Ivan Nourdin
email: inourdin@gmail.com

Murad S Taqqu

Central and non-central limit theorems in a free probability setting

Keywords: AMS subject classifications: 46L54, 60H05, 60H07 Central limit theorem, Non-central limit theorem, Convergence in distribution, Fractional Brownian motion, Free Brownian motion, Free probability, Rosenblatt process, Wigner integral

Long-range dependence in time series may yield non-central limit theorems. We show that there are analogous time series in free probability with limits represented by multiple Wigner integrals, where Hermite processes are replaced by non-commutative Tchebycheff processes. This includes the non-commutative fractional Brownian motion and the non-commutative Rosenblatt process.

Introduction and main result

Normalized sums of i.i.d. random variables satisfy the usual central limit theorem. But it is now well-known that this is not necessarily the case if the i.i.d. random variables are replaced by a stationary sequence with long-range dependence, that is, with a correlation which decays slowly as the lag tends to infinity. We want to investigate whether similar non-central results hold in the free probability setting.

We are motivated by the fact that there is often a close correspondence between classical probability and free probability. For example, the Gaussian law has the semicircular law as an analogue, hence the notion of a stationary semicircular sequence with a given correlation function. Multiple Wiener integrals which span the so-called Wiener chaos have multiple Wigner integrals as an analogue, with Hermite polynomials being replaced by Tchebycheff polynomials. We will see that the notion of Hermite rank is to be replaced by the Tchebycheff rank and that long-range dependence in the free probability setting also yields non-standard limits which are somewhat analogous to those in classical probability. In classical probability, the limits can be represented by multiple Wiener integrals. In free probability, they are represented by multiple Wigner integrals with similar kernels. Finally, the Hermite processes that appear in the limit in the usual probability setting are replaced, in the free probability setting, by non-commutative Tchebycheff processes.

Before stating our main results, let us describe their analogue in the classical probability framework. Let Y = {Y k : k ∈ Z} be a stationary Gaussian sequence on a probability space (Ω, F, P ), with E[Y k ] = 0 and E[Y 2 k ] = 1, and let ρ(kl) = E[Y k Y l ] be its correlation kernel. (Observe that ρ is symmetric, that is, ρ(n) = ρ(-n) for all n 1.) Let H 0 (y) = 1, H 1 (y) = y, H 2 (y) = y 2 -1, H 3 (y) = y 3 -3y, . . ., denote the sequence of Hermite polynomials (determined by the recursion yH k = H k+1 + kH k-1 ), and consider a real-valued polynomial Q of the form Q(y) = s q a s H s (y), (1.1) with q 1 and a q = 0, and where only a finite number of coefficients a s are non zero. The integer q is called the Hermite rank of Q. Finally, set

W n (Q, t) = [nt] k=1 Q(Y k ) = s q a s [nt] k=1 H s (Y k ), t 0. (1.2)
The following theorem is a summary of the main findings in Breuer and Major [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF], Dobrushin and Major [START_REF] Dobrushin | Non-central limit theorems for non-linear functions of Gaussian fields[END_REF] and Taqqu [START_REF] Taqqu | Weak Convergence to Fractional Brownian Motion and to the Rosenblatt Process[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF]. Here and throughout the sequel, the notation 'f.d.d.' stands for the convergence in finite-dimensional distribution.

Theorem 1.1. Let Q be the polynomial defined by (1.1) and let W n (Q, •) be defined by (1.2).

1. If k∈Z |ρ(k)| q is finite, then, as n → ∞,

W n (Q, •) √ n f.d.d. → s q s!a 2 s k∈Z ρ(k) s × B, (1.3)
with B a classical Brownian motion.

2. Let L : (0, ∞) → (0, ∞) be a function which is slowly varying at infinity and bounded away from 0 and infinity on every compact subset of [0, ∞). If ρ has the form

ρ(k) = k -D L(k), k 1, (1.4)
with 0 < D < 1 q , then, as n → ∞,

W n (Q, •) n 1-qD/2 L(n) q/2 f.d.d. → a q (1 -qD 2 )(1 -qD) × R 1-qD/2,q , (1.5)
where R 1-qD/2,q is the qth Hermite process of parameter 1 -qD/2.

For background on the notions of Hermite processes, see Dobrushin and Major [START_REF] Dobrushin | Non-central limit theorems for non-linear functions of Gaussian fields[END_REF], Embrechts and Maejima [START_REF] Embrechts | Selfsimilar Processes[END_REF], Peccati and Taqqu [START_REF] Peccati | Wiener Chaos: Moments, Cumulants and Diagrams[END_REF], and Taqqu [START_REF] Taqqu | Weak Convergence to Fractional Brownian Motion and to the Rosenblatt Process[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF][START_REF] Taqqu | Fractional Brownian motion and long-range dependence[END_REF]. A slowly varying function at infinity L(x) is such that lim x→∞ L(cx)/L(x) = 1 for all c > 0. Constants and logarithm are slowly varying. A useful property is the Potter's bound (see [START_REF] Bingham | Regular Variation[END_REF]Theorem 1.5.6, (ii)]): for every δ > 0, there is C = C(δ) > 1 such that, for all x, y > 0,

L(x) L(y) C max x y δ , x y -δ . (1.6)
When ρ is given by (1.4), then

k∈Z |ρ(k)| q < ∞ if D > 1/q = ∞ if 0 < D < 1/q .
When D > 1/q, one says that the process X has short-range dependence. When 0 < D < 1/q, one says that it has long-range dependence. In the critical case D = 1/q, the series may be finite or infinite, depending on the precise value of L; this is the reason why we do not investigate this case further in Theorem 1.1 nor in the forthcoming Theorem 1.2. We now state our main result, which may be regarded as a non-commutative counterpart of Theorem 1.1. Some of the terms and concepts of free probability used here are defined in Section 2 which introduces free probability in a nutshell.

Let X = {X k : k ∈ Z} be a stationary semicircular sequence on a non-commutative probability space (A , ϕ), assume that ϕ(X k ) = 0 and ϕ(X

2 k ) = 1 and let ρ(k -l) = ϕ(X k X l ) be its correlation kernel. (Observe that ρ is symmetric, that is, ρ(n) = ρ(-n) for all n 1.) Let U 0 (x) = 1, U 1 (x) = x, U 2 (x) = x 2 -1, U 3 (x) = x 3 -2x, .
. ., denote the sequence of Tchebycheff polynomials of second kind (determined by the recursion xU k = U k+1 + U k-1 ). For a presentation of polynomials in the classical and free probability setting, see Anshelevich [START_REF] Anshelevich | Appell polynomials and their relatives[END_REF]. Consider a real-valued polynomial

Q of the form Q(x) = s q a s U s (x), (1.7)
with q 1 and a q = 0, and where only a finite number of coefficients a s are non zero. The integer q is called the Tchebycheff rank of Q. Finally, set

V n (Q, t) = [nt] k=1 Q(X k ) = s q a s [nt] k=1 U s (X k ), t 0. (1.8)
Our main result goes as follows:

Theorem 1.2. Let Q be the polynomial defined by (1.7) and let V n (Q, •) be defined by (1.8).

