bone marrow-derived mesenchymal stem cells: consequences for cell therapies[END_REF]. However, the difference between the two studies may be related to the age characteristics of the donors employed. The change in the number of MSC with age was observed in the studies when a higher number of young donors were included, suggesting that most of the observed decline in the number of CFU-F occurs in early adulthood when changes in the skeletal dynamics from a modelling mode characteristic of skeletal growth and consolidation to a remodelling dynamics characteristic of the adult skeleton occurs. This is true for the study by Stolzing et al [START_REF] Stolzing | Age-related changes in human bone marrow-derived mesenchymal stem cells: consequences for cell therapies[END_REF]where the major difference in CFU-F numbers occur between the adolescent group (age below 20) and adult group (age above 20). This may also explain why experiments employing rats showed a decline in the CFU-F number as they continue to grow throughout their lifespan. Changes later in life may be very small if at all, possibly present only in certain pathological conditions. Similar observations have been reported in HSC where the highest proliferative demands are present in the first year of life and changes after that are very small and difficult to detect unless in condition of high proliferative stress such as following transplantation (reviewed in [START_REF] Bellantuono | Stem cell ageing: does it occur and can we intervene?[END_REF]).

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Age-related changes in proliferative capacity of MSC.

Due to the very property of self-renewal, the proliferation ability of stem cells is expected to be extensive or even indefinite. The effect of donor age on the maximal proliferative potential of plastic adherent bone marrow-derived MSC has been examined by several investigators.

An age-related decline in the maximal life span from 30-40 population doublings in younger donors to 20 PD in older donors was observed [START_REF] Baxter | Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion[END_REF][START_REF] Stenderup | Aging is associated with decreased maximal life span and accelerated senescence of bone marrow stromal cells[END_REF] suggesting that MSC lose proliferative capacity in vivo as well as in vitro. This is mirrored by telomere shortening and increased number of MSC showing signs associated with senescence such as a change in morphology (from spindle shape to large flat cells) [START_REF] Baxter | Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion[END_REF][START_REF] Stolzing | Age-related changes in human bone marrow-derived mesenchymal stem cells: consequences for cell therapies[END_REF][START_REF] Stenderup | Aging is associated with decreased maximal life span and accelerated senescence of bone marrow stromal cells[END_REF], expression of cell senescence markers e.g.

β Galactosidase, tumor suppressor TP53 and cell cycle regulator protein p21 [START_REF] Stolzing | Age-related changes in human bone marrow-derived mesenchymal stem cells: consequences for cell therapies[END_REF][START_REF] Stenderup | Aging is associated with decreased maximal life span and accelerated senescence of bone marrow stromal cells[END_REF] and an increase in the levels of oxidized and glycated protein and lipofuscin, metabolites present in senescent cells [START_REF] Stolzing | Age-related changes in human bone marrow-derived mesenchymal stem cells: consequences for cell therapies[END_REF].

Several studies have demonstrated a significant decrease in the mean telomere restriction fragment (mTRF) in MSC obtained from younger donors between the primary passage and the end of the culture, which significantly correlated with the number of population doublings [18, 32, 37][38]. Most importantly, when MSC were cultured for an equal number of population doublings, the telomere length in younger donors was significantly longer compared to older donors [START_REF] Baxter | Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion[END_REF][START_REF] Stenderup | Aging is associated with decreased maximal life span and accelerated senescence of bone marrow stromal cells[END_REF]. If one assumes that all MSC undergo telomere shortening at similar rates in vitro, these data imply that the difference in telomere length seen between younger and older donors is due to loss of mTRF occurring in vivo and can be estimated to be about 17bp/year [START_REF] Baxter | Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion[END_REF]. Measurement of telomere length in uncultured MSC is now required to exclude that culturing of MSC of young and old donors does not introduce a bias and that the difference in telomere length seen is truly a reflection of a loss occurring in vivo. However

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the fact that the telomere length of MSC obtained from fetal blood, liver and bone marrow was significantly higher (p<0.01) compared to adult MSC [START_REF] Guillot | Human first-trimester fetal MSC express pluripotency markers and grow faster and have longer telomeres than adult MSC[END_REF], shows loss of telomere length with developmental age and is well in agreement with these data.

