S Favaro

S G Walker

A generalized constructive definition for the Dirichlet process

Keywords: Random probability measures, Dirichlet process, Blackwell and MacQueen urn scheme, Bayesian Nonparametrics MSC: Primary 60G57; secondary 62F15

In this paper we provide an alternative constructive definition for the Dirichlet process which generalizes the one given by Sethuraman

Introduction

The Dirichlet process is a random probability measure (r.p.m.) whose characterization and properties were presented by [START_REF] Ferguson | A Bayesian analysis of some nonparametric problems[END_REF] and [START_REF] Ferguson | Prior distributions on spaces of probability measures[END_REF] and further investigated by [START_REF] Blackwell | Discretness of Ferguson selections[END_REF] and Blackwell and Mac-Queen (1973).

In order to define the Dirichlet process, let (X , T ) be a Polish space endowed with the Borel σ-field X and consider the following associated spaces of measures M X and P X . In particular, M X is the space of locally finite nonnegative measures on (X , X ) endowed with the σ-field M X generated by the vague topology V which makes (M X , V) a Polish space, and P X is the space of probability measures on (X , X ) endowed with its σ-field P X generated by the weak convergence topology W which makes (P X , W) a Polish space. Let (Ω, F , p) be a probability space and let α ∈ M X , with total mass a := α(X ).

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

ACCEPTED MANUSCRIPT

A Dirichlet process on (X , X ) with parameter α is a r.p.m. P such that, for any finite measurable partition (B 1 , . . . , B k ) of X , α(B j ) > 0, for j = 1, . . . , k, (P (B 1 ; ω), . . . , P (B k ; ω)) is a random vector absolutely continuous with respect to the Lebesgue measure on R k-1 with n i=1 P (B i , ω) = 1 and with Dirichlet distribution with parameter (α(B 1 ), . . . , α(B k )), k ≥ 2.

Various authors have considered other characterizations and properties of the Dirichlet process. In particular, in this paper we are interested in the series representation of the Dirichlet process provided by [START_REF] Sethuraman | A constructive definition of Dirichlet priors[END_REF]. Moving from the constructive definition of the Dirichlet process in [START_REF] Sethuraman | A constructive definition of Dirichlet priors[END_REF] we provide a more general construction. We will prove that the new construction is still a random measure on (X , X ) giving probability one to the subset of discrete probability measures on (X , X ) and its finite dimensional distributions are Dirichlet distributions. This establishes that the new construction is a Dirichlet process.

The generalized Sethuraman construction

Let α ∈ M X with total mass a, let {n i , i ≥ 1} be a fixed integer value sequence and let (Ω, F , p) be a probability space supporting three independent sequences of r.v.s θ

:= {θ i , i ≥ 1}, q := {(q i,1 , . . . , q i,n i ), i ≥ 1} and Y := {(Y i,1 , . . . , Y i,n i ), i ≥ 1}.
The sequence θ is a sequence of independent r.v.s distributed according to a Beta d.f. with parameter (n i , a), the sequence q is a sequence of independent random vectors identically distributed according to a Dirichlet d.f. with parameter (1, . . . , 1) and the sequence Y is a sequence of independent random vectors from a Pólya sequence with parameter α, i.e. if P i are independent Dirichlet processes with parameter α, then for every

i ≥ 1, Y i,1 , . . . , Y i,n i |P i are i.i.d. from P i .
The condition of independence between the sequence of r.v.s θ, q and Y and the usual construction of a product measure implies the existence of the probability space (Ω, F, p) supporting the sequence of r.v.s θ, q and Y and does not require any restrictions on (X , X ), such as it being a Polish space. We now consider the sequence of r.v.s {p i , i ≥ 1} obtained from the sequence of r.v.s θ, by the usual stick breaking construction p 1 = θ 1 and

p i = θ i i-1 j=1 (1 -θ j ) for i ≥ 2. In particular, the stick breaking construction implies that n i=1 p i = 1 -n i=1 (1 -θ i ) → 1 a.s. as n → ∞.
For any ω ∈ Ω and B ∈ X define the map

P (ω, B) := i≥1 p i (ω) n i j=1 q i,j (ω)δ Y i,j (ω) (B) (1)
which is clearly a measurable map from (Ω, F) into (P X , P X ) and takes values in the subset of discrete probability measures on (X , X ).

In Theorem 2 we prove that the finite dimensional distributions of P are Dirichlet distributions. This establishes that P is a Dirichlet process with parameter α. We first cosider the following lemma which recall a distributional equation having as unique solution the Dirichlet process with parameter α. As pointed out by a referee this type of distributional equations for the Dirichlet process appear already in [START_REF] James | Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and beta-gamma processes[END_REF] and for more general r.p.m., using the duality with the posterior distribution, are discussed explicitely in [START_REF] James | Poisson Process Partition Calculus with applications to Exchangeable models and Bayesian Nonparametrics[END_REF]. We leave a short proof for clearity.

Lemma 1 Let α ∈ M X with total mass a, let {n i , i ≥ 1} be a fixed integer value sequence and let θ, q and Y the three sequences of r.v.s above described. Then, for every i ≥ 1, the distributional equation

Q i d = θ i n i j=1 q i,j δ Y i,j + (1 -θ i )Q i (2)
has as its unique solution the Dirichlet process with parameter α.

Proof. For evey i ≥ 1, let n independent r.v.s ξ i,1 , . . . , ξ i,n such that ξ i,j is distributed according to a Beta d.f. with parameter (1, n -j). From Theorem 1 in [START_REF] Jambunathan | Some properties of beta and gamma distributions[END_REF] it follows that q i,1

d = ξ i,1 and q i,j d = ξ i,j j-1 l=1 (1 -ξ i,l
), for j = 2, . . . , n. By this stick breaking construction it follows by induction that 1

-n-1 j=1 q i,j = n-1 j=1 (1 -ξ i,j ). Let B 1 , . . . , B k any measurable partition of X , then it follows that given Y i,n , ξ i,n (δ Y i,n (B 1 ), . . . , δ Y i,n (B k )) has Dirichlet distribution with parameter (δ Y i,n (B 1 ), . . . , δ Y i,n (B k )).
Using the stick breaking construction of q i,j , for j = 1, . . . , n

n j=1 q i,j (δ Y i,j (B 1 ), . . . , δ Y i,j (B k )) = n-1 j=1 q i,j (δ Y i,j (B 1 ), . . . , δ Y i,j (B k )) +   1 - n-1 j=1 q i,j   [ξ i,n (δ Y i,n (B 1 ), . . . , δ Y i,n (B k ))]
and, since by construction n j=1 q i,j = 1, it easily follows that given

Y i,1 , . . . , Y i,n , n j=1 q i,j (δ Y i,j (B 1 ), . . . , δ Y i,j (B k )) has Dirichlet distribution with updated pa- rameter (α(B 1 ) + n j=1 δ Y i,j (B 1 ), . . . , α(B k ) + n j=1 δ Y i,j (B k ))
. This shows that the Dirichlet process with parameter α satisfies the distributional equation ( 2). The uniqueness of the solution follows by Lemma 3.3 in [START_REF] Sethuraman | A constructive definition of Dirichlet priors[END_REF].

Theorem 2 Let α ∈ M X with total mass a, let {n i , i ≥ 1} be a fixed integer value sequence and let P defined by (1). Then, for any measurable partition B 1 , . . . , B k of X , (P (B 1 ), . . . , P (B k )) is distributed according to a Dirichlet distribution with parameter (α(B 1 ), . . . , α(B k )).

Proof. From Lemma 1, the distributional equation

P = θ 1 n 1 j=1 q 1,j (δ Y 1,j (B 1 ), . . . , δ Y 1,j (B k )) + (1 -θ 1 )P.
has unique solution the Dirichlet process with parameter α. Then, to prove the theorem, we can use arguments similar to those used in Lemma 1. In particular, if we define P := (P (B 1 ), . . . , P (B k )), then

m i=1 p i n i j=1 q i,j (δ Y i,j (B 1 ), . . . , δ Y i,j (B k )) + 1 - m i=1 p i P = m-1 i=1 p i n i j=1 q i,j (δ Y i,j (B 1 ), . . . , δ Y i,j (B k )) + 1 - m-1 i=1 p i   θ m nm j=1 q m,j (δ Y m,j (B 1 ), . . . , δ Y m,j (B k )) + (1 -θ m ) P   and θ m n j=1 q m,j (δ Y m,j (B 1 ), . . . , δ Y m,j (B k )) + (1 -θ m ) P has Dirichlet distribution with parameter (α(B 1 ), . . . , α(B k )). Then, it follows that m i=1 p i n i j=1 q i,j (δ Y i,j (B 1 ), . . . , δ Y i,j (B k )) +   1 - m j=1 p i   P
has Dirichlet distribution with parameter (α(B 1 ), . . . , α(B k )), and the result follows by taking the limit for m → ∞.

The following remark underlines the connection between the constructive definition of the Dirichlet process that we have proposed and the one given in [START_REF] Sethuraman | A constructive definition of Dirichlet priors[END_REF].

Remark 3 The measurable map (1) generalizes the measurable map used in [START_REF] Sethuraman | A constructive definition of Dirichlet priors[END_REF]. In particular, if in (1) we set n i = 1 for all i ≥ 1, then

P (ω, B) = i≥1 p i (ω) 1 j=1 q i,j (ω)δ Y i,j (ω) (B) = i≥1 p i (ω)δ Y i,1 (ω) (B) (3)
which corresponds to the measurable map used in [START_REF] Sethuraman | A constructive definition of Dirichlet priors[END_REF]. Moreover, if we fix n i = n for all i ≥ 1 then, as a consequence of Theorem 2 and of convergence results in [START_REF] Sethuraman | Convergence of Dirichlet measures and the interpretation of their parameter[END_REF] if follows P → P * weakly as n → ∞, where P * is a Dirichlet process with parameter α.

Acknowledgements

The authors are grateful to the Associate Editor and to the referees for their valuable comments and suggestions.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 3 Discussion

The series representation provided by [START_REF] Sethuraman | A constructive definition of Dirichlet priors[END_REF] has been widely used in several areas of Bayesian nonparametrics. In particular, it has been used to simulate the Dirichlet process via deterministic and random approximations (see [START_REF] Ishwaran | Exact and approximate sumrepresentation for the Dirichlet process[END_REF], [START_REF] Muliere | Approximating distributions of random functional of Ferguson-Dirichlet priors[END_REF] and the references therein) and to define new approaches for computing functionals of the Dirichlet process (see [START_REF] Diaconis | Some new tools for Dirichlet priors[END_REF], [START_REF] Guglielmi | MCMC estimation of the law of the mean of a Dirichlet process[END_REF]).