1. If k∈Z |ρ(k)| q is finite, then, as n → ∞, V n (Q, •) √ n f.d.d. → s q a 2 s k∈Z ρ(k) s × S, (1.9)
with S a free Brownian motion, defined in Section 2.7.

2. Let L : (0, ∞) → (0, ∞) be a function which is slowly varying at infinity and bounded away from 0 and infinity on every compact subset of [0, ∞). If ρ has the form (1.4) with 0 < D < 1 q , then, as n → ∞,

V n (Q, •) n 1-qD/2 L(n) q/2 f.d.d. → a q (1 -qD 2 )(1 -qD) × R 1-qD/2,q , (1.10)
where R 1-qD/2,q is the qth non-commutative Tchebycheff process of parameter 1 -qD/2, defined by a multiple Wigner integral of order q. It is given in Definition 3.5.

Let us compare Theorem 1.1 and Theorem 1.2. The fact that (1.9) relies on the free Brownian motion S implies for example that the marginal distribution of S is not Gaussian but is the semicircular law defined in Section 2.6, which has, in particular, a compact support. The notion of free independence is very different from the classical notion of independence as noted in Section 2.5. The resulting multiple integrals are then of a very different nature. Also, since the Hermite and Tchebycheff polynomials are different, the decomposition of the polynomial Q in Tchebycheff polynomials is different from its decomposition in Hermite polynomials and, consequently, its Tchebycheff rank can be different from its Hermite rank. This implies that even the order of the multiple integral in the limit may not be the same.

In the non-commutative probability context, the exact expression of the underlying law is of relatively minor importance. Moments are essential. In fact, convergence in law can be defined in our setting through convergence of moments. Also, in the classical probability setting, one can consider a function Q(x) in (1.2) which is not necessarily a polynomial. One could possibly do so as well in the non-commutative probability setting.

The methods of proof of Theorems 1.1 and 1.2 are different. The proof of the central limit theorem in Theorem 1.2 does not need to rely on cumulants and diagrams as in the classical case (see [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF]). It uses a transfer principle established recently in [START_REF] Nourdin | Multidimensional semicircular limits on the free Wigner chaos[END_REF] and stated in Proposition 5.1. The proof of the non-central limit theorem in Theorem 1.2 is much simpler than in the classical proof of Dobrushin and Major [START_REF] Dobrushin | Non-central limit theorems for non-linear functions of Gaussian fields[END_REF] and Taqqu [START_REF] Taqqu | Weak Convergence to Fractional Brownian Motion and to the Rosenblatt Process[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF] because it is sufficient here to establish the convergence of joint moments. Such an approach breaks down in the classical case when q 3 because, there, the joint moments do not characterize the target distribution anymore.

We proceed as follows. In Section 2, we present free probability in a nutshell and define multiple Wigner integrals. In Section 3, we introduce the non-commutative fractional Brownian motion, the non-commutative Rosenblatt process and more generally the non-commutative Tchebycheff processes, and study some of their basic properties. We compute their joint moments in Section 4. Theorem 1.2 is proved in Section 5.

2 Free probability in a nutshell 2.1 Random matrices Let (X ij ) i,j 1 be a sequence of independent standard Brownian motions, all defined on the same probability space (Ω, F, P ). Consider the random matrices

A n (t) = X ij (t) √ n 1 i,j n ,
and

M n (t) = A n (t) + A n (t) * √ 2 , (2.11)
where A n (t) * denotes the transpose or adjoint of the matrix A n (t). Thus, M n (t) is self-adjoint. For each t, A n (t) and M n (t) both belong to A n , the set of random matrices with entries in

L ∞-(Ω) = ∩ p>1 L p (Ω)
, that is, with all moments. On A n , consider the linear form τ n : A n → R defined by

τ n (M ) = E 1 n Tr(M ) , (2.12)
where E denotes the mathematical expectation associated to P , whereas Tr(•) stands for the usual trace operator. The space (A n , τ n ) is the prototype of a non-commutative probability space. Let t k > . . . > t 1 > t 0 = 0. A celebrated theorem by Voiculescu [START_REF] Voiculescu | Limit laws for random matrices and free product[END_REF] asserts that the increments

M n (t 1 ), M n (t 2 ) -M n (t 1 ), . . . , M n (t k ) -M n (t k-1
) are asymptotically free, meaning that

τ n (Q 1 (M n (t i 1 ) -M n (t i 1 -1 )) . . . Q m (M n (t im ) -M n (t im-1 ))) → 0 as n → ∞,
for all m 2, all i 1 , . . . , i m ∈ {1, . . . , k} with i 1 = i 2 , i 2 = i 3 , . . ., i m-1 = i m , and all real-valued polynomials Q 1 , . . . , Q m such that τ n (Q i (M n (t i ) -M n (t i-1 ))) → 0 as n → ∞ for each i = 1, . . . , m.

Let t > 0. The celebrated Wigner theorem [START_REF] Wigner | On the distribution of the roots of certain symmetric matrices[END_REF] can be formulated as follows: as n → ∞, M n (t) converges in law to the semicircular law of variance t, that is, for any real-valued polynomial Q,

τ n (Q(M n (t))) → 1 2πt 2 √ t -2 √ t Q(x) 4t -x 2 dx.
In the same way as calculus provides a nice setting for studying limits of sums and as classical Brownian motion provides a nice setting for studying limits of random walks, free probability provides a convenient framework for investigating limits of random matrices. In the setting of free probability, free independence (see Section 2.5) replaces independence, increments have a semicircular marginal law (see Section 2.6) and thus, free Brownian motion (see Section 2.7) which has these properties, can be used to study random matrices M n (t) for large n. Conversely, one may visualize free Brownian motion as a large random matrix M n (t).

Non-commutative probability space

In this paper, we use the phrase "non-commutative probability space" to indicate a von Neumann algebra A (that is, an algebra of operators on a complex separable Hilbert space, closed under adjoint and convergence in the weak operator topology) equipped with a trace ϕ, that is, a unital linear functional (meaning preserving the identity) which is weakly continuous, positive (meaning ϕ(X) ≥ 0 whenever X is a non-negative element of A ; i.e. whenever X = Y Y * for some Y ∈ A ), faithful (meaning that if ϕ(Y Y * ) = 0 then Y = 0), and tracial (meaning that ϕ(XY ) = ϕ(Y X) for all X, Y ∈ A , even though in general XY = Y X). We will not need to use the full force of this definition, only some of its consequences. See [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF] for a systematic presentation.