Theoretically, long-term growth of MSC requires maintenance of telomere length that is usually accomplished by the presence of telomerase activity an enzyme which adds telomeric repeats at the end of chromosomes In HSC expression of telomerase has been shown to delay the process of aging although it did not prevent it [START_REF] Allsopp | Telomerase is required to slow telomere shortening and extend replicative lifespan of HSCs during serial transplantation[END_REF]. It is controversial whether MSC express telomerase to sufficient levels to act in a similar way. Most of the studies agree that telomerase is not expressed by MSC in culture [START_REF] Zimmermann | Lack of telomerase activity in human mesenchymal stem cells[END_REF][START_REF] Simonsen | Telomerase expression extends the proliferative life-span and maintains the osteogenic potential of human bone marrow stromal cells[END_REF]. However, only one study suggested that telomerase may be expressed in a sub-population of MSC expressing STRO-1 bright VCAM+ marker before their exposure to in vitro culture conditions [START_REF] Gronthos | Molecular and cellular characterisation of highly purified stromal stem cells derived from human bone marrow[END_REF]. It is possible that, similarly to HSC, a very primitive subpopulation of MSC express telomerase in vivo.

However, upon isolation in culture, MSC lose part of their stemness and telomerase is switched off. In support of this a very recent study demonstrated that culture expanded human MSC underwent commitment towards osteoblast lineage and expressed high levels of osteogenic regulatory gene Cbfa1, and alkaline phosphatase. Low levels hTERT transcripts and telomerase activity were detected at the start but repressed by increased Cbfa1 expression [START_REF] Isenmann | hTERT transcription is repressed by Cbfa1 in human mesenchymal stem cell populations[END_REF].

Age-related changes in differentiation potential of MSC

Several studies have examined the effects of donor age on the differentiation potential of MSC. One caveat to these studies is that, in contrast to HSC, where assessment occur following in vivo transplantation, in vitro culture and expansion of MSC is a prerequisite for performing differentiation studies and in several of these studies the degree of in vitro culture 12

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prior to differentiation studies has not been controlled. Their differentiation potential to the osteogenic lineage has been the most investigated in relation to aging. Stenderup et al [33] and Justesen et al [START_REF] Justesen | Maintenance of osteoblastic and adipocytic differentiation potential with age and osteoporosis in human marrow stromal cell cultures[END_REF], cultured MSC from younger, older and osteoporotic patients and found maintained osteoblast differentiation potential at early passage in culture (low population doublings). Similarly, Muraglia et al tested the osteogenic, chondrogenic and adipogenic differentiation capacity of MSC in donors of various ages and found that the number of tripotent clones did not change with age in culture at early passage. However with time in culture the tripotent clones lost their adipogenic potential [START_REF] Muraglia | Clonal mesenchymal progenitors from human bone marrow differentiate in vitro according to a hierarchical model[END_REF].

Some studies have examined the effect of donor age on bone forming capacity using in vivo bone formation assays. In this assay MSC mixed with hydroxyapatite/tricalcium phosphate (HA/TCP) as a carrier were implanted subcutaneously in syngeneic animals. Both cells from younger and older donors were tested in both younger and older recipient animals. MSC derived from aged rats exhibited decreased bone formation capacity compared with MSC from younger donors [START_REF] Inoue | The effect of aging on bone formation in porous hydroxyapatite: biochemical and histological analysis[END_REF]. Stenderup et al [START_REF] Stenderup | Aged human bone marrow stromal cells maintaining bone forming capacity in vivo evaluated using an improved method of visualization[END_REF] employed similar methodology to test the difference between MSC obtained at early passage (few population doublings) from younger and older human donors. After implantation in immune-deficient mice, no difference was found in the amount of bone formed between the two age groups. On contrast, Mendes et al. [START_REF] Mendes | Bone tissue-engineered implants using human bone marrow stromal cells: effect of culture conditions and donor age[END_REF]

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are subtle and not consistent and may need sensitive assays for its detection. However, the effect of donor age becomes clearer upon in vitro expansion of the cells in culture due to an accelerated senescence phenotype. These findings are of relevance to regenerative medicine protocols where cells from elderly donor need to be used for therapy.