In this paper, we have introduced a more general constructive definition of the Dirichlet process. From the original constructive definition in [START_REF] Sethuraman | A constructive definition of Dirichlet priors[END_REF], for every i ≥ 1, we replaced the random measures δ Y i involved in the series representation, by an appropriate random convex linear combination of dependent random measures δ Y i,1 , . . . , δ Y i,n i , where Y i,1 , . . . , Y i,n i are from a Pólya sequence.

The new series representation has more flexibility due to the introduction of the integer value sequence {n i , i ≥ 1}. In particular, on the basis of the large number of applications of the series representation it seems natural and interesting to investigate the consequences of the new series representation in Bayesian nonparametrics.


AN ESTIMATE FOR THE PROBABILITY OF DEPENDENT EVENTS ART ŪRAS DUBICKAS

Keywords: 2000 Mathematics Subject Classification. 60A99, 60C05, 11B25 dependent events, Lovász local lemma, covering systems

In this note we prove an estimate for the probability that none of several events will occur provided that some of those events are dependent. This estimate (essentially due to Filaseta, Ford, Konyagin, Pomerance and Yu) can be applied to coverings of Z by systems of congruences, coverings of Z d by lattices and similar problems. Although this result is similar to the Lovász local lemma, it is independent of it. We will also prove a corollary in the style of the local lemma and show that in some situations our lower bound is stronger than that given by the Lovász lemma. As an illustration, we shall make some computations with an example considered earlier by Chen.

Introduction

Let (Ω, F , P) be a probability space. Throughout, we shall use the notation E c = Ω\E for every event E. If the events E 1 , . . . , E ℓ are all independent of one another (more precisely, e.g., for every j = 2, . . . , ℓ, the events E j and ∪ j-1 i=1 E i are independent), then P(∩ ℓ k=1 E c k ) = P(E c 1 )P(E c 2 ) . . . P(E c ℓ ) = ℓ k=1

(1 -P(E k )).

This implies that if P(E k ) < 1 for each k, then there is a positive probability that none of the events E 1 , . . . , E ℓ will occur. The Lovász local lemma [START_REF] Erdős | Problems and results of 3-chromatic hypergraphs and some related questions[END_REF] gives the same conclusion when the mutual independence of events is replaced by their 'rare' dependence. The 'symmetric' version of the local lemma states that if P(E k ) p for every k = 1, . . . , ℓ, where each of the events E k is independent of all other events except for at most d of them, then P(∩ ℓ k=1 E c k ) > 0 provided that ep(d + 1) 1. See, for instance, [START_REF] Alon | The Probabilistic Method[END_REF], [START_REF] Chen | Revisit the Lovász local lemma[END_REF] for more precise and more general versions of this statement, although, in general, the constant e which occurs in this inequality cannot be replaced by any smaller constant [START_REF] Shearer | On a problem of Spencer[END_REF]. There are many useful applications of the Lovász local lemma in combinatorics, graph theory, number theory as well as in other fields of mathematics, statistics and computer science [START_REF] Czumaj | Coloring nonuniform hypergraphs: a new algorithmic approach to the general Lovász local lemma[END_REF], [START_REF] Deng | The Lovász local lemma and its applications to some combinatorial arrays[END_REF], [START_REF] Dubickas | An approximation by lacunary sequence of vectors[END_REF], [START_REF] Grytczuk | Pattern avoidance on graphs[END_REF], [START_REF] Scott | On dependency graphs and the lattice gas[END_REF], [START_REF] Srinivasan | An extension of the Lovász local lemma, and its applications to integer programming[END_REF], [START_REF] Szabó | An application of Lovász Local Lemma -a new lower bound for the van der Waerden Number[END_REF].

Recently, in a paper of [START_REF] Filaseta | Sieving by large integers and covering systems of congruences[END_REF] devoted to the study of covering systems of congruences, another lower bound for the quantity

P(∩ ℓ k=1 E c k )
, where some of the events E i and E j are dependent, was obtained. Although their lemma (see Lemma 2.1 in [START_REF] Filaseta | Sieving by large integers and covering systems of congruences[END_REF]) is given in terms of densities of covering systems, a corresponding probabilistic statement asserts that

P(∩ ℓ k=1 E c k ) ℓ k=1 (1 -P(E k )) - 1 i<j ℓ E i ,E j dependent P(E i )P(E j ) (1)
if E 1 , . . . , E ℓ are events in a probability space with the property that if E k is independent individually of the events E j 1 , . . . , E jt , then it is independent of every event in the sigma algebra generated by E j 1 , . . . , E jt (see Remark 2 in [START_REF] Filaseta | Sieving by large integers and covering systems of congruences[END_REF]). Indeed, let r k , n k , k = 1, 2, . . . , ℓ, be a collection of pairs of integers, where n k 2 and r k ∈ {0, 1, . . . , n k -1}. Here n k , k = 1, 2, . . . , ℓ, are not necessarily distinct, but r i = r j if n i = n j . Let N be the least common multiple of the numbers n 1 , . . . , n ℓ . Put Ω = {1, 2, . . . , N}. Suppose that the sigma algebra F consists of all 2 N subsets of Ω, and that the distribution is uniform, namely, P(n) = 1/N for every n ∈ Ω. Assume that E k is the event that an integer n ∈ Ω belongs to the residue class r k (mod n k ). It is easy see that the events E i and E j are dependent if and only if gcd(n i , n j ) > 1. Note that the density of integers (in Z) which are not of the form r k (mod n k ), where k = 1, . . . , ℓ, is equal to the probability [START_REF] Filaseta | Sieving by large integers and covering systems of congruences[END_REF] stating that this density is at least

P(∩ ℓ k=1 E c k ). Using P(E k ) = |E k |/|Ω| = 1/n k , one derives from (1) Lemma 2.1 of
ℓ k=1 (1 -1/n k ) - 1 i<j ℓ gcd(n i ,n j )>1 1/(n i n j ).
In this note, we shall prove the following: Theorem 1. Let E j , j = 1, 2, . . . , ℓ, be events in a probability space (Ω, F , P) such that if, for every j = 2, . . . , ℓ, the event E j is independent of each of the events E k 1 , . . . , E kt , where k 1 , . . . , k t are some indices of the set {1, 2, . . . , j -1}, then the events E j and

E k 1 ∪ • • •∪ E kt are also independent. Set F j = ∪ * i<j E i
, where the union is taken over every index i < j for which the events E i and E j are dependent. Then

P(∩ ℓ k=1 E c k ) ℓ k=1 (1 -P(E k )) - ℓ j=2 P(E j )(1 -P(E j+1 )) . . . (1 -P(E ℓ ))P(F j ). (2)
Clearly, P(F j ) * i<j P(E i ), where the sum is taken over every i < j for which the events E i and E j are dependent. So (2) implies that

P(∩ ℓ k=1 E c k ) ℓ k=1 (1 -P(E k )) - 1 i<j ℓ E i ,E j dependent P(E i )P(E j )(1 -P(E j+1 )) . . . (1 -P(E ℓ )). (3) From (1 -P(E j+1 )) . . . (1 -P(E ℓ )) 1, we see that (3) yields (1).
Note that (2) is stronger than (3) and (1). In fact,

F c j = ∩ * i<j E c i ,
where the intersection is taken over every index i < j for which the events E i and E j are dependent. Writing in ( 2)

P(F j ) = 1 -P(F c j ) = 1 -P(∩ * i<j E c i )
, one can estimate each P(F j ) from above, using an estimate on P(∩ * i<j E c i ) from below, as in (2). The latter intersection contains strictly less than j ℓ terms, so one can continue in this way step by step.

Corollary 2. Let E 1 , . . . , E ℓ be events in a probability space (Ω, F , P) satisfying the condition of Theorem 1, and let α and β be two positive real numbers satisfying α(e β -1) 1.

(4)

If each event is independent of all the other events except for at most d of them and P(E j ) p for every j = 1, . . . , ℓ, where p min{α/d, 1e -β/ℓ }, then there is a nonzero probability that none of the events occur.

Note that if p = max 1 j ℓ P(E j ) < 1/ℓ, then

P(∩ ℓ k=1 E c k ) = 1 -P(∪ ℓ k=1 E k ) 1 - ℓ k=1 P(E k ) 1 -ℓp > 0.
If p 1/(e(d + 1)), then P(∩ ℓ k=1 E c k ) > 0, by the Lovász local lemma. The same conclusion P(∩ ℓ k=1 E c k ) > 0 follows from Corollary 2, if, say, ℓ = 14, d = 5, p = 0.08. In this case, ep(d + 1) > 1.3 and ℓp = 1.12 > 1, so neither the trivial bound nor the Lovász local lemma are applicable. Even the stronger version of Lovász local lemma given by p(1 + 1/d) d (d + 1) 1 (see (3.1) in [START_REF] Chen | Revisit the Lovász local lemma[END_REF])) is not applicable, because 0.08(1 + 1/5) 5 6 = 1.1943 • • • > 1. However, taking α = 0.43 and β = 0.2 in Corollary 2, we see that the conditions (4) and p min{α/d, 1e -β/ℓ } hold, because 0.43(e 1.2 -1) = 0.9976 • • • < 1 and 0.08 < min{0.43/5, 1e -1.2/14 } = 0.0821 . . . . So, for this choice of parameters, we have P(∩ 14 k=1 E c k ) > 0. The proofs of Theorem 1 and Corollary 2 are given in the next section. We stress that our proof is just a probabilistic interpretation of a number-theoretic argument given in [START_REF] Filaseta | Sieving by large integers and covering systems of congruences[END_REF], although the inequality (2) is slightly stronger than (1) and the condition of the theorem is weaker. One may expect that the inequalities (1) -(3) can be applied not only to covering systems of congruences of Z as in Theorem 1 of [START_REF] Filaseta | Sieving by large integers and covering systems of congruences[END_REF], but also to covering systems of lattices that cover (or do not cover) the lattice Z d , where d is a positive integer, and also in some situations similar to those, where the Lovász local lemma is applicable.