Random variables

In a non-commutative probability space, we refer to the self-adjoint elements of the algebra as random variables. Any random variable X has a law: this is the unique probability measure µ on R with the same moments as X; in other words, µ is such that

R Q(x)dµ(x) = ϕ(Q(X)), (2.13)
for any polynomial Q. (The existence and uniqueness of µ follow from the positivity of ϕ, see [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Proposition 3.13].) Thus ϕ acts as an expectation. The τ n in (2.12), for example, play the role of ϕ. Also, while there is a classical random variable X ∈ R with law µ, there is, a priori, no direct relationship between X and X.

Convergence in law

We say that a sequence (X 1,n , . . . , X k,n ), n 1, of random vectors converges in law to a random vector (X 1,∞ , . . . , X k,∞ ), and we write

(X 1,n , . . . , X k,n ) law → (X 1,∞ , . . . , X k,∞ ),
to indicate the convergence in the sense of (joint) moments, that is,

lim n→∞ ϕ (Q(X 1,n , . . . , X k,n )) = ϕ (Q(X 1,∞ , . . . , X k,∞ )) , (2.14)
for any polynomial Q in k non-commuting variables. In the case of vectors, there may be no corresponding probability law µ (X 1 ,...,Xn) as in (2.13), see [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Lecture 4]. We say that a sequence (F n ) of non-commutative stochastic processes (that is, each F n is a oneparameter family of self-adjoint operators F n (t) in the non-commutative probability space (A , ϕ)) converges in the sense of finite-dimensional distributions to a non-commutative stochastic process F ∞ , and we write

F n f.d.d. → F ∞ ,
to indicate that, for any k 1 and any t 1 , . . . , t k 0,

(F n (t 1 ), . . . , F n (t k )) law → (F ∞ (t 1 ), . . . , F ∞ (t k )).

Free independence

In the non-commutative probability setting, the notion of independence (introduced by Voiculescu in [START_REF] Voiculescu | Symmetries of some reduced free product C * -algebras. Operator algebras and their connection with topology and ergodic theory[END_REF]) goes as follows. Let A 1 , . . . , A p be unital subalgebras of A . Let X 1 , . . . , X m be elements chosen from among the A i 's such that, for 1 ≤ j < m, two consecutive elements X j and X j+1 do not come from the same A i , and such that ϕ(X j ) = 0 for each j. The subalgebras A 1 , . . . , A p are said to be free or freely independent if, in this circumstance,

ϕ(X 1 X 2 • • • X m ) = 0. (2.15)
Random variables are called freely independent if the unital algebras they generate are freely independent. Freeness is in general much more complicated than classical independence. Nevertheless, if X, Y are freely independent, then their joint moments are determined by the moments of X and Y separately as in the classical case. For example, if X, Y are free and m, n 1, then by (2.15),

ϕ (X m -ϕ(X m )1)(Y n -ϕ(Y n )1) = 0.
By expanding (and using the linear property of ϕ), we get

ϕ(X m Y n ) = ϕ(X m )ϕ(Y n ), (2.16)
which is what we would expect under classical independence. But, by setting

X 1 = X 3 = X -ϕ(X)1 and X 2 = X 4 = Y -ϕ(Y ) in (2.
15), we note that two consecutive X j do not belong to the same subalgebra and hence, by (2.15), we also have

ϕ (X -ϕ(X)1)(Y -ϕ(Y )1)(X -ϕ(X)1)(Y -ϕ(Y )1) = 0.
By expanding, using (2.16) and the tracial property of ϕ (for instance ϕ(XY X) = ϕ(X 2 Y )) we get

ϕ(XY XY ) = ϕ(Y ) 2 ϕ(X 2 ) + ϕ(X) 2 ϕ(Y 2 ) -ϕ(X) 2 ϕ(Y ) 2 , which is different from ϕ(X 2 )ϕ(Y 2 )
, which is what one would have obtained if X and Y were classical independent random variables. Let us note, furthermore, that the relation between moments and cumulants 1 is different from the classical case (see [11, identity (11.8)]).

Semicircular distribution

The semicircular distribution S(m, σ 2 ) with mean m ∈ R and variance

σ 2 > 0 is the probability distribution S(m, σ 2 )(dx) = 1 2πσ 2 4σ 2 -(x -m) 2 1 {|x-m|≤2σ} dx.
(2.18)

If m = 0, this distribution is symmetric around 0, and therefore its odd moments are all 0. A simple calculation shows that the even centered moments are given by (scaled) Catalan numbers:

for non-negative integers k, m+2σ m-2σ (x -m) 2k S(m, σ 2 )(dx) = C k σ 2k ,
where [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Lecture 2]). In particular, the variance is σ 2 while the centered fourth moment is 2σ 4 . The semicircular distribution plays here the role of the Gaussian distribution. It has the following similar properties:

C k = 1 k + 1 2k k (see, e.g.,
1. If S ∼ S(m, σ 2 ) and a, b ∈ R, then aS + b ∼ S(am + b, a 2 σ 2 ).
1 Cumulants have the following property which linearizes independence:

κn(X + Y, . . . , X + Y ) = κn(X, . . . , X) + κn(Y, . . . , Y ), n 1.
(2.17) Relation (2.17) holds in classical probability if X and Y are independent random variables and it holds in free probability if X and Y are freely independent (see [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Proposition 12.3]). Since the classical notion of independence is different from the notion of free independence, the cumulants κn in classical probability are different from those in free probability.

If

S 1 ∼ S(m 1 , σ 2 1 ) and S 2 ∼ S(m 2 , σ 2 2 ) are freely independent, then S 1 +S 2 ∼ S(m 1 +m 2 , σ 2 1 + σ 2 2 ).
The second property can be readily verified using the R-transform. The R-transform of a random variable X is the generating function of its free cumulants. It is such that, if X and Y are freely independent, then [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Formula (11.13)]).

R X+Y (z) = R X (z) + R Y (z). The R-transform of the semicircular law is R S(m,σ 2 ) (z) = m + σ 2 z, z ∈ C (see

Free Brownian Motion

A one-sided free Brownian motion S = {S(t)} t 0 is a non-commutative stochastic process with the following defining characteristics:

(1) S(0) = 0.

(2) For t 2 > t 1 0, the law of S(t 2 )-S(t 1 ) is the semicircular distribution of mean 0 and variance t 2t 1 .