As mentioned above MSC can differentiate into osteoblasts and other lineages including adipocytes. In vivo, several investigators have demonstrated that with age there is a decrease in trabecular bone volume (TBV), which is associated with increased bone marrow adipocytic volume [START_REF] Justesen | Adipocyte tissue volume in bone marrow is increased with aging and in patients with osteoporosis[END_REF][START_REF] Meunier | Osteoporosis and the replacement of cell populations of the marrow by adipose tissue. A quantitative study of 84 iliac bone biopsies[END_REF] and these changes are more pronounced in osteoporotic patients. Furthermore, in a mouse model of accelerated senescence (SAM-6 mouse) decreased bone formation was associated with enhanced adipogenesis in vivo and in vitro [START_REF] Kajkenova | Increased adipogenesis and myelopoiesis in the bone marrow of SAMP6, a murine model of defective osteoblastogenesis and low turnover osteopenia[END_REF]. All these findings led to the hypothesis that, MSC acquired a bias in the differentiation ability in favour of the adipocyte lineage instead of the osteoblastic lineage. Although initial studies on the in vitro differentiation potential of MSC have reported that osteoblast differentiation was inversely correlated to adipocyte cell differentiation [START_REF] Beresford | Evidence for an inverse relationship between the differentiation of adipocytic and osteogenic cells in rat marrow stromal cell cultures[END_REF][START_REF] Bennett | Adipocytic cells cultured from marrow have osteogenic potential[END_REF], these were superseded by numerous other studies where either no difference was seen or MSC from older donors exhibited a decreased differentiation ability to the adipogenic lineage [START_REF] Baxter | Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion[END_REF][START_REF] Stolzing | Age-related changes in human bone marrow-derived mesenchymal stem cells: consequences for cell therapies[END_REF][START_REF] Justesen | Maintenance of osteoblastic and adipocytic differentiation potential with age and osteoporosis in human marrow stromal cell cultures[END_REF] when compared to younger donors. Thus, these studies suggest that adipogenic differentiation from MSC can decline with age in vitro but in vivo adipogenesis may be regulated independently and may not be necessarily related to impaired differentiation of MSC.

Age-related changes in bone microenvironment

Growth factors and cytokines are secreted in the bone microenvironment, sequestrated in bone matrix and exert their main effects locally as a controller of osteoblast activity during did not change with donor age [START_REF] Bismar | Transforming growth factor beta (TGF-beta) levels in the conditioned media of human bone cells: relationship to donor age, bone volume, and concentration of TGF-beta in human bone matrix in vivo[END_REF].

Demineralised bone matrix (DBM) powder was implanted subcutaneously in animals of different ages and the amount of bone formed was quantitated using histomorphometric or biochemical techniques. Irving et al [START_REF] Irving | Ectopic bone formation and aging[END_REF] found that bone induction capacity of bone matrix is impaired in older animals. Bone powder obtained from tibiae and femurs of 6-month-old rats was implanted subcutaneously in young (6 weeks), adult (6 months), and old (2 years) rat recipients. Bone was formed after 14 days in young rats, after 15 days in adult rats and after 25 days in old rats. Similarly, the bone induction of powder obtained from bones of 4-monthold rats was dependent on recipient age with gradual decline in rats aged 1 months old to 16 months old [START_REF] Nishimoto | The effect of aging on bone formation in rats: biochemical and histological evidence for decreased bone formation capacity[END_REF]. Also, decreased in rates of bone formation and amount of bone obtained after subcutaneous implantation of DBM powder were observed in old rats (4 months old) compared with young rats (4 weeks old) [START_REF] Hosny | Osteoinduction in young and old rats using demineralized bone powder allografts[END_REF]. The decreased bone formation found in rats was confirmed in rabbits where 3, 6, 18, and 28 months old rabbit recipients received intra- [START_REF] Ryu | Identification of senescence-associated genes in human bone marrow mesenchymal stem cells[END_REF]. It is therefore possible that MSC exert an influence on the changes in the microenvironment and in turn on the decreased bone formation. Recently, we have employed serum as a surrogate for the microenvironment "seen" by MSC. We compared the effect of sera obtained from young and old donors on MSC proliferation and differentiation [START_REF] Abdallah | Inhibition of osteoblast differentiation but not adipocyte differentiation of mesenchymal stem cells by sera obtained from aged females[END_REF]. We found that sera of elderly donors were inhibitory to osteoblast and not adipocyte differentiation of MSC. The putative factor(s) responsible for these effects remain to be determined.