We remark that in most applications of the local lemma the condition of Theorem 1 (and that of Corollary 2) that the pairwise independence of events implies their union-wise independence is satisfied, so these statements can be applied. In order to establish mutual independence one can use, for example, a so-called Mutual Independence Principle stated on p. 41 in [START_REF] Molloy | Graph Coloring and the Probabilistic Method[END_REF], which asserts that if X = X 1 , . . . , X m is a sequence of random experiments and E 1 , . . . , E ℓ is a set of events, where each E k is determined by

G k ⊆ X, and G k ∩ (G k 1 , . . . , G kt ) = ∅, then E k is mutually independent of {E k 1 , . . . , E kt }.
In conclusion, we shall give three simple examples. The first uses covering systems of congruences and shows that in some cases the inequality (2) is sharp, although (1) and (3) are not. The second shows how Theorem 1 can be applied to covering systems of Z d by lattices. The third example is taken from Chen's paper (1997). It is given in [START_REF] Chen | Revisit the Lovász local lemma[END_REF] to demonstrate that a better version of the Lovász local lemma beats all the previous bounds. We show that the bound (3) is even stronger for the same choice of probabilities and the same 'dependency digraph'.

Proofs

Proof of Theorem 1: The inequality (2) clearly holds for ℓ = 1, because P(E c 1 ) = 1 -P(E 1 ). Suppose that it holds for ℓ = l -1, namely,

P(∩ l-1 k=1 E c k ) l-1 k=1 (1 -P(E k )) - l-1 j=2 P(E j )(1 -P(E j+1 )) . . . (1 -P(E l-1
))P(F j ). ( 5)

Set B j = ∩ k j E c k .
Clearly, B l = B l-1 ∩E c l , so B l ⊆ B l-1 and P(B l-1 ) = P(B l )+P(B l-1 ∩E l ). Let J(j) be the subset of all indices of the set {1, 2, . . . , j -1} for which the events E j and E i , where i is a fixed element of J(j), are independent. Let I(j) = {1, 2, . . . , j -1} \ J(j), so that F j = ∪ i∈I(j) E j . By the condition of the theorem, the events E l and ∪ i∈J(l) E i are independent, so the events E l and Ω \ ∪ i∈J(l) E i = ∩ i∈J(l) E c i are also independent. Hence, using B l-1 ⊆ ∩ i∈J(l) E c i , we deduce that

P(B l-1 ∩ E l ) P(∩ i∈J(l) E c i ∩ E l ) = P(∩ i∈J(l) E c i )P(E l ). Note that ∩ i∈J(l) E c
i is contained in the union of B l-1 and F l , so P(∩ i∈J(l) E c i ) P(B l-1 ) + P(F l ). It follows that P(B l-1 ) = P(B l ) + P(B l-1 ∩ E l ) P(B l ) + P(E l )(P(B l-1 ) + P(F l )), giving P(B l ) P(B l-1 )(1 -P(E l )) -P(E l )P(F l ). Since B l-1 = ∩ l-1 k=1 E c k , inserting (5) into the right hand side of this inequality, we find that

P(B l ) = P(∩ l k=1 E c k ) is at least l k=1 (1 -P(E k )) -P(E l )P(F l ) -(1 -P(E l )) l-1 j=2 P(E j )(1 -P(E j+1 )) . . . (1 -P(E l-1 ))P(F j ), which is equal to l k=1 (1 -P(E k )) - l j=2 P(E j )(1 -P(E j+1 )) . . . (1 -P(E l ))P(F j ).
So (2) holds for ℓ = l. This completes the proof of the theorem by induction.

Proof of Corollary 2: Without loss of generality we may increase each P(E j ) to p, so assume that P(E j ) = p for every j = 1, . . . , ℓ. Note that each E j is independent of the event ∪ (d) i<j E i , where (d) means that at most d elements of the union are omitted (those dependent with E j ), so for each fixed j the sum on the right hand side of (3) contains at most d elements. Hence, in order to show that P(∩ ℓ k=1 E c k ) > 0, by (3), it suffices to prove the inequality

(1 -p) ℓ > dp 2 (1 + (1 -p) + (1 -p) 2 + • • • + (1 -p) ℓ-2 ). ( 6
)
The right hand side of ( 6) is strictly smaller than

dp 2 (1 + (1 -p) + (1 -p) 2 + • • • + (1 -p) ℓ-1 ) = dp 2 1 -(1 -p) ℓ 1 -(1 -p) = dp(1 -(1 -p) ℓ ).
Dividing both sides of ( 6) by (1p) -ℓ , we see that

P(∩ ℓ k=1 E c k ) > 0 if dp((1 -p) -ℓ -1) 1. From p 1 -e -β/ℓ it follows that (1 -p) -ℓ e β , hence (1 -p) -ℓ -1 e β -1
. Moreover, we have pd α. Multiplying these two inequalities and using (4), we obtain dp((1-p) -ℓ -1) α(e β -1) 1, which is the required inequality.

Examples

Suppose that E 1 , E 2 , E 3 are the events that n ∈ Ω = {1, 2, . . . , 15} belongs to the the residue classes 1 (mod 3), 1 (mod 5) and 8 (mod 15), respectively. Then

E c 1 ∩ E c 2 ∩ E c
3 is the event that n ∈ Ω is one of the numbers 2, 3, 5, 9, 12, 14, 15. It follows that

P(E c 1 ∩ E c 2 ∩ E c 3 ) = 7/15.
The events E 1 and E 2 are independent, whereas the events E 1 and E 3 (and also E 2 and E 3 ) are dependent. Clearly, P(E 1 ) = 1/3, P(E 2 ) = 1/5, P(E 3 ) = 1/15. Note that F 3 = E 1 ∪ E 2 , so P(F 3 ) = P(E 1 ) + P(E 2 ) -P(E 1 ∩ E 2 ) = 1/3 + 1/5 -1/15 = 7/15. Thus the right hand side of (2) is equal to (1-P(E 1 ))(1-P(E 2 ))(1-P(E 3 ))-P(E 3 )P(F 3 ) = (1-1/3)(1-1/5)(1-1/15)-7/225 = 7/15. So we have an equality in (2). In other words, 7/15 is the density of integers which do not belong to the arithmetic progressions 3k 1 + 1, 5k 2 + 1, 15k 3 + 8, where k 1 , k 2 , k 3 ∈ Z. The right hand sides of both (1) and (3) are equal to

(1 -P(E 1 ))(1 -P(E 2 ))(1 -P(E 3 )) -P(E 1 )P(E 3 ) -P(E 2 )P(E 3 ) = 104/225, which is 0.46222 • • • < 0.466666 • • • = 7/15.
Next, suppose that m, d and q ij 2, where 1 i m and 1 j d, are positive integers. Let r ij ∈ {0, 1, . . . , q ij -1}, and let Λ i = (q i1 Z + r i1 , . . . , q id Z + r id ), i = 1, . . . , m, be some m lattices in Z d . Suppose that, for every j = 1, . . . , d, the numbers q 1j , . . . , q mj are pairwise coprime. Set Q k = q k1 q k2 . . . q kd for k = 1, . . . , m. We claim that, for any s lattices of the form L t = (Q 1 Z + m t1 , . . . , Q d Z + m td ), t = 1, . . . , s, where m tj ∈ {0, 1, . . . , Q j -1}, the lattices Λ 1 , . . . , Λ m , L 1 , . . . , L s do not cover

Z d if s < (Q 1 -1)(Q 2 -1) . . . (Q m -1). ( 7
)
Indeed, by the same argument as in the introduction, we can describe E i as the event that a point in Z d belongs to the lattice Λ i for i = 1, . . . , m. Let E m+1 be the event that a point in

Z d belongs to the union L 1 ∪• • •∪L s .
The events E i and E j are independent if 1 i < j m, whereas the events E i and E m+1 are dependent for each i = 1, . . . , m. Clearly, P(E i ) = 1/Q i for i = 1, . . . , m and

P(E m+1 ) s/(Q 1 . . . Q m ). Since F m+1 = ∪ m i=1 E i , we have 1 -P(F m+1 ) = 1 -P(∪ m i=1 E i ) = P(∩ m i=1 E c i ) = m i=1 (1 -P(E i )) = (1 -1/Q 1 ) . . . (1 -1/Q m ). Assume that Z d ⊆ Λ 1 ∪ . . . Λ m ∪ L 1 ∪ . . . L s . Then P(E c 1 ∩ • • • ∩ E c m+1 ) = 0. Thus (2) with ℓ = m + 1 implies that (1 -1/Q 1 )(1 -1/Q 2 ) . . . (1 -1/Q m )(1 -P(E m+1 )) P(E m+1 )P(F m+1 ) = P(E m+1 )(1 -(1 -1/Q 1 )(1 -1/Q 2 ) . . . (1 -1/Q m )). Hence (1 -1/Q 1 )(1 -1/Q 2 ) . . . (1 -1/Q m ) P(E m+1 ). Using P(E m+1 ) s/(Q 1 . . . Q m ), we deduce that s (Q 1 -1)(Q 2 -1) . . . (Q m -1), contrary to (7).
Finally, we shall consider Example 1 of [START_REF] Chen | Revisit the Lovász local lemma[END_REF], where the events E 1 , . . . , E 7 have a dependency digraph D = {(1, 2), (1, 3), (2, 1), (2, 4), (2, 5), (3, 1), (3, 6), (3, 7), (4, 2), (5, 2), (6, 3), (7, 3)} ∪{(i, i) : 1 i 7}. This means that, for each i = 1, . . . , 7, the event E i is mutually independent of the events {E j : (i, j) / ∈ D}. Assume, in addition, that E i is independent of ∪ j<i,(i,j) / ∈D E j , so that Theorem 1 is applicable.

As in (3.13) of [START_REF] Chen | Revisit the Lovász local lemma[END_REF]), suppose first that p i = P(E i ) are given by (p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 ) = (1/9, 9/64, 9/64, 1/8, 1/8, 1/8, 1/8).

It was observed in [START_REF] Chen | Revisit the Lovász local lemma[END_REF] (1p k )p 1 p 2 (1p 3 )(1p 4 )(1p 5 )(1p 6 )(1p 7 )

-p 1 p 3 (1p 4 )(1p 5 )(1p 6 )(1p 7 )p 2 p 4 (1p 5 )(1p 6 )(1p 7 ) -p 2 p 5 (1p 6 )(1p 7 )p 3 p 6 (1p 7 )p 3 p 7 = 46745849/150994944 = 0.3095 . . . , which is better.