(3) For all n and t n > • • • > t 2 > t 1 > 0, the increments S(t 1 ), S(t 2 ) -S(t 1 ), . . . , S(t n ) -S(t n-1 ) are freely independent.

A two-sided free Brownian motion S = {S(t)} t∈R is defined to be

S(t) = S 1 (t) if t 0 S 2 (-t) if t < 0 ,
where S 1 and S 2 are two freely independent one-sided free Brownian motions.

Wigner integral

From now on, we suppose that L 2 (R p ) stands for the set of all real-valued square-integrable functions on R p . When p = 1, we only write L 2 (R) to simplify the notation. Let S = {S(t)} t∈R be a two-sided free Brownian motion. Let us quickly sketch out the construction of the Wigner integral of f with respect to S. For an indicator function

f = 1 [u,v] , the Wigner integral of f is defined by R 1 [u,v] (x)dS(x) = S(v) -S(u).
We then extend this definition by linearity to simple functions of the form f =

k i=1 α i 1 [u i ,v i ] , where [u i , v i ] are disjoint intervals of R. Simple computations show that ϕ R f (x)dS(x) = 0 (2.19) ϕ R f (x)dS(x) × R g(x)dS(x) = f, g L 2 (R) . (2.20)
By approximation, the definition of R f (x)dS(x) is extended to all f ∈ L 2 (R), and (2. [START_REF] Voiculescu | Symmetries of some reduced free product C * -algebras. Operator algebras and their connection with topology and ergodic theory[END_REF])-(2.20) continue to hold in this more general setting. As anticipated, the Wigner process

Semicircular sequence and semicircular process

Let k 2. A random vector (X 1 , . . . , X k ) is said to have a k-dimensional semicircular distribution if, for every λ 1 , . . . , λ k ∈ R, the random variable λ 1 X 1 + . . . + λ k X k has a semicircular distribution.

In this case, one says that the random variables X 1 , . . . , X k are jointly semicircular or, alternatively, that (X 1 , . . . , X k ) is a semicircular vector. As an example, one may visualize the components of the random vector (X 1 , . . . , X k ) as, approximatively, normalized random matrices (M n (t 1 ), . . . , M n (t k )) in (2.11) with large n.

Let I be an arbitrary set. A semicircular family indexed by I is a collection of random variables {X i : i ∈ I} such that, for every k 1 and every (i 1 , . . . , i k ) ∈ I k , the vector (X i 1 , . . . , X i k ) has a k-dimensional semicircular distribution. When X = {X i : i ∈ I} is a semicircular family for which I is denumerable (resp. for which I = R + ), we say that X is a semicircular sequence (resp. semicircular process).

The distribution of any centered semicircular family {X i : i ∈ I} turns out to be uniquely determined by its covariance function Γ : I 2 → R given by Γ(i, j) = ϕ(X i X j ). (This is an easy consequence of [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Corollary 9.20].) When I = Z, the family is said to be stationary if Γ(i, j) = Γ(|i -j|) for all i, j ∈ Z.

Let X = {X k : k ∈ Z} be a centered semicircular sequence and consider the linear span H of X, called the semicircular space associated to X. It is a real separable Hilbert space and, consequently, there exists an isometry Φ :

H → L 2 (R). For any k ∈ Z, set e k = Φ(X k ); we have, for all k, l ∈ Z, R e k (x)e l (x)dx = ϕ(X k X l ) = Γ(k, l).
Thus, since the covariance function Γ of X characterizes its distribution, we have

{X k : k ∈ Z} law = R e k (x)dS(x) : k ∈ Z ,
with the notation of Section 2.8.

Multiple Wigner integral

Let S = {S(t)} t∈R be a two-sided free Brownian motion, and let p 1 be an integer. When f belongs to L 2 (R p ) (recall from Section 2.8 that it means, in particular, that f is real-valued), we write f * to indicate the function of L 2 (R p ) given by f * (t 1 , . . . , t p ) = f (t p , . . . , t 1 ).

Following [START_REF] Biane | Stochastic analysis with respect to free Brownian motion and analysis on Wigner space[END_REF], let us quickly sketch out the construction of the multiple Wigner integral of f with respect to S. Let ∆ q ⊂ R q be the collection of all diagonals, i.e. ∆ q = {(t 1 , . . . , t q ) ∈ R q : t i = t j for some i = j}.

(2.21)

For a characteristic function f = 1 A , where A ⊂ R q has the form A = [u 1 , v 1 ] × . . . × [u q , v q ] with A ∩ ∆ q = ∅
, the qth multiple Wigner integral of f is defined by

I S q (f ) = (S(v 1 ) -S(u 1 )) . . . (S(v q ) -S(u q )).
We then extend this definition by linearity to simple functions of the form f = k i=1 α i 1 A i , where

A i = [u i 1 , v i 1 ] × . . . × [u i q , v i q ]
are disjoint q-dimensional rectangles as above which do not meet the diagonals. Simple computations show that

ϕ(I S q (f )) = 0 (2.22) ϕ(I S q (f )I S q (g)) = f, g * L 2 (R q ) . (2.23)
By approximation, the definition of I S q (f ) is extended to all f ∈ L 2 (R q ), and (2.22)-(2.23) continue to hold in this more general setting. If one wants I S q (f ) to be a random variable in the sense of Section 2.3, it is necessary that f be mirror symmetric, that is, f = f * , in order to ensure that I S q (f ) is self-adjoint, namely (I S q (f )) * = I S q (f ) (see [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF]). Observe that I S 1 (f ) = R f (x)dS(x) (see Section 2.8) when q = 1. We have moreover ϕ(I S p (f )I S q (g)) = 0 when p = q, f ∈ L 2 (R p ) and g ∈ L 2 (R q ). (2.24)

When r ∈ {1, . . . , p ∧ q}, f ∈ L 2 (R p ) and g ∈ L 2 (R q ), let us write f r ⌢ g to indicate the rth contraction of f and g, defined as being the element of L 2 (R p+q-2r ) given by

f r ⌢ g(t 1 , . . . , t p+q-2r ) (2.25) = R r f (t 1 , .
. . , t p-r , x 1 , . . . , x r )g(x r , . . . , x 1 , t p-r+1 , . . . , t p+q-2r )dx 1 . . . dx r .