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MSC aging and intervention to abolish age-related bone loss

Replacing or "rejuvenating" resident MSC populations by either MSC transplantation or by activation of resident MSC by bioactive molecules are potentially useful approaches to enhance the functional capacity of MSC and represent a novel approach to increase in vivo

Telomerase and rejuvenation of MSC

As we described above, cultured MSC lack telomerase activity and exhibit telomere shortening in culture. Also, telomere shortening was associated with expression of a senescence phenotype in MSC cultures [START_REF] Stolzing | Age-related changes in human bone marrow-derived mesenchymal stem cells: consequences for cell therapies[END_REF][START_REF] Stenderup | Aging is associated with decreased maximal life span and accelerated senescence of bone marrow stromal cells[END_REF] . We have tested the possibility of "rejuvenation" of MSC by forced expression of hTERT [START_REF] Simonsen | Telomerase expression extends the proliferative life-span and maintains the osteogenic potential of human bone marrow stromal cells[END_REF] and found that these telomerized MSC exhibit enhanced bone-forming activity both in vitro and in vivo besides promoting extended proliferation and differentiation ability of MSC [START_REF] Simonsen | Telomerase expression extends the proliferative life-span and maintains the osteogenic potential of human bone marrow stromal cells[END_REF][START_REF] Abdallah | Inhibition of osteoblast differentiation but not adipocyte differentiation of mesenchymal stem cells by sera obtained from aged females[END_REF]. We have also reported recently [START_REF] Holmes | Hematopoietic progenitor cell deficiency in fetuses and children affected by Down's syndrome[END_REF] that telomerase activity can be restored transiently to MSC by treatment with TSA (trichostatin A) that changes the epigenetic status of hTERT protomer. Unfortunately, long-term telomerization of the MSC lead to genetic instability and tumor formation [START_REF] Serakinci | Telomerase promoter reprogramming and interaction with general transcription factors in the human mesenchymal stem cell[END_REF].

Thus, alternative, short-term or conditional activation of telomerase may be a better approach for clinical intervention that remains to be examined.

Identification of Notch signaling in MSC aging

The identification of new molecular players involved in MSC aging came from the studies on accelerates osteoclastogenesis [START_REF] Rauner | Inhibition of Lamin A/C Attenuates Osteoblast Differentiation and Enhances RANKL-Dependent Osteoclastogenesis[END_REF]. Gene expression profiling of MSC overexpressing the abnormal form of laminin A showed alterations in the activation of the Notch signaling pathway [START_REF] Scaffidi | Lamin A-dependent misregulation of adult stem cells associated with accelerated ageing[END_REF]. Moreover, the abnormal form of laminin A has been seen to increase in cells with physiological aging [START_REF] Scaffidi | Lamin A-dependent nuclear defects in human aging[END_REF], suggesting that, at least in certain circumstances, with age MSC may be affected in a similar way to MSC from patients affected by HGPS. Indeed Notch signaling has been shown to be important for the maintenance of the MSC pool in physiological conditions [START_REF] Hilton | Notch signaling maintains bone marrow mesenchymal progenitors by suppressing osteoblast differentiation[END_REF]. The disruption of Notch signalling in the limb skeletogenic mesenchyme markedly increased trabecular bone mass in adolescent mice whilst in aged mice caused osteopenia [START_REF] Chen | Self-renewal of embryonic stem cells by a small molecule[END_REF].