Another choice p 1 = p 2 = • • • = p 7 = p = 27/256 (see (3.17) in [START_REF] Chen | Revisit the Lovász local lemma[END_REF])) with the same dependency digraph D as above gave the following lower bounds on P(∩

  that the above mentioned symmetric version of the Lovász lemma is inapplicable. The trivial bound P(∩ 7

	k=1 E c k ) 1 -7 k=1 p i gives
	P(∩ 7 k=1 E c k ) 31/288 = 0.1076 . . . .
	The improved version of the Lovász lemma (see Theorem 1 and (3.16) in (Chen, 1997))
	yields
	P(∩ 7 k=1 E c k ) 0.1456 . . . .
	However, our inequality (3) gives
	7
	P(∩ 7 k=1 E c k )
	k=1

  7 k=1 E c k ): 0.1334 . . . , 0.2617 . . . and 0.3507 . . . (see (3.18) and (3.19) in[START_REF] Chen | Revisit the Lovász local lemma[END_REF]). Once again, a simple computation shows that our bound (3)P(∩ 7 k=1 E c k ) (1p) 7p 2 ((1p) 5 + (1p) 4 + (1p) 3 + (1p) 2 + (1p) + 1)= 114529002852169/281474976710656 = 0.4068 . . . gives a slightly better numerical value.This research was supported in part by the Lithuanian State Studies and Science Foundation. The author thanks a referee for pointing out several inaccuracies and supplying with a useful reference.


Nicholas C Turner
email: nicholas.turner@icr.ac.uk

Alan Ashworth
email: alan.ashworth@icr.ac.uk

Biomarkers of PARP inhibitor sensitivity

Text

PARP inhibitors represent one of the most exciting recent developments in cancer therapy. Substantial efficacy has been shown with PARP inhibitors in the treatment of hereditary BRCA1/2 related Breast and Ovarian cancer as single agents [START_REF] Audeh | Oral poly(ADP-ribose) polymerase inhibitor olaparib in patients with BRCA1 or BRCA2 mutations and recurrent ovarian cancer: a proofof-concept trial[END_REF][START_REF] Tutt | Oral poly(ADP-ribose) polymerase inhibitor olaparib in patients with BRCA1 or BRCA2 mutations and advanced breast cancer: a proof-ofconcept trial[END_REF][START_REF] Sandhu | First-in-human of a poly(ADP-ribose) polymerase (PARP) inhibitor MK-4827 in advanced cancer patients (pts) with antitumor activity in BRCA-deficient and sporadic ovarian cancers[END_REF] and in combination with temozolomide [START_REF] Isakoff | A phase II trial of the PARP inhibitor veliparib (ABT888) and temozolomide for metastatic breast cancer[END_REF]. Similarly, encouraging activtity has been shown in sporadic ovarian cancer with a PARP inhibitor as a single agent [START_REF] Gelmon | Can we define tumors that will respond to PARP inhibitors? A phase II correlative study of olaparib in advanced serous ovarian cancer and triple-negative breast cancer[END_REF], and in sporadic triple negative breast cancer in combination with gemcitabine/carboplatin chemotherapy [START_REF] O'shaughnessy | Efficacy of BSI-201, a poly (ADPribose) polymerase-1 (PARP1) inhibitor, in combination with gemcitabine/carboplatin (G/C) in patients with metastatic triple-negative breast cancer (TNBC): Results of a randomized phase II trial[END_REF]. Yet the picture is not universally positive. Negative studies have been reported in heavily pre-treated sporadic triple negative cancer, with PARP inhibitor as a single agent [START_REF] Gelmon | Can we define tumors that will respond to PARP inhibitors? A phase II correlative study of olaparib in advanced serous ovarian cancer and triple-negative breast cancer[END_REF] and no evidence of activity in combination with temozolomide [START_REF] Isakoff | A phase II trial of the PARP inhibitor veliparib (ABT888) and temozolomide for metastatic breast cancer[END_REF]. To understand the reasons some studies have succeeded, and others failed, the development of biomarkers that will predict the sensitivity, or resistance, to PARP inhibitors is required.

There are two conceptually independent ways in which PARP inhibitors are thought to act as anti-cancer agents. First, PARP inhibitors work as single agents targeting homologous recombination (HR) deficient cancers through synthetic lethality. Second, PARP inhibitors also act as chemotherapy or radiotherapy sensitizers in the absence of single agent activity. For example, PARP inhibitors substantially increase the potency of temozolomide in vitro. Whether this translates to an increased therapeutic window in cancers with normal DNA repair in unclear; substantial bone marrow toxicity has been demonstrated with PARP inhibitor and chemotherapy combinations. In reality many PARP inhibitors are being developed assuming a combination of these two strategies, on the assumption that a combination of a PARP inhibitor with chemotherapy may target HR deficient cancers more effectively than PARP inhibitor alone.

Single gene biomakers for PARP inhibitor sensitivity

In sporadic cancers there are multiple mechanisms through which HR may be lost, and PARP inhibitors, at least in vitro, target HR defective cancers independent of the underlying molecular defect [START_REF] Mccabe | Deficiency in the repair of DNA damage by homologous recombination and sensitivity to poly(ADP-ribose) polymerase inhibition[END_REF]. There is in vitro evidence, of varying strength, to support as potentially biomarkers of PARP inhibitor sensitivity, BRCA1/2 sporadic mutation [START_REF] Farmer | Targeting the DNA repair defect in BRCA mutant cells as a therapeutic strategy[END_REF],

BRCA1 promoter methylation [START_REF] Esteller | Promoter hypermethylation and BRCA1 inactivation in sporadic breast and ovarian tumors[END_REF][START_REF] Yu | Basic fibroblast growth factor (FGF-2): the high molecular weight forms come of age[END_REF], BRCA1 suppression in the absence of methylation [START_REF] Yu | Basic fibroblast growth factor (FGF-2): the high molecular weight forms come of age[END_REF], PTEN deficient cancers [START_REF] Mendes-Pereira | Synthetic lethal targeting of PTEN mutant cells with PARP inhibitors[END_REF], ATM mutation [START_REF] Schaffner | Somatic ATM mutations indicate a pathogenic role of ATM in B-cell chronic lymphocytic leukemia[END_REF][START_REF] Stilgenbauer | Biallelic mutations in the ATM gene in T-prolymphocytic leukemia[END_REF], MRE11 dominant negative mutations in mismatch repair deficient cancers [START_REF] Wen | A mutant allele of MRE11 found in mismatch repair-deficient tumor cells suppresses the cellular response to DNA replication fork stress in a dominant negative manner[END_REF], and FANCF promoter methylation [START_REF] Olopade | FANCF methylation contributes to chemoselectivity in ovarian cancer[END_REF]. This list can likely be simplified in that each tumour type differs in the potential mechanism of HR deficiency, and therefore each tumour type may ultimately be defined by a panel of biomarkers. However, there is a pressing need to identify unifying biomarkers of HR deficiency that detect the underlying defect in HR that all these diverse mechanisms share. Such a marker of "BRCAness" [START_REF] Turner | Hallmarks of 'BRCAness' in sporadic cancers[END_REF] may predict the benefit from PARP inhibitors in multiple cancer types, regardless of the underlying molecular mechanism.

Set against this background Concalves et al report an assessment of PARP1 mRNA levels in sporadic breast cancer. Reanalysing data from their own previously published whole genome gene expression arrays they show that PARP1 mRNA expression is highly heterogenous, and related to proliferative rate of the tumour (REF Concalves et al). This association with proliferation has been suggested previously [START_REF] Brustmann | Poly(adenosine diphosphate-ribose) polymerase expression in serous ovarian carcinoma: correlation with p53, MIB-1, and outcome[END_REF], and is a pattern characteristic of multiple DNA repair genes where expression is higher in S phase with active DNA repair required to facilitate DNA replication. This is particularly true in cancer cells as oncogene driven replication is abnormal and prone to the generation of DNA damage [START_REF] Bartkova | Oncogene-induced senescence is part of the tumorigenesis barrier imposed by DNA damage checkpoints[END_REF]. Concalves et al also show that PARP1 levels are associated with genomic copy number, yet the functional significance of this is unclear.

They go on to "meta-analyse" multiple data sets, and combining such data sets is technically challenging and fraught with pitfalls. A particular difficulty is that and the authors attribute tumour subtype using predictors that are not robust [START_REF] Weigelt | Breast cancer molecular profiling with single sample predictors: a retrospective analysis[END_REF], and therefore for this reason alone this data should be treated with caution.

The ultimate question posed by Concalves et al is whether PARP1 levels, assessed either at the mRNA or protein level, predict benefit from PARP inhibitors. Others have previously suggested that high PARP1 levels may predict for the benefit of cytotoxic DNA damaging chemotherapy [START_REF] Von Minckwitz | PARP is expressed in all subtypes of early breast cancer and is a predictive factor for response to neoadjuvant chemotherapy[END_REF][START_REF] Loibl | PARP expression in early breast cancer and its predictive value for response to neoadjuvant chemotherapy[END_REF], providing some support for PARP1 levels as a potential biomarker, although this may possibly reflect the association of PARP1 levels with proliferation. So what evidence is there that high PARP1 levels would predict benefit from a PARP inhibitor? It is first important to recall that there is no evidence that PARP1 acts as an oncogene or promotes tumourigenesis, and that biomarkers to predict benefit of DNA repair inhibitors are unlikely to follow the same paradigms as biomakers of therapies that target oncogenic drivers such as trastuzumab. In vitro, sensitivity to PARP inhibitors as single agents is limited to cell lines with HR deficiency. This suggests that PARP1 levels are only likely to be a biomarker for PARP inhibitor sensitivity if HR deficient cancers have higher PARP1 levels, or to put it another way if HR deficiency results in compensatory increase in PARP1 levels and that results in an increase in PARP1 function. Yet at present there is little evidence to support this being the case. PARP1 protein levels do not differ between isogenic pairs of HR deficient and proficient cancers [START_REF] Gottipati | Poly(ADP-ribose) polymerase is hyperactivated in homologous recombination-defective cells[END_REF], and moreover there is in vitro evidence that PARP1 levels do not correlate with PARP1 activity [START_REF] Zaremba | Poly(ADP-ribose) polymerase-1 polymorphisms, expression and activity in selected human tumour cell lines[END_REF]. In contrast, PARP activity is substantially increased in HR deficient cancers; In vitro PARP1 is hyperactive in HR deficient cell lines [START_REF] Gottipati | Poly(ADP-ribose) polymerase is hyperactivated in homologous recombination-defective cells[END_REF]. So rather than total PARP1 levels, a functional assay such as assessment of PAR (poly(ADP)ribose) polymer levels [START_REF] Yang | Immunohistochemical detection of poly(ADP-ribose) polymerase inhibition by ABT-888 in patients with refractory solid tumors and lymphomas[END_REF] may potentially assay HR function and possibly be a biomarker of BRCAness.