By convention, set f 0 ⌢ g = f ⊗ g as being the tensor product of f and g. Since f and g are not necessarily symmetric functions, the position of the identified variables x 1 , . . . , x r in (2.25) is important, in contrast to what happens in classical probability (see [START_REF] Peccati | Wiener Chaos: Moments, Cumulants and Diagrams[END_REF]Section 6.2]). Observe moreover that f

r ⌢ g L 2 (R p+q-2r ) f L 2 (R p ) g L 2 (R q ) (2.26)
by Cauchy-Schwarz, and also that f p ⌢ g = f, g * L 2 (R p ) when p = q. We have the following product formula (see [2, Proposition 5.3.3]), valid for any f ∈ L 2 (R p ) and g ∈ L 2 (R q ):

I S p (f )I S q (g) = p∧q r=0 I S p+q-2r (f r ⌢ g).
(2.27)

We deduce (by a straightforward induction) that, for any e ∈ L 2 (R) and any q 1,

U q R e(x)dS x = I S q (e ⊗q ), (2.28)
where

U 0 (x) = 1, U 1 (x) = x, U 2 (x) = x 2 -1, U 3 (x) = x 3 -2x, .
. ., is the sequence of Tchebycheff polynomials of second kind (determined by the recursion xU k = U k+1 + U k-1 ), R e(x)dS(x) is understood as a Wigner integral (as defined in Section 2.8), and e ⊗q is the qth tensor product of e, i.e., the symmetric element of L 2 (R q ) given by e ⊗q (t 1 , . . . , t q ) = e(t 1 ) . . . e(t q ).

Non-commutative Tchebycheff processes and basic properties

We define first the non-commutative fractional Brownian motion, then the non-commutative Rosenblatt process, which is a multiple Wigner integral of order 2, and then we introduce the general non-commutative Tchebycheff processes involving multiple Wigner integrals of arbitrary order.

Non-commutative fractional Brownian motion

The classical fractional Brownian motion was introduced by Kolmogorov [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum[END_REF] and developed by Mandelbrot and Van Ness [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF]. It is readily checked that S 1/2 is nothing but a one-sided free Brownian motion. Immediate properties of S H , proved in Corollary 4.4, include the selfsimilarity property (that is, for all a > 0 the process {a -H S H (at) : t 0} is a ncfBm of parameter H) and the stationary property of the increments (that is, for all h > 0 the process {S H (t + h) -S H (h) : t 0} is a ncfBm of parameter H). Conversely, ncfBm of parameter H is the only standardized semicircular process to verify these two properties, since they determine the covariance (3.29).

It is interesting to notice that ncfBm may be easily represented as a Wigner integral as follows:

S H f.d.d. = 2H (H -1 2 )β(H -1 2 , 2 -2H) R (• -x) H-1 2 + -(-x) H-1 2 + dS(x).
Here, β stands for the usual Beta function. In the classical probability case, one has a similar representation with S replaced by a Brownian motion.

As an illustration, we will now show that normalized sums of semicircular sequences can converge to ncfBm. Let {X k : k ∈ Z} be a stationary semicircular sequence with ϕ(X k ) = 0 and ϕ(X 2 k ) = 1, and suppose that its correlation kernel ρ(k

-l) = ϕ(X k X l ) verifies n k,l=1 ρ(k -l) ∼ Kn 2H L(n) as n → ∞, (3.30)
with L : (0, ∞) → (0, ∞) slowly varying at infinity, 0 < H < 1 and K a positive constant. Consider the non-commutative stochastic process

Z n (t) = 1 n H L(n) [nt] k=1 X k , t 0.
For any t s 0, we have, as n → ∞,

ϕ [Z n (t)Z n (s)] = 1 2 ϕ Z n (t) 2 + 1 2 ϕ Z n (s) 2 - 1 2 ϕ (Z n (t) -Z n (s)) 2 = 1 2n 2H L(n) [nt] i,j=1 ρ(i -j) + 1 2n 2H L(n) [ns] i,j=1 ρ(i -j) - 1 2n 2H L(n) [nt] i,j=[ns]+1 ρ(i -j) = 1 2n 2H L(n) [nt] i,j=1 ρ(i -j) + 1 2n 2H L(n) [ns] i,j=1 ρ(i -j) - 1 2n 2H L(n) [nt]-[ns] i,j=1 ρ(i -j) → K 2 t 2H + s 2H -(t -s) 2H = K ϕ(S H (t)S H (s)).
Let p 1 as well as t 1 , . . . , t p 0. Since the X k 's are centered and jointly semicircular, the process Z n is centered and semicircular as well, and we have shown that, as n → ∞,

Z n f.d.d. → √ K S H .

Non-commutative Rosenblatt process

The Hermite process indexed by q 1 appeared as a limit in Theorem 1.1 in the classical probability setting. When q = 1, it is fractional Brownian motion. When q = 2, it is the Rosenblatt process, introduced in Taqqu [15] and which appears as a limit in many statistical tests. See Taqqu [START_REF] Taqqu | The Rosenblatt process[END_REF] for a recent overview. We introduce here the non-commutative Rosenblatt process.

Definition 3.2. Let H ∈ ( 1 2 , 1). The non-commutative Rosenblatt process of parameter H is the non-commutative stochastic process defined by the double Wigner integral

R H (t) = R H,2 (t) = I S 2 f H (t, •) , t 0, (3.31)
where

f H (t, x, y) = H(2H -1) β( H 2 , 1 -H) t 0 (s -x) H 2 -1 + (s -y) H 2 -1 + ds, (3.32)
with β the usual Beta function.

Using the relations

R (t -x) H 2 -1 + (s -x) H 2 -1 + dx = β( H 2 , 1 -H)|t -s| H-1 ,
as well as

H(2H -1) [0,T ] 2 |t -s| 2H-2 dsdt = T 2H , T > 0, (3.33) it is straightforward to check that ϕ(R H (t) 2 ) = R 2 f H (t, x, y) 2 dxdy = t 2H .
The Rosenblatt process at (fixed) time t is a double Wigner integral whose kernel f H (t, •) is symmetric, see (3.32). As such, it enjoys useful properties, that we derive now in full generality. Assume then that f ∈ L 2 (R 2 ) is a given symmetric kernel. One of the most effective ways of dealing with I S 2 (f ) is to associate to f the following Hilbert-Schmidt operator:

A f : L 2 (R) → L 2 (R); g → R f (•, y)g(y)dy. (3.34)
In other words, A f transforms an element g of L 2 (R) into the contraction f 1 ⌢ g ∈ L 2 (R). We write {λ f,j : j 1} and {e f,j : j 1}, respectively, to indicate the eigenvalues of A f and the corresponding eigenvectors (forming an orthonormal system in L 2 (R)).

Some useful relations between all these objects are explained in the next proposition. The proof, which is omitted here, relies on elementary functional analysis (see e.g. Section 6.2 in [START_REF] Hirsch | Elements of Functional Analysis[END_REF]). Proposition 3.3. Let f be a symmetric element of L 2 (R 2 ), and let the above notation prevail.