The identification of Notch signalling as a pathway involved in self-renewal of MSC opens up opportunities for drug discovery and strategies to identify compounds which can be used to manipulate stem cells for therapeutic or biotechnology purposes. Whilst considerably more information is needed, progress is well underway with proof-of-concept approaches. The ability of intervening and increasing even modestly the stem cell pool may have great

Concluding remarks

The inadequate and very variable technology available at present for studying MSC and the diversity of mechanisms leading to aging pose a major challenge to studying the contribution of MSC to the aging phenotype and has led to the very variable results obtained in different laboratories. Studying larger samples of the healthy aging population is needed for an accurate analysis. Also, identification of better markers for the authentic MSC and a more controlled procedure for bone marrow harvest and culture are important pre-requisite for obtaining accurate data. Irrespective of these difficulties, the idea of "rejuvenation" of stem cells either through transplantation or activation of endogenous stem cells are highly relevant and may provide a novel strategy for abolishing some of the age-related diseases. In this context, studying the molecular mechanism underlying several of the progeria (acceleratedsenescence) syndromes can enhance our understanding of the mechanisms of aging and MSC aging and thus provide novel targets for intervention.

  C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPTStem cells are defined by their ability to reconstitute in vivo the tissue of origin giving rise to all the differentiated mature cell types present in that tissue and contributing to its long term maintenance and repair. Stem cells differ from progenitor cells (also known as transient amplifying cells). Progenitor cells exhibit extended proliferative capacity and multipotent differentiation ability in vitro, but have very little ability to contribute to long term tissue regeneration in vivo. In contrast, to demonstrate "stemness" properties, cells have to show the ability to regenerate the tissue of origin following transplantation and possibly in serial transplantation studies. There is very little evidence for long term skeletal regeneration capacity of MSC in vivo. Transplantation of MSC by intravenous (iv) infusion has shown disappointing results with low or no engraftment of the cells to bone and bone marrow[START_REF] Bensidhoum | Homing of in vitro expanded Stro-1-or Stro-1+ human mesenchymal stem cells into the NOD/SCID mouse and their role in supporting human CD34 cell engraftment[END_REF][START_REF] Bentzon | Tissue distribution and engraftment of human mesenchymal stem cells immortalized by human telomerase reverse transcriptase gene[END_REF][START_REF] Chapel | Mesenchymal stem cells home to injured tissues when co-infused with hematopoietic cells to treat a radiation-induced multi-organ failure syndrome[END_REF][START_REF] Devine | Mesenchymal stem cells distribute to a wide range of tissues following systemic infusion into nonhuman primates[END_REF].

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  C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPTbone remodelling. Several investigators measured the effect of donor age on the amount of growth factors present in the bone matrix directly or indirectly through bone inductive effects of demineralised bone matrix powder. IGF-I and IGF-II are important regulator of osteoblastic cell proliferation and differentiation. Seck et al[START_REF] Seck | Age-related changes in insulin-like growth factor I and II in human femoral cortical bone: lack of correlation with bone mass[END_REF]examined the relationship between donor age, cortical bone content of IGF-I and IGF-II as well as cortical bone remodeling in a large number of donors at different ages. Bone samples were obtained from the proximal femur neck. The authors found an age-related decrease in bone matrix concentration of IGF-I in both men and women and of IGF-II in men only. However, there was no significant correlation between bone content of IGF-I, IGF-II and any parameter of bone remodeling. From the same group, production of transforming growth factor (TGF)-β

16 A

 16 C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT bone formation. During the recent years specific signaling pathways have been identified that provide possible target for intervention. Insight into these pathways has been obtained from studying human diseases of progeria (or accelerated senescence) and through combination of genetic studies and studying gene modified animal models.