Biomarkers of BRCAness

Where are we with other potential markers of BRCAness? Potential signatures of BRCAness have been reported based on the differences in gene expression between hereditary BRCA1/2 related cancers and sporadic cancers [START_REF] Konstantinopoulos | Gene expression profile of BRCAness that correlates with responsiveness to chemotherapy and with outcome in patients with epithelial ovarian cancer[END_REF][START_REF] Rodriguez | DNA repair signature is associated with anthracycline response in triple negative breast cancer patients[END_REF]. These gene expression signatures may enrich for a group of "hereditary-like" sporadic cancers, and these cancers are more responsive to DNA damaging chemotherapy [START_REF] Konstantinopoulos | Gene expression profile of BRCAness that correlates with responsiveness to chemotherapy and with outcome in patients with epithelial ovarian cancer[END_REF][START_REF] Rodriguez | DNA repair signature is associated with anthracycline response in triple negative breast cancer patients[END_REF]. At present these studies are interesting proofs of principle, but validation of these signatures is insufficient for use in the clinic. Perhaps the most promising potential biomarker of BRCAness would be if the genomic "scar" of HR deficiency could be identified. In a way analogous to microsatellite instability in mismatch repair deficient cancers, there is likely to be a genomic mutational pattern that predicts for underlying HR deficiency. In part this may be a characteristic degree of gross genomic instability, but there is also some indication that certain complex mutations may be a marker of HR deficiency. For example, BRCA1 mutant breast cancers frequently have characteristic mutations in TP53 that are infrequent in sporadic cancer [START_REF] Holstege | High incidence of protein-truncating TP53 mutations in BRCA1-related breast cancer[END_REF]. The presence of these complex mutations in sporadic triple negative breast cancers may predict benefit to neoadjuvant cisplatin [START_REF] Silver | Efficacy of neoadjuvant Cisplatin in triple-negative breast cancer[END_REF], providing some support for their use as a biomarkers of HR deficiency, although further validation work is required.

PARP inhibitors in combination or as single agents?

PARP1 inhibitors have been proposed to target HR deficient cells through inhibiting single strand break repair, perhaps trapping PARP1 on the break inhibiting further repair [START_REF] Godon | PARP inhibition versus PARP-1 silencing: different outcomes in terms of single-strand break repair and radiation susceptibility[END_REF]. Ultimately this may cause the collapse of replication forks leading to a DNA double strand breaks that are only effectively repaired by HR [START_REF] Farmer | Targeting the DNA repair defect in BRCA mutant cells as a therapeutic strategy[END_REF]. Yet recent data has added layers of complexity to the story. PARP1 itself may be required for the repair of stalled replications forks [START_REF] Bryant | PARP is activated at stalled forks to mediate Mre11dependent replication restart and recombination[END_REF], but not for replication forks that have collapsed to generate a double strand break. Whether the involvement of PARP1 in stalled fork repair is significant to our understanding of the potency of PARP inhibitors is unclear, but this data does emphasise that the repair of stalled replication forks is complex, and that PARP inhibition may force repair down a particular pathway.

A major issue in biomarker development for PARP inhibitors is to try and dissect the issue of whether PARP inhibitors should be developed as single agents or in combination with chemotherapy. At present it is generally assumed that same biomarkers, of homologous recombination deficiency, will predict for the benefit of PARP inhibitors as single agents, as well as PARP inhibitors in combination with chemotherapy. However, it is probable that different molecular defects in HR may predict sensitivity to PARP inhibitors alone, whereas other defects for benefit to PARP inhibitors in combination with chemotherapy. For example, cancers with a "hard" defect in HR, such as BRCA1/2 mutations, may be responsive to PARP inhibitors as single agents, whereas potentially cancers with a "soft" or mild defect in HR may not show sensitivity to single agent PARP inhibition, but could show substantial benefit when the cell is stressed by the combination of PARP inhibition and more complex damage created by chemotherapy.

Biomarkers for resistance to therapy

Mechanisms of resistance to PARP inhibitor therapy are beginning to emerge. In hereditary breast cancers reversion mutation of BRCA1/2 gene, either directly to wild type [START_REF] Swisher | Secondary BRCA1 mutations in BRCA1-mutated ovarian carcinomas with platinum resistance[END_REF] or by intragenic deletion [START_REF] Edwards | Resistance to therapy caused by intragenic deletion in BRCA2[END_REF], restoring normal BRCA1/2 protein function is highly likely to be a major mechanism of resistance, with the mutation selected by prior platinum therapy [START_REF] Fong | Poly(ADP)-ribose polymerase inhibition: frequent durable responses in BRCA carrier ovarian cancer correlating with platinum-free interval[END_REF]. Further potential resistance mechanisms identified include 53BP1 loss as a mechanism of resistance in BRCA1 deficiency [START_REF] Bouwman | 53BP1 loss rescues BRCA1 deficiency and is associated with triple-negative and BRCA-mutated breast cancers[END_REF][START_REF] Bunting | 53BP1 inhibits homologous recombination in Brca1-deficient cells by blocking resection of DNA breaks[END_REF], that partly rescues the deficiency in HR seen BRCA1 deficient cells. Overexpression of RAD51 has also been suggested to partially rescue HR deficiency [START_REF] Schild | Overexpression of RAD51 suppresses recombination defects: a possible mechanism to reverse genomic instability[END_REF], but it is uncertain if this is clinically relevant. It is also important to recall that many of the PARP inhibitors in current clinical development are P-glycoprotein substrates.

Finally the issue of tumour heterogeneity, and the influence of prior therapy, on a biomarker is a likely to be a major confounding factor in predicting response. PARP inhibitors share potential resistance mechanisms with conventional chemotherapy, and this emphasises the importance of not relying fully on biomarkers present in the primary tissue, or germline, when treating patients in the metastatic setting. Therefore, to maximise clinical potential it will be important to assess PARP inhibitors and companion biomarkers either in previously untreated cancers, such as the neoadjuvant setting, or with metastatic disease biopsies immediately prior to treatment.

Conflict of Interest Statement

Professor Ashworth may benefit financially from the development of PARP inhibitors through patents held jointly with AstraZeneca through the Institute of Cancer Research "rewards to inventors" scheme.


Anja Frömberg

Daniela Gutsch

Daniel Schulze

Claudia Vollbracht

Gabriele Weiss

Frank Czubayko

Dr Achim Aigner

Ascorbate exerts anti-proliferative effects through cell cycle inhibition and sensitizes tumor cells towards cytostatic drugs

Keywords: Ascorbate, dehydroascorbate, 5-FU, Paclitaxel, prostate carcinoma

Purpose: While the benefits of ascorbic acid (vitamin C, ascorbate) as an essential nutrient are well established, its effects on tumor cells and in tumor treatment are controversial. In particular, conflicting data exist whether ascorbate may increase the cytotoxic effects of antineoplastic drugs or may rather exert adverse effects on drug sensitivity during cancer treatment. Findings are further obscured regarding the distinction between ascorbate and dehydroascorbate (DHA). Thus, the purpose of this study was to evaluate and directly compare the cytotoxic efficacy of ascorbate compared to DHA, and to analyse if ascorbate at pharmacological concentrations affects the efficacy of antineoplastic agents in prostate carcinoma cells.

Methods: We directly compare the effects of ascorbate (supplied as 'Pascorbin® solution for injection') and DHA on tumor cell viability, and determine IC 50 values for various cell lines. At concentrations well below the IC 50 , ascorbate effects on cell proliferation and cell cycle are analysed. We furthermore determine changes in cellular sensitivity towards various cytostatic drugs upon pre-treatment of cells with ascorbate.

Results:

We demonstrate higher therapeutic efficacy of ascorbate over DHA in various cell lines, independent of cell line-specific differences in ascorbate sensitivity, and identify the extracellular generation of H 2 O 2 as critical mechanism of ascorbate action. We furthermore show that, in addition to pro-apoptotic effects described previously, ascorbate treatment already at concentrations well below the IC 50 exerts anti-proliferative effects on tumor cells. Those are based on interference with the cell cycle, namely by inducing a G 0 /G 1 arrest. Pre-treatment of tumor cells with ascorbate leads to increased cellular sensitivity towards Docetaxel, Epirubicin, Irinotecan and 5-FU, but not towards Oxaliplatin and Vinorelbin. For Docetaxel and 5-FU, a linear correlation between this sensitizing effect and the ascorbate dosage is observed.

Conclusions:

The redox-active form of vitamin C, ascorbate, shows therapeutic efficacy in tumor cells. These antitumor effects of ascorbate are mainly based on its extracellular action and, in addition to the induction of apoptosis, also include an anti-proliferative effect by inducing cell cycle arrest. Furthermore, ascorbate treatment specifically enhances the cytostatic potency of certain chemotherapeutics, which implicates therapeutic benefit during tumor treatment.

Introduction

Ascorbate (vitamin C) is an essential nutrient acting as anti-oxidant and co-factor for various enzymes. The effects of ascorbate on tumor cells and in tumor therapy, however, are controversial since studies by Cameron and Pauling reported that high amounts of ascorbic acid prolonged the life of terminal cancer patients [START_REF] Cameron | Supplemental ascorbate in the supportive treatment of cancer: reevaluation of prolongation of survival times in terminal human cancer[END_REF], while trials performed by the Mayo Clinic could not confirm these results [START_REF] Creagan | Failure of high-dose vitamin C (ascorbic acid) therapy to benefit patients with advanced cancer. A controlled trial[END_REF][START_REF] Moertel | Highdose vitamin C versus placebo in the treatment of patients with advanced cancer who have had no prior chemotherapy. A randomized double-blind comparison[END_REF]. Because of new insights into the pharmacokinetics of intravenous and oral vitamin C applications, the role of intravenous ascorbic acid in cancer treatment had to be newly assessed [START_REF] Padayatty | Vitamin C pharmacokinetics: implications for oral and intravenous use[END_REF]. Oral ascorbate produces concentrations in plasma and tissue which are tightly controlled (<0.2 mM), while pharmacological concentrations of ascorbate in plasma (>>0.2 mM), which may lead to tumor cytotoxicity, can only be achieved by parenteral administration. Furthermore, there is some scientific debate regarding the therapeutic relevance and efficacy of ascorbate versus dehydroascorbate ( [START_REF] Espey | Comment re: Vitamin C antagonizes the cytotoxic effects of chemotherapy[END_REF][START_REF] Heaney | Vitamin C antagonizes the cytotoxic effects of antineoplastic drugs[END_REF][START_REF] Levine | Losing and finding a way at C: new promise for pharmacologic ascorbate in cancer treatment[END_REF]; see below).