1. The series ∞ j=1 λ p f,j converges for every p 2, and f admits the expansion

f = ∞ j=1 λ f,j e f,j ⊗ e f,j , (3.35)
where the convergence takes place in L 2 (R 2 ).

For every p 2, one has the relations

Tr(A p f ) = R p f (x 1 , x 2 ) . . . f (x p-1 , x p )f (x p , x 1 )dx 1 . . . dx p = ∞ j=1 λ p f,j , (3.36)
where Tr(A p f ) stands for the trace of the pth power of A f .

In the following statement we collect some facts concerning the law of a random variable of the type I S 2 (f ). Proposition 3.4. Let F = I S 2 (f ), where f is a symmetric element of L 2 (R 2 ). 1. The following equality holds:

F Law = ∞ j=1 λ f,j S 2 j -1 , (3.37)
where (S j ) j 1 is a sequence of freely independent S(0, 1) random variables, and the series converges in L 2 (A , ϕ).

2. For every p 2, the pth free cumulant of F is given by the following formula:

κ p (F, . . . , F ) = R p f (x 1 , x 2 ) . . . f (x p-1 , x p )f (x p , x 1 )dx 1 . . . dx p . (3.38)
Proof. Relation (3.37) is an immediate consequence of (3.35), of the identity

I S
2 (e f,j ⊗ e f,j ) = I S 1 (e f,j ) 2 -1, as well as of the fact that the {e f,j } are orthonormal (implying that the sequence {I S 1 (e f,j ) : j 1} is composed of freely independent S(0, 1) random variables). To prove (3.38), it suffices to use the linearization property (2.17) of free cumulants, as well as the fact 2 that κ p (S 2 -1, . . . , S 2 -1) = κ p (S 2 , . . . , S 2 ) + κ p (-1, . . . , -1) = κ p (S 2 , . . . , S 2 ) = 1 for all p 2, see e.g. [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]Proposition 12.13] for the last equality. We thus obtain the desired conclusion by means of (3.36).

In the classical probability setting where S j is N (0, 1), there is an additional factor of 2 p-1 (p-1)! in (3.38). See e.g. Taqqu [START_REF] Taqqu | Weak Convergence to Fractional Brownian Motion and to the Rosenblatt Process[END_REF].

Non-commutative Tchebycheff processes

The classical probability versions of these processes are the Hermite processes. See e.g. Peccati-Taqqu [14, Section 9.5]. Definition 3.5. Let H ∈ ( 1 2 , 1). The qth non-commutative Tchebycheff process of parameter H is the non-commutative stochastic process defined by the Wigner integral R H,q (t) = I S q f H,q (t, •) , t 0, (3.39) where

f H,q (t, x 1 , . . . , x q ) = H(2H -1) β( 1 2 -1-H q , 2-2H q ) q/2 t 0 (s -x 1 ) -( 1 2 + 1-H q ) + . . . (s -x q ) -( 1 2 + 1-H q ) + ds, (3.40)
with β the usual Beta function.

2 Since all the free cumulants of S 2 are equal to 1, S 2 has a free Poisson law with mean 1.

The process R H,q becomes the non-commutative fractional Brownian motion and the noncommutative Rosenblatt process when q = 1 and q = 2 respectively. Note however that when q = 1 the process is defined for H between 0 and 1.

Using the relations

R (t -x) γ-1 + (s -x) γ-1 + dx = β(γ, 1 -2γ)|t -s| 2γ-1 , 0 < γ < 1/2, (3.41)
and (3.33), we easily get that

ϕ(R H,q (t) 2 ) = R q f H,q (t, x 1 , . . . , x q ) 2 dx 1 . . . dx q = t 2H . (3.42)
The process R H,q has stationary increments and is selfsimilar with parameter H, as stated in Corollary 4.4 below.

Computation of joint moments

Let p 2 be a given integer. Let f 1 , . . . , f p be real functions of q 1 , . . . , q p variables respectively. Write q p = (q 1 , . . . , q p ). We want to compute

(. . . ((f 1 r 1 ⌢ f 2 ) r 2 ⌢ f 3 ) . . .) r p-1 ⌢ f p (4.43)
for some functions f 1 , . . . , f p of interest. The contraction operator r ⌢ is defined in (2.25). The expression (4.43) makes sense if and only if r = (r 1 , . . . , r p-1 ) ∈ A(q p ), where A(q p ) is the set of those (r 1 , . . . , r p-1 ) ∈ {0, . . . , q 2 } × . . . × {0, . . . , q p } such that r 1 q 1 , r 2 q 1 + q 2 -2r 1 , r 3 q 1 + q 2 + q 3 -2r 1 -2r 2 , . . . , r p-1 q 1 + . . . + q p-1 -2r 1 -. . . -2r p-2 , (4.44) and (4.43) equals a real number if and only if r ∈ B(q p ), where B(q p ) = r = (r 1 , . . . , r p-1 ) ∈ A(q p ) : 2r 1 + . . . + 2r p-1 = q 1 + . . . + q p . (4.45) Indeed, for f 1 r 1 ⌢ f 2 to make sense, we need 0 r 1 q 1 ∧ q 2 . Then, for (f 1

r 1 ⌢ f 2 ) r 2
⌢ f 3 to make sense, we need 0 r 2 q 3 ∧ (q 1 + q 2 -2r 1 ); this is because f 1 r 1

⌢ f 2 has q 1 + q 2 -2r 1 variables and f 3 has q 3 variables. It follows, by induction, that for (4.43) to make sense, it is necessary that (r 1 , . . . , r p-1 ) ∈ A(q p ). In order for (4.43) to be a scalar, we need (r 1 , . . . , r p-1 ) ∈ B(q p ), since the number of variables of (4.43) is given by q 1 + . . .

+ q p -2r 1 -. . . -2r p-1 .
The following lemma gives the value of (4.43) for functions f 1 , . . . , f p of interest. The result involves an array of non-negative integers α ij (r), 1 i < j p, (4.46) which are defined as follows for r ∈ B(q p ). Consider the following figure, where there are q 1 +. . .+q p dots. The first q 1 corresponds to the q 1 variables of f 1 , . . ., the last q p dots corresponds to the q p variables of f p .

We shall associate pairs of dots according to the mechanism described below, and we say that a dot is available if it has not been associated so far. The association rule, call it (A), involves block 1 block 2 block 3 block 4

Figure 1: p = 4, q 1 = 3, q 2 = 2, q 3 = 4 and q 4 = 3 block 1 block 2 block 3 block 4

Figure 2:

r 1 = 1
associating the left most available dot in a block with the right most available dot in preceding blocks.

To perform these associations, proceed as follows. Start with block j = 2 and do r 1 associations with block 1, following the association rule (A).