  progeroid syndromes. Progeroid syndromes (or accelerated aging syndromes) are caused by a single gene defects which results in accelerating aging phenotypes. The advantage is that aging features seen in such syndromes may result from the acceleration of mechanisms postulated to play causal role in aging and thus can give information on the contribution of a A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT given mechanism in normal aging. In progeroid syndromes where stem cells have been studied a clear association among accelerated telomere shortening, decrease number of stem cells, organ failure and lifespan has been shown[63, 64] [66,67]. Werner syndrome (WS), dyskeratosis congenital (DC) and Hutchinson-Gilford Progeria syndrome (HGPS), are good example of this in the case of MSC [65, 66]. In HGPS, a rare, fatal genetic disorder that is characterized by segmental accelerated aging with osteoporosis and accelerated telomere shortening, a link between BMSC and osteoporosis has been made. The major causal mutation associated with HGPS triggers abnormal messenger RNA splicing of the laminin A gene. Knock down of laminin A in BMSC has been shown to impair osteoblastogenesis and

A

  C C E P T E D M A N U S C R I P TACCEPTED MANUSCRIPT therapeutic impact. Small molecule manipulation of both ES cells and adult tissue stem/progenitor cells has shown that such an approach is feasible[START_REF] Chen | Self-renewal of embryonic stem cells by a small molecule[END_REF][START_REF] Sato | Maintenance of pluripotency in human and mouse embryonic stem cells through activation of Wnt signaling by a pharmacological GSK-3-specific inhibitor[END_REF][START_REF] Trowbridge | Glycogen synthase kinase-3 is an in vivo regulator of hematopoietic stem cell repopulation[END_REF]. A better understanding of the molecular basis of MSC aging may uncover novel molecule capable of extending tissue survival and repair.

  demonstrated that when human MSC from 53 donors of various age were seeded on calcium phosphate scaffolds and implanted under the skin of nude mice, the ability of cultures for forming bone declined with age with 67% of the culture able to form bone at 41-50 years of age, 50% at 51-70 years of age and less that 46% beyond 70 years of age. In conclusion, all these studies suggest that the age-related changes of MSC capacity for osteoblast

differentiation and bone formation using early passage (low population doublings) of MSC 13

  The amount of bone formed by older donor DBM powder was decreased compared with younger donors but the difference was not statistically significant. These studies suggest that age-related changes in the bone microenvironment may play a role in the decreased bone formation occurring with age. However, it is unclear what cells are responsible for the change in growth factors secretion. It is possible that stem cells as such change their cytokine profile secretion to modify the environment to suite their survival, proliferation and differentiation needs. In an expression profiling study Cairney et al (in this issue) has shown increased secretion of chemokines such as IL8 in HSC with age. MSC are known to secrete cytokines such as IL6, or IL11 which have been shown to change with age

	ACCEPTED MANUSCRIPT
	muscular implantation of DBM powder from 6 month-old male rabbit donors [58]. The
	amount of new bone formed was negatively correlated with age of the recipient. Similar
	experiments were performed using human DBM powder from younger (8 donors, 18-46 years) and older donors (9 donor, 62-90 years) that were implanted subcutaneously in mice R I P T [59].

Table 2

 2 Summary of studies assessing age-related changes in mesenchymal stem cell numbers, Y=young donor, O=older donors, F=Female, M=Male, N=Number, mo=months, y=years, OA=Osteoarthritis,

	Specie	Age	Assay	Outcome	Reference
	Rats 6/12 mo (Y), 24 mo (O)	CFU-O	Decrease	[29]
	Rats 4 mo (Y) 21mo (O)	CFU-F CFU-O	30% decrease CFU-F	[28]
				no difference in CFU-O
	Mice 4/6 mo (Y), 24mo (O)	CFU-F CFU-O	Decrease in CFU-F and CFU-O	[82]
	Mice 3 to 15 mo		CFU-F (> 50 cells)	Increase	[83]
	Mice 3 to 15 mo		CFU-F (> 50 cells)	No increase if corrected for increase in
				cellularity

Marker

Cross reaction % cells generating CFU-F

In vivo bone formation

Re-isolation following transplant

Telomerase expression

Stro-1 [START_REF] Simmons | Identification of stromal cell precursors in human bone marrow by a novel monoclonal antibody, STRO-1[END_REF][START_REF] Gronthos | Molecular and cellular characterisation of highly purified stromal stem cells derived from human bone marrow[END_REF][START_REF] Gronthos | The STRO-1+ fraction of adult human bone marrow contains the osteogenic precursors[END_REF]