At pharmacological concentrations, ascorbate exerts cytotoxic effects selectively on tumor cells due to the induction of hydrogen peroxide (H 2 O 2 )-dependent cytotoxicity after ascorbate exposure. The cellular consequences in malignant cells seem to be manifold. More specifically, ascorbate has been shown to influence tumor cells by interfering with iron uptake [START_REF] Carosio | Sodium ascorbate induces apoptosis in neuroblastoma cell lines by interfering with iron uptake[END_REF], by releasing apoptosis-inducing factor (AIF) from mitochondria [START_REF] Hong | Ascorbate (vitamin C) induces cell death through the apoptosisinducing factor in human breast cancer cells[END_REF], by increasing the expression of p53, p21 and cellular calcium, by decreasing the mitochondrial membrane potential and activation of caspase 3 [START_REF] Hahm | The molecular mechanisms of vitamin C on cell cycle regulation in B16F10 murine melanoma[END_REF][START_REF] Lin | Sodium ascorbate inhibits growth via the induction of cell cycle arrest and apoptosis in human malignant melanoma A375.S2 cells[END_REF], by reducing the levels of hypoxia-inducible factor (HIF-1) thus leading to reduced myc-mediated tumorigenesis [START_REF] Gao | HIF-dependent antitumorigenic effect of antioxidants in vivo[END_REF], by the inhibition of cyclooxygenase-2 (COX-2) expression and the modulation of insulin-like growth factor II (IGF-II) production [START_REF] Lee | Vitamin C suppresses proliferation of the human melanoma cell SK-MEL-2 through the inhibition of cyclooxygenase-2 (COX-2) expression and the modulation of insulin-like growth factor II (IGF-II) production[END_REF] and/or by caspase-independent autophagy [START_REF] Du | Mechanisms of ascorbate-induced cytotoxicity in pancreatic cancer[END_REF]. It is noteworthy that some of these effects are mediated, at least in part, by the redox-state of the cell [START_REF] Malorni | Oxidative stress leads to a rapid alteration of transferrin receptor intravesicular trafficking[END_REF][START_REF] Qian | Erlotinib activates mitochondrial death pathways related to the production of reactive oxygen species in the human non-small cell lung cancer cell line A549[END_REF], and although Lin and colleagues reported a reduction in the level of reactive oxygen species (ROS) of ascorbate-treated cells [START_REF] Lin | Sodium ascorbate inhibits growth via the induction of cell cycle arrest and apoptosis in human malignant melanoma A375.S2 cells[END_REF], the majority of data supports the notion that the basis of tumor cytotoxicity may also rely on the generation of H 2 O 2 [START_REF] Chen | Pharmacologic ascorbic acid concentrations selectively kill cancer cells: action as a pro-drug to deliver hydrogen peroxide to tissues[END_REF][START_REF] Du | Mechanisms of ascorbate-induced cytotoxicity in pancreatic cancer[END_REF].

During the extracellular production of H 2 O 2 , ascorbate is oxidized to dehydroascorbate (DHA), and accumulates especially in tumor cells upon facilitated DHA uptake by glucose transporters (GLUTs) and intracellular reduction to ascorbate [START_REF] Agus | Stromal cell oxidation: a mechanism by which tumors obtain vitamin C[END_REF][START_REF] Nualart | Recycling of vitamin C by a bystander effect[END_REF][START_REF] Vera | Mammalian facilitative hexose transporters mediate the transport of dehydroascorbic acid[END_REF][START_REF] Vera | Human HL-60 myeloid leukemia cells transport dehydroascorbic acid via the glucose transporters and accumulate reduced ascorbic acid[END_REF]. Despite this efficient internalization, only limited information about the cytotoxicity of DHA on tumor cells is available. In order to evaluate the efficacy of ascorbate compared to DHA, we determined and directly compare in this study the IC 50 values of both redox states of vitamin C for different tumor cell lines. The results of three recent tumor xenograft studies also demonstrated that pharmacologic doses of ascorbate, which are achievable in humans through i.v. application [START_REF] Hoffer | Phase I clinical trial of i.v. ascorbic acid in advanced malignancy[END_REF], may have potential for therapy in cancer [START_REF] Casciari | Effects of high dose ascorbate administration on L-10 tumor growth in guinea pigs[END_REF][START_REF] Chen | Pharmacologic doses of ascorbate act as a prooxidant and decrease growth of aggressive tumor xenografts in mice[END_REF][START_REF] Du | Mechanisms of ascorbate-induced cytotoxicity in pancreatic cancer[END_REF]. Still, although these preclinical data in animal models showed a significant reduction of tumor growth, the use of pharmacologic ascorbate as a single agent was not curative. These results suggest that, rather than ascorbate single treatment, the combination of pharmacologic ascorbate with other therapies deserves further exploration for treatment of cancers, and emphasize that a future trend may lie in combination of ascorbate with chemotherapeutics [START_REF] Hoffer | Phase I clinical trial of i.v. ascorbic acid in advanced malignancy[END_REF].

Based on the anti-oxidant role of ascorbate and in the light of previous publications, however, the effects of ascorbate on chemotherapy remain very controversial. While some studies suggest that ascorbate may increase the effects of some antineoplastic agents including arsenic trioxide [START_REF] Dai | Malignant cells can be sensitized to undergo growth inhibition and apoptosis by arsenic trioxide through modulation of the glutathione redox system[END_REF][START_REF] Grad | Mitochondria as targets for established and novel anti-cancer agents[END_REF], Doxorubicin, Cisplatin and Paclitaxel in human breast carcinoma cells [START_REF] Kurbacher | Ascorbic acid (vitamin C) improves the antineoplastic activity of doxorubicin, cisplatin, and paclitaxel in human breast carcinoma cells in vitro[END_REF], 5-FU and Cisplatin in esophageal cancer cell lines [START_REF] Raouf | Vitamin C enhances chemosensitization of esophageal cancer cells in vitro[END_REF] and 5-FU in mouse lymphoma cells [START_REF] Nagy | Chemosensitizing effect of vitamin C in combination with 5-fluorouracil in vitro[END_REF], other scientists assume the contrary [START_REF] Lawenda | Should supplemental antioxidant administration be avoided during chemotherapy and radiation therapy?[END_REF]. More specifically, Heaney et al. found antagonistic effects upon pre-treatment of tumor cells with DHA, with dose-dependently decreased cytotoxicity of various antineoplastic agents in vitro and in vivo [START_REF] Heaney | Vitamin C antagonizes the cytotoxic effects of antineoplastic drugs[END_REF]. In this study, an only modest effect on intracellular ROS was observed. Since DHA, as described above, is the commonly transported form of vitamin C via facilitated transport prior to being reduced to ascorbic acid and trapped intracellularly, they conclude that vitamin C supplementation may exert adverse effects during cancer treatment [START_REF] Heaney | Vitamin C antagonizes the cytotoxic effects of antineoplastic drugs[END_REF]. On the other hand, ascorbate effects on tumor cells have been linked to extracellular rather than intracellular ascorbate [START_REF] Chen | Pharmacologic ascorbic acid concentrations selectively kill cancer cells: action as a pro-drug to deliver hydrogen peroxide to tissues[END_REF][START_REF] Chen | Ascorbate in pharmacologic concentrations selectively generates ascorbate radical and hydrogen peroxide in extracellular fluid in vivo[END_REF].

Thus, the aim of the second part of this study was to analyse if ascorbate in pharmacological concentrations affects the efficacy of antineoplastic agents in prostate carcinoma cells. To this end, we used 5-FU which had already been tested previously in other cell lines regarding the ascorbate dependence of its cytostatic effect [START_REF] Raouf | Vitamin C enhances chemosensitization of esophageal cancer cells in vitro[END_REF][START_REF] Kurbacher | Ascorbic acid (vitamin C) improves the antineoplastic activity of doxorubicin, cisplatin, and paclitaxel in human breast carcinoma cells in vitro[END_REF][START_REF] Nagy | Chemosensitizing effect of vitamin C in combination with 5-fluorouracil in vitro[END_REF] and furthermore included a set of chemotherapeutic reagents which are based on different modes of action, i.e. the topoisomerase inhibitor Irinotecan, the anti-mitotic drugs Docetaxel and Vinorelbine, the anthracycline Epirubicin and the platinum Oxaliplatin.

Materials and methods

Reagents and cell lines

Cell lines were obtained from the American type culture collection (ATCC/LGC Promochem, Wesel, Germany) and cultivated under standard conditions (37°C, 5 % CO 2 ) in IMDM (PAA, Cölbe, Germany) supplemented with 10 % fetal calf serum (FCS) unless indicated otherwise.

Ascorbate (Pascorbin® solution for injection) was provided by Pascoe, Giessen, Germany, and DHA and catalase were purchased from Sigma (Deisenhofen, Germany).

Ascorbate / DHA treatment and assessment of cell viability and apoptosis

For the determination of ascorbate / DHA effects on cell viability, cells were seeded in 96 well plates at 1*10 4 cells/well in serum-containing medium. The next day, cells were treated, in the absence or presence of 100 µg/ml catalase, with ascorbate or DHA at the concentrations indicated in Figs. 1 and2 in serum-free medium. After 24h, the medium was changed again and the cells were cultivated for further 48h in IMDM/10% FCS medium. The numbers of viable cells in eight wells were then determined using a colorimetric assay according to the manufacturer's protocol (Cell Proliferation Reagent WST-1, Roche Molecular Biochemicals, Mannheim, Germany). For the determination of apoptosis, 3*10 5 cells/well were seeded in a sixwell plate and treated with ascorbate or DHA in serum-free medium as described above. Caspase-3/-7 activity was determined 24 h after treatment start unless stated otherwise, with Ac-DEVD-AMC (BD PharMingen) as substrate. Cells were harvested in 300 µl lysis buffer (50 mM Tris/HCl, pH 7.4, 150 mM NaCl, 2 mM MgCl 2 , 10 mM DTT, protease inhibitor-cocktail set III (Calbiochem/EMD Biosciences)), and the lysates were sonicated in an ultrasonic bath for 5 min prior to centrifugation. 40 µl supernatant was transferred to white walled 96 well plates, the reaction was started by adding 10 µl substrate solution (10 mM Ac-DEVD-AMC (Acetyl-Asp-Glu-Val-Asp-7-Amino-4-methylcoumarin) stock solution in DMSO, dissolved in 500 µl PBS) to a final concentration of 40 µM Ac-DEVD-AMC and measured after 60 min using a FLUOstar OPTIMA microplate reader. In parallel, total cell numbers after treatment under identical conditions as described above were determined by WST-1 assay, and readings from caspase-3/-7 measurements were normalized for cell densities.

Determination of cell proliferation and cell cycle measurements

Studies of anchorage-dependent proliferation were carried out essentially as described previously [START_REF] Abuharbeid | Cytotoxicity of the novel anti-cancer drug rViscumin depends on HER-2 levels in SKOV-3 cells[END_REF] in the presence of IMDM / 10 % FCS. Cells were seeded in 96 well plates at 1.5 *10 3 cells/well, and the next day the medium was changed to fresh serum-containing medium supplemented with 0, 0.1, 0.3 or 1 mM ascorbate. At the time points indicated, the numbers of viable cells were determined by WST-1 as described above.