Proceed to block j = 3, 4, . . . , p. In block j 3, associate r j-1 dots with available dots in the preceding blocks following the association rule (A). Once block j = p is done, all dots have been associated pairwise.

Figure 3 below illustrates an example of associations.

block 1 block 2 block 3 block 4

Figure 3: p = 4, q 1 = 3, q 2 = 2, q 3 = 4, q 4 = 3, r 1 = 1, r 2 = 2 and r 3 = 3 Definition 4.1. When r = (r 1 , . . . , r p-1 ) ∈ B(q p ), denote by α ij (r) the number of associations between dots of j with dots of block i, 1 i < j p.

For instance, in Figure 3, we have

  α 12 (r) α 13 (r) α 14 (r) α 23 (r) α 24 (r) α 34 (r)   =   1 1 1 1 0 2   .
The following lemma gives an explicit expression for (4.43) for functions f 1 , . . . , f q which appear in the sequel. Lemma 4.2. Let T ⊂ R, and let e : T × R → R be a measurable function. Fix also integers p 2 and q 1 , . . . , q p 1, and let ν 1 , . . . , ν p be given signed measures on R. Assume further that

R q T |e(s, x 1 )| . . . |e(s, x q i )| |ν i |(ds) 2 dx 1 . . . dx q i < ∞, i = 1, . . . , p.
Finally, for any i = 1, . . . , p, define f i : R q i → R to be f i (x 1 , . . . , x q i ) = T e(s, x 1 ) . . . e(s, x q i )ν i (ds).

Then, for any r = (r 1 , . . . , r p-1 ) ∈ B(q p ) (recall the definition (4.45) of B(q p )), we have

(. . . ((f 1 r 1 ⌢ f 2 ) r 2 ⌢ f 3 ) . . .) r p-1 ⌢ f p = T p 1 i<j p R e(s i , x)e(s j , x)dx α ij (r)
ν 1 (ds 1 ) . . . ν p (ds p ).

Proof. Fix r = (r 1 , . . . , r p-1 ) ∈ B(q p ). We have

(. . . ((f 1 r 1 ⌢ f 2 ) r 2 ⌢ f 3 ) . . .) r p-1 ⌢ f p = T p (. . . ((e(s 1 , •) ⊗q 1 r 1 ⌢ e(s 2 •) ⊗q 2 ) r 2 ⌢ e(s 3 , •) ⊗q 3 ) . . .) r p-1 ⌢ e(s p , •) ⊗qp ν 1 (ds 1 ) . . . ν p (ds p ),
where we expressed

f i as f i (•) = T e(s, •) ⊗q i ν i (ds). What matters in the computation of (. . . ((e(s 1 , •) ⊗q 1 r 1 ⌢ e(s 2 , •) ⊗q 2 ) r 2 ⌢ e(s 3 , •) ⊗q 3 ) . . .) r p-1
⌢ e(s p , •) ⊗qp is, for each i < j, the number α ij (r) of associations between the block j and the block i. Hence

(. . . ((e(s 1 , •) ⊗q 1 r 1 ⌢ e(s 2 , •) ⊗q 2 ) r 2 ⌢ e(s 3 , •) ⊗q 3 ) . . .) r p-1 ⌢ e(s p , •) ⊗qp = 1 i<j p R e(s i , x)e(s j , x)dx α ij (r)
, and the desired formula follows.

We shall apply Lemma 4.2 with a discrete ν = δ k where δ k is Dirac mass, and with a Lebesgue-type ν. The following result plays an important role in the proof of the non-central limit theorem (1.10). Proposition 4.3. Fix two integers q 1 and p 2, and let t 1 , . . . , t p be positive real numbers. Set q p = (q 1 , . . . , q p ) = (q, . . . , q). Recall the definition (4.46) of α ij (r) when r = (r 1 , . . . , r p-1 ) ∈ B(q p ). Then, the following two assertions hold:

1. Let X = {X k : k ∈ Z} be a stationary semicircular sequence with ϕ(X k ) = 0 and ϕ(X 2 k ) = 1, and let ρ(kl) = ϕ(X k X l ) be its correlation kernel. Write U q to indicate the qth Tchebycheff polynomial, and define

G n (t) = [nt] k=1 U q (X k ), t 0. (4.47) Then ϕ(G n (t 1 ) . . . G n (t p )) = [nt 1 ] k 1 =1 . . . [ntp] kp=1 (r 1 ,...,r p-1 )∈B(qp) 1 i<j p ρ(k i -k j ) α ij (r) . (4.48)
2. Let R H,q be a qth non-commutative Tchebycheff process of parameter H ∈ (1/2, 1). Then

ϕ(R H,q (t 1 ) . . . R H,q (t p )) (4.49) = H p/2 (2H -1) p/2 t 1 0 ds 1 . . . tp 0 ds p (r 1 ,...,r p-1 )∈B(qp) 1 i<j p |s i -s j | -α ij (r)× 2-2H q .
Proof. 1. Let {e k : k ∈ Z} ⊂ L 2 (R) be defined as in Section 2.9. In the definition (4.47) of G n , we can assume without loss of generality that

X k = I S 1 (e k ) = R e k (x)dS(x), k ∈ Z.
By virtue of (2.28), we have G n (t) = I S q (g n (t, •)), with

g n (t, x 1 , . . . , x q ) = [nt] k=1 e k (x 1 ) . . . e k (x q ).
By the product formula (2.27),

G n (t 1 )G n (t 2 ) = I S q g n (t 1 , •) I S q g n (t 2 , •) = q r 1 =0 I S 2q-2r 1 g n (t 1 , •) r 1 ⌢ g n (t 2 , •) .
Iterative applications of the product formula (2.27) lead to

G n (t 1 ) . . . G n (t p ) (4.50) = (r 1 ,...,r p-1 )∈A(qp) I S pq-2r 1 -...-2r p-1 (. . . ((g n (t 1 , •) r 1 ⌢ g n (t 2 , •)) r 2 ⌢ g n (t 3 , •)) . . .) r p-1 ⌢ g n (t p , •) ,
where A(q p ) is defined by (4.44). By applying ϕ to (4.50) and using (2.22), we deduce that the only non-zero terms occur when pq -2r 1 -. . .

-2r p-1 = 0. Since I 0 (c) = c, c ∈ R, we get ϕ(G n (t 1 ) . . . G n (t p )) (4.51) = (r 1 ,...,r p-1 )∈B(qp) (. . . ((g n (t 1 , •) r 1 ⌢ g n (t 2 , •)) r 2 ⌢ g n (t 3 , •)) . . .) r p-1 ⌢ g n (t p , •),
with B(q p ) defined by (4.45). Now, recall from Section 2.9 the following property of e k : R e k (x)e l (x)dx = ϕ(X k X l ) = ρ(kl). (4.52)

Hence, using Lemma 4.2 with T = Z, ν i =

[nt i ]
k=1 δ k (δ k being the Dirac mass at k) and e(k, x) = e k (x), we obtain the formula (4.48) for ϕ(G n (t 1 ) . . . G n (t p )).