For cell cycle analysis, 1.5 x 10 5 cells were seeded in sixwell plates and treated as described above. 24 h after treatment start, cells were harvested and fixed in 70% ethanol in PBS for 1 h on ice, incubated with 50 µg/ml RNase A at 37°C overnight, stained with 50 µg/ml Propidium iodide and subjected to flow cytometric analysis using FACSCalibur.

Treatment of cells with chemothe rapeutics

For the determination of cellular sensitivity towards cytostatics, DU-145 cells were seeded in 96well plates at 4*10 3 /well 1 day prior to pretreatment with 0 mM, 1 mM or 2 mM ascorbate in serum-free medium for 24h as described above. The medium was then changed again for serumcontaining medium + cytostatics at the concentrations detailed in Fig. 3. After 48 -72 h, the number of viable cells was determined by WST-1 as described above.

Statistics

Each experiment was performed in triplicates or quadruplicates for every time point / concentration. All curves represent an overlay of at least three independent experiments, with errors bars depicting standard errors of the mean (s.e.m.). Curve fitting was performed using SigmaPlot and the 4 parameter Hill equation [START_REF] Goutelle | The Hill equation: a review of its capabilities in pharmacological modelling[END_REF].

Results

Determination of ascorbate and DHA sensitivities of various cell lines

In initial experiments, we assessed the ascorbate sensitivity of various tumor cell lines from different tumor entities. While ascorbate dose-dependent effects on the cells were observed in all cell lines (see Suppl. Fig. 1B for two representative examples), the sensitivities towards ascorbate varied between different cancer cell lines which is reflected by IC 50 values in the range of 1.7 mM to > 60 mM (Suppl. Fig. 1a).

For the direct comparison between ascorbate and DHA effects, carcinoma cell lines LS174T (colon), SKOV-3 (ovarian), DU-145 (prostate) and A549 (breast) were selected and treated with various amounts of the compounds as indicated in the Fig. 1. Variations observed between different cell lines were seen regarding the ascorbate sensitivity, but not for DHA. More importantly, independent of these variations cells were considerably less sensitive towards DHA as compared to ascorbate in all experiments, with DHA IC 50 values between 12.7 mM (DU-145) and 14.9 mM (LS174T) and profound inhibitory effects being observed only at 30 mM DHA.

When comparing the different cell lines no direct correlation between ascorbate and DHA sensitivities were observed, i.e. high sensitivity towards ascorbate did not translate into higher sensitivity towards DHA and vice versa (compare e.g. Fig. 1a vs. 1d). It should also be noted that, while the assessment of DHA effects at 100 mM was not possible due to poor solubility, the pH indicator in the medium changed its color already at 30 mM indicating some medium acidification by DHA which may add to its cytotoxicity and thus the over-estimation of DHA anti-tumor effects. Notably, this effect was not observed for ascorbate. We also confirmed the stability of ascorbate under comparable experimental conditions by detecting > 99% ascorbate after 24h (data not shown). Taken together, despite its more efficient uptake [START_REF] Heaney | Vitamin C antagonizes the cytotoxic effects of antineoplastic drugs[END_REF], tumor cells are significantly less sensitive towards DHA as compared to ascorbate.

To gain some more insight into the mechanistic details of ascorbate cytotoxicity, LS174T cells were treated as described above in the presence or absence of 100 µg/ml catalase. Treatment of the cells with catalase largely prevented ascorbate-mediated cytotoxicity, as indicated by a >30fold shift of the ascorbate dose-response curve to the right (Fig. 2a). In contrast, no effects of catalase on dehydroascorbate cytotoxicity were observed (Fig. 2b). The comparison between Figs. 2a and 2b also revealed that the addition of catalase during the treatment quenched the ascorbate cytotoxicity even below dehydroascorbate effects. Identical results were obtained in SKOV-3 cells (data not shown). From these data we conclude tha t the effects of ascorbic acid are the result of the extracellular generation of hydrogen peroxide, which is destroyed in the presence of catalase.

Assessment of anti-proliferative and pro-apoptotic effects of ascorbate treatment

For subsequent experiments, DU-145 prostate carcinoma cells were selected as representative tumor cell line. To assess the effects of ascorbate at concentrations below the IC 50 in more detail, proliferation assays were performed. Under subconfluent conditions, untreated DU-145 cells showed an exponential growth curve over at least 5d in the presence of serum, which was indistinguishable from the cellular proliferation in the presence of 0.1 mM or 0.3 mM ascorbate.

Notably, however, upon addition of 1 mM ascorbate, which is still 5-fold below the IC 50 in this cell line, tumor cell growth curves were markedly reduced (Fig. 3a).

To analyse whether this reduction in the number of viable cells was due to increased apoptosis or reduced proliferation, or both, we determined ascorbate effects on cellular apoptosis by measuring the activity of caspase-3, which is the mutual / shared effector caspase of both caspases-8/-10 (TNF / TRAIL / Fas ligand pathway) and caspase-9 (stress / DNA damageinduced cytochrom C release). When the cells were treated with ascorbate for 24h under serumfree conditions, an ascorbate-induced increase in caspase activity was observed (Fig. 3b). While this effect was ascorbate dose-dependent, it was only moderate with a maximum ~ 1.5-fold increase at ascorbate concentrations above the IC 50 . Similar effects were observed upon treatment of the cells with DHA, although cells appeared again to be less sensitive towards DHA as compared to ascorbate (note the absence of an induction of apoptosis at 1 mM DHA). To monitor the time course, cells were treated with ascorbate for only 4h. A slight (~ 1.2-fold) induction of apoptosis was observed which was most profound at 2-4 h after the end of the treatment and decreased again thereafter. In contrast, no induction of apoptosis was observed in the presence of serum (data not shown).

Next, we determined ascorbate effects on the cell cycle. When proliferating cells were treated with 1 mM ascorbate, which was the concentration leading to an efficient reduction in the number of viable cells (see Fig. 3a), marked changes in the cell cycle distribution were observed.

More specifically, the percentage of the cells in G0/G1 phase increased from 40% to 56% upon ascorbate treatment, with a corresponding decrease of the number of cells in t he G2/M phase (Fig. 3c). The detection of these changes did not require synchronization of the cells, but in nocodazole pre-treated cells similar results were observed (data not shown). When cell cycle data were analysed for sub-G0/G1 peak levels representing the number of cells which display a latestage cell death phenotype, a ~ 2.5 fold increase was observed upon ascorbate treatment (Fig. 3d).

Taken together, these data show that anti-tumor effects of ascorbate in vitro rely on an increase in apoptosis as well as effects on cell cycle, i.e. an increased percentage of cells in the quiescent state in the presence of ascorbate.

Increased cellular sensitivities towards cytostatics upon ascorbate pre -treatment

Since ascorbate would be hardly used in a single treatment protocol in cancer therapy and could be more promising in combination with established cytostatics, we next assessed the effects of ascorbate pre-treatment on cellular sensitivity towards cytostatic drugs. To this end, cytostatic drugs were selected which are used in prostate carcinoma therapy and cover different mechanisms of action. Assays of anchorage-dependent proliferation revealed the inhibition of DU-145 cell proliferation starting at Docetaxel concentrations above 1 nM with the IC 50 being 10 nM (Fig. 4a). Remarkably, this dose-response curve shifted significantly, to ~ 3-fold lower Docetaxel concentrations (IC 50 = 3.5 nM), upon pretreatment of DU-145 cells with 2 mM ascorbate 24 h prior to addition of Docetaxel. The comparison with cells pre-treated with 1 mM ascorbate also revealed that the increase in cell sensitivity towards Docetaxel (IC 50 = 6.7 nM) is ascorbate dose-dependent (Fig. 4a).

Likewise, ascorbate pre-treatment led to increased sensitivity towards Epirubicin. Here, a shift of the dose-response curves towards > 2.5-fold lower Epirubicin concentrations was already observed for 1 mM ascorbate (IC 50 0.26 mM vs. 0.10 mM), with no further increase in sensitivity at higher ascorbate concentrations (Fig. 4b).

In contrast, no effects of ascorbate pre-treatment were observed on cellular sensitivities towards Oxaliplatin or Vinorelbin (Figs. 4c,d). Independent of ascorbate, cytotoxic effects started above 1 µM Oxaliplatin with the IC 50 being ~ 8.5 µM, or above 0.1 µM Vinorelbin (IC 50 ~ 0.9 µM).

Likewise, Irinotecan IC 50 values were not markedly altered (11.2 µM, 10.5 µM and 10.3 µM for 0 mM, 1 mM and 2 mM ascorbate pre-treatment, respectively). However, while Irinotecan did not result in 100 % cell death even at highest concentrations (1000 µM), ascorbate pre-treatment further decreased the number of viable cells, thus enhancing the maximum cytotoxic effect of Irinotecan (Fig. 4e). Finally, > 2-fold increased sensitivities upon ascorbate pre-treatment were observed towards 5-FU, as indicated by the shift of the dose-response curves to the left. This ascorbate effect on cell sensitivity was again dose-dependent with IC 50 values for 5-FU being decreased from ~ 9.0 µM to 6.5 µM or 4.0 µM upon pre-treatment with 1 mM or 2 mM ascorbate, respectively (Fig. 4f).

To analyse the dose-dependence in more detail, IC 50 values were correlated to the concentrations of ascorbate pre-treatment. For both cytostatics, 5-FU and Docetaxel, a direct linear correlation between the dose-response curves, as defined by the IC 50 , and the ascorbate concentration was observed (Fig. 4g,h).

Discussion

In this study, we show anti-tumor effects of ascorbate in vitro and, through the direct comparison with DHA, we demonstrate that ascorbate effects are much more profound. Since DHA is taken up by cells more readily due to facilitated transport through the glucose transporters, mainly GLUT1, and is then intracellularly reduced with the ascorbate be ing trapped intracellularly [START_REF] Vera | Mammalian facilitative hexose transporters mediate the transport of dehydroascorbic acid[END_REF][START_REF] Vera | Human HL-60 myeloid leukemia cells transport dehydroascorbic acid via the glucose transporters and accumulate reduced ascorbic acid[END_REF], this also shows that the anti-tumor effects rely on extracellular rather than intracellular ascorbate. This is supported by our experiments which demonstrate the inhibition of ascorbate cytotoxicity upon addition of catalase, thus identifying extracellular H 2 O 2 formation as critical step. Also, this conclusion is in line with previous publications which have shown that ascorbateinduced cell death in vitro and anti-tumor effects in vivo are dependent on extrac ellular H 2 O 2 formation [START_REF] Chen | Pharmacologic ascorbic acid concentrations selectively kill cancer cells: action as a pro-drug to deliver hydrogen peroxide to tissues[END_REF][START_REF] Chen | Ascorbate in pharmacologic concentrations selectively generates ascorbate radical and hydrogen peroxide in extracellular fluid in vivo[END_REF], while the effects of DHA on intracellular ROS (despite superior uptake; see above) are only moderate [START_REF] Heaney | Vitamin C antagonizes the cytotoxic effects of antineoplastic drugs[END_REF]. High extracellular ascorbate concentrations are consequently a prerequisite for selective tumor cytotoxicity, which does not appear to be fulfilled in the case of DHA.

Pro-apoptotic effects of ascorbate have been described previously (see e.g. [START_REF] Chen | Pharmacologic ascorbic acid concentrations selectively kill cancer cells: action as a pro-drug to deliver hydrogen peroxide to tissues[END_REF]) and are confirmed in this paper. The determination of increased caspase activity in this study, however, may rather underestimate pro-apoptotic effects since it has been shown that ascorbate also induces caspaseindependent necrotic cell death [START_REF] Du | Mechanisms of ascorbate-induced cytotoxicity in pancreatic cancer[END_REF][START_REF] Ohtani | Tumor suppressor 101F6 and ascorbate synergistically and selectively inhibit non-small cell lung cancer growth by caspase-independent apoptosis and autophagy[END_REF]. The presence of caspase-independent effects in the case of ascorbate may also explain why caspase activation is comparable between ascorbate and dehydroascorbate while cytotoxic effects are not. This notion is further supported by the fact that the assessment of sub-G 0 /G 1 peak levels, representing the number of cells which display a latestage cell death phenotype, revealed a more profound increase upon ascorbate treatment.

More interestingly, in addition to the induction of apoptosis we observe inhibitory effects on tumor cells which are based on decreased cell proliferation. We show that ascorbate, already at concentrations well below its IC 50 , interferes with the cell cycle by inducing a G 0 /G 1 arrest. To date there is only limited evidence in melanoma and Raji cells with regard to the consequences of ascorbate on cell cycle, suggesting that ascorbate in low concentrations (0.05-0.2 mM) inhibits the proliferation by inducting growth arrest in a dose-dependent manner without cytotoxic effects, while higher concentrations (2-16 mM) seem to cause both cell cycle arrest and apoptosis, and still higher concentrations result in necrosis [START_REF] Belin | Antiproliferative effect of ascorbic acid is associated with the inhibition of genes necessary to cell cycle progression[END_REF][START_REF] Hahm | The molecular mechanisms of vitamin C on cell cycle regulation in B16F10 murine melanoma[END_REF][START_REF] Lin | Sodium ascorbate inhibits growth via the induction of cell cycle arrest and apoptosis in human malignant melanoma A375.S2 cells[END_REF]. Our studies confirm the anti-proliferative effects of ascorbate for a tumor entity with high incidence rate, i.e. prostate carcinoma, and demonstrate that these effects require only ascorbate concentrations well below the IC 50 . Furthermore, our data establish that pre-treatment of tumor cells with ascorbate increases their sensitivity towards certain cytostatics. The fact that this is true for Docetaxel, Epirubicin, Irinotecan and 5-FU, while IC 50 values for Oxaliplatin or Vinorelbin remain unchanged, emphasizes the specificity of this effect with regard to the choice of the cytostatic drug and its mode of action. While in the case of all 'ascorbate-sensitive' drugs the mechanism of action seems to be mainly related to interfering with DNA-/ RNA synthesis and/or inhibition of cell cycle progression, the precise basis of the specificity of this ascorbate effect for certain cytostatics remains to be elucidated. This is particularly true since Docetaxel, but not Vinorelbin sensitivity is affected, despite the fact that both drugs act on microtubule (although at different steps in microtubule formation / breakdown). There is no correlation between 'ascorbate sensitivity' and the drugs being a p-glycoprotein substrate (Docetaxel, Epirucibin, Vinorelbin, Irinotecan) or not (Oxaliplatin, 5-FU) [START_REF] Li | Overexpression and involvement of special AT-rich sequence binding protein 1 in multidrug resistance in human breast carcinoma cells[END_REF][START_REF] Takara | Cytotoxic effects of 27 anticancer drugs in HeLa and MDR1-overexpressing derivative cell lines[END_REF], and thus our data already firmly exclude an effect of (changes in) pglycoprotein as underlying mechanism. This is important since in many cases increased resistance of tumor cells is due to p-glycoprotein overexpression [START_REF] Takara | Cytotoxic effects of 27 anticancer drugs in HeLa and MDR1-overexpressing derivative cell lines[END_REF].

The comparison with the data from Heaney et al. [START_REF] Heaney | Vitamin C antagonizes the cytotoxic effects of antineoplastic drugs[END_REF] demonstrate that the cellular localization as well as the dosage of ascorbate appears to be crucial for the determination of synergistic / antagonistic effects of an ascorbate + cytostatic combination: while DHA, which is readily internalized by the cells, was shown to display antagonistic effects [START_REF] Heaney | Vitamin C antagonizes the cytotoxic effects of antineoplastic drugs[END_REF], pretreatment with ascorbate, which acts extracellularly as a pro-oxidant, leads to increased sensitivity towards several antineoplastic drugs (see our data; [START_REF] Raouf | Vitamin C enhances chemosensitization of esophageal cancer cells in vitro[END_REF][START_REF] Dai | Malignant cells can be sensitized to undergo growth inhibition and apoptosis by arsenic trioxide through modulation of the glutathione redox system[END_REF][START_REF] Grad | Mitochondria as targets for established and novel anti-cancer agents[END_REF][START_REF] Kurbacher | Ascorbic acid (vitamin C) improves the antineoplastic activity of doxorubicin, cisplatin, and paclitaxel in human breast carcinoma cells in vitro[END_REF][START_REF] Nagy | Chemosensitizing effect of vitamin C in combination with 5-fluorouracil in vitro[END_REF]. Also, while Heaney used DHA in concentrations between 0 and 750 µM, comparable to physiological and supraphysiological blood levels after oral supplementation, we applied pharmacological concentrations in the low millimolar range (1-2 mM), which require i.v. injection. Furthermore, the fact that Heaney et al. used leukemia cells is noteworthy since it is known that some leukemia cell lines display growth stimulation in the presence of ascorbate. These proliferative effects clearly depend on the ascorbate concentration, with physiological and subphysiological ascorbate concentrations stimulating proliferation, while pharmacological ascorbate concentrations in the mM range exert anti-proliferative effects [START_REF] Bode | Ascorbic acid recycling in Nb2 lymphoma cells: implications for tumor progression[END_REF][START_REF] Park | Biological nature of the effect of ascorbic acids on the growth of human leukemic cells[END_REF][START_REF] Park | Growth modulation of human leukemic, preleukemic, and myeloma progenitor cells by L-ascorbic acid[END_REF]. Thus, the determination of possible synergistic effects with chemotherapeutics needs to include the tumor entity (solid or haematological) as well as the concentration (physiological or pharmacological) and redox state (ascorbate or DHA) of vitamin C.

While our data support the notion that ascorbate at pharmacological concentrations may be beneficial during cancer treatment, this would require i.v. injection, and caution is warranted regarding whether or not sufficient concentrations are achievable and regarding renal function, since oxalate calculus and renal failure are known and accepted contraindications for intravenous ascorbate. However, the tolerability of high-dose intravenous ascorbate is documented in a dosefinding phase I and pharmacokinetic study in 24 patients with advanced untreatable cancer. A dose of 1,5g/kg body weight, three times weekly, appeared to be safe and free of important toxicity [START_REF] Hoffer | Phase I clinical trial of i.v. ascorbic acid in advanced malignancy[END_REF].

Taken together, by the direct comparison of ascorbate and DHA we demonstrate that tumor cell cytotoxicity requires the application of ascorbate in mM concentrations and that ascorbate effects rely on the generation of H 2 O 2 (and perhaps other ROS). Still, although preclinical data in animal models showed a significant reduction of tumor growth, the use of pharmacologic ascorbate as a single agent was not curative. In the light of these findings, our and previous data demonstrating increased efficacy of cytostatics already at ascorbate concentrations in the low mM range deserve attention. While the potentiation of cytostatic effects is only three-fold and ascorbate cytotoxicity will also be dependent on the tumor microenvironment e.g. regarding free metal ions, this will be worthwhile to be further explored in vivo. This is particularly true since a three-fold reduction in the IC 50 of an antineoplastic agent is most likely relevant in cancer treatment because of the high general toxicity of these agents, which are often dose-limiting or the reason for interruption of chemotherapy.

Fig. 1

 1 Fig. 1Sensitivities of various carcinoma cell lines (a, LS174T colon; b, SKOV-3 ovarian; c, DU-145 prostate; d, A549 breast) towards ascorbate (red) versus DHA (black).

Fig. 2

 2 Fig. 2 Sensitivities of LS174T colon carcinoma cells towards ascorbate (a) or DHA (b) in the presence (closed circles) or absence (open circles) of 100 µg/ml catalase for H 2 O 2 degradation.

Fig. 3 (

 3 Fig. 3 (a) Anti-proliferative effects of ascorbate treatment on DU-145 prostate carcinoma cells. (b) Determination of caspase-3 activity as a marker for induction of apoptosis upon ascorbate or DHA treatment. (c) Effects of ascorbate treatment on the cell cycle and on (d) the sub-G0/G1 peak levels representing the number of cells which display a late-stage cell death phenotype.

Fig. 4 1 (

 41 Fig. 4Sensitivities of DU-145 prostate carcinoma cells towards Docetaxel (a), Epirubicin (b), Oxaliplatin (c), Vinorelbin (d), Irinotecan (e), and 5-Fluorouracil (5-FU, f) in the presence of no (black) versus 1 mM (red) or 2 mM (green) ascorbate. For Docetaxel (g) and 5-FU (h), a direct correlation between ascorbate pre-treatment concentrations (x-axis) and the decrease in IC 50 values is observed.

Cancer Chemotherapy and Pharmacology

Acknowledgme nts

We are grateful to Andrea Wüstenhagen for expert technical assistance, and to Dr. Udo Bakowsky for helpful support. This work was in part supported by funding from Pascoe pharmazeutische Präparate GmbH, Giessen, Germa ny, where C.V. and G.W. are employed.