2. Reasoning as in Point 1 above, we get here that ϕ(R H,q (t 1 ) . . . R H,q (t p )) (4.53) = (r 1 ,...,r p-1 )∈B(qp)

(. . . ((f H,q (t 1 , •) r 1 ⌢ f H,q (t 2 , •)) r 2 ⌢ f H,q (t 3 , •)) . . .) r p-1 ⌢ f H,q (t p , •),
with B(q p ) defined by (4.45). Lemma 4.2, with T = R + , measures

ν i (ds) = H(2H -1) β( 1 2 -1-H q , 2-2H q ) q/2 1 [0,t i ] (s)ds
and the function e(s, x) = (sx)

-( 1 2 + 1-H q ) +
, then yields, because of (3.41), (. . . ((f H,q (t 1 , •)

r 1 ⌢ f H,q (t 2 , •)) r 2 ⌢ f H,q (t 3 , •)) . . .) r p-1 ⌢ f H,q (t p , •) (4.54) = H p/2 (2H -1) p/2 t 1 0 ds 1 . . . tp 0 ds p 1 i<j p |s i -s j | -α ij (r)× 2-2H q .
By inserting (4.54) in (4.53), we obtain the formula (4.49) for ϕ(R H,q (t 1 ) . . . R H,q (t p )).

The celebrated Breuer-Major theorem (see [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF], see also [START_REF] Nourdin | Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality[END_REF]Chapter 7] for a modern proof, and see Theorem 1.1, part 1, for the statement) asserts that

W n (H q , •) √ n , . . . , W n (H N , •) √ n (5.55)
converges as n → ∞ in the sense of finite-dimensional distributions to

σ q q! B q , . . . , σ N √ N ! B N ,
where σ 2 s := k∈Z ρ(k) s (s = q, . . . , N ), and B q , . . . , B N are independent standard Brownian motions. (The fact that k∈Z ρ(k) s 0 is part of the conclusion.) On the other hand, using (2.28) as well as its Gaussian counterpart (where one replaces, in (2.28), the Tchebycheff polynomial U q and the free Brownian motion S by the Hermite polynomial H q and the standard Brownian motion B respectively; see, e.g., [START_REF] Nourdin | Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality[END_REF]Theorem 2.7.7]), we get, for any s = q, . . . , N , that

V n (U s , t) = I S s   [nt] k=1 e ⊗s k   and W n (H s , t) = I B s   [nt] k=1 e ⊗s k   ,
where the sequence {e k : k ∈ Z} is as in Section 2.9 and I B s (•) stands for the multiple Wiener integral of order s with respect to B. We observe that the kernel

[nt]
k=1 e ⊗s k is a symmetric function of L 2 (R s ). Therefore, according to Proposition 5.1 (transfer principle), we deduce that the noncommutative counterpart of (5.55) holds as well, that is, we have that

V n (U q , •) √ n , . . . , V n (U N , •) √ n ,
converges as n → ∞ in the sense of finite-dimensional distributions to (σ q S q , . . . , σ N S N ) ,

where S q , . . . , S N denote freely independent free Brownian motions. The desired conclusion (1.9) follows then as a consequence of this latter convergence, together with the decomposition (1.7) of Q and the identity in law (see Section 2.6):

a q σ q S q + . . . + a N σ N S N law = a 2 q σ 2 q + . . . + a 2 N σ 2 N × S.
5.2 Proof of the non-central limit theorem (1.10)

The proof is more delicate than the one for (1.9). This is because the limit in the usual probability setting is not Gaussian, since it is given by a multiple Wiener integral of order greater than 1. Therefore, we cannot use the transfer principle as in (1.9). We need to focus on the detailed structure of

V n (Q, t) = a q [nt] k=1 U q (X k ) + N s=q+1 a s [nt] k=1 U s (X k ).
(5.56)

where we have again chosen N large enough so that a s = 0 for all s N . The idea of the proof is to show that, after normalization, the second term in (5.56) is asymptotically negligible, so that it is sufficient to focus on the first term

G n (t) = [nt] k=1 U q (X k ).
We can therefore apply Proposition 4.3 which provides, in (4.48), an expression for ϕ(G n (t 1 ) . . . G n (t p )) involving multiple sums. We show that the diagonals in these multiple sums can be excluded (this is step 1 below). We express the remainder as integrals (step 2 below) and apply the dominated convergence theorem to obtain the expression (4.49) which caracterizes the qth non-commutative Tchebycheff process R H,q . More specifically, fix ε ∈ (0, 1]. By virtue of (1.4), there exists an integer M > 0 large enough so that, for all j > M , |ρ(j)| = j -D L(j) ε 1.

For any real t > 0 and any integer s larger than or equal to q + 1, we can write

ϕ     1 n 1-qD/2 L(n) q/2 [nt] k=1 U s (X k )   2   = 1 n 2-qD L(n) q [nt] k,l=1 ρ(k -l) s 1 n 2-qD L(n) q     [nt] k,l=1 |k-l| M 1 + 2ε [nt] k,l=1 k>l+M |ρ(k -l)| q     t n 1-qD L(n) q   (2M + 1) + 2ε [nt] j=1 j -qD L(j) q   .
(5.57)

Since qD < 1, we have that n qD-1 L(n) -q → 0 (to see this, use (1.6) with 1/L instead of L, 1/L being slowly varying as well). The following lemma is useful at this stage.

Lemma 5.2. When t > 0 is fixed, we have

[nt] j=1 j -qD L(j) q ∼ [nt] 1-qD L([nt]) q 1 -qD as n → ∞. ( 5

.58)

Proof. Although this is a somehow standard result in the theory of regular variation (Karamata's type theorem), we prove (5.58) for sake of completeness. First, observe that

1 L([nt]) q [nt] 1-qD [nt] j=1 j -qD L(j) q = 1 0 l n (x)dx, where l n (x) = [nt] j=1 L(j) L([nt]) q j [nt] -qD 1 [ j-1 [nt] j [nt] ) (x). Since L(j)/L([nt]) = L([nt]×(j/[nt]))/L([nt]) → 1 for fixed j/[nt] as n → ∞, one has l n (x) → l ∞ (x) for x ∈ (0, 1)
, where l ∞ (x) = x -qD . By choosing a small enough δ > 0 so that q(D + δ) < 1 (this is possible because qD < 1) and L(j)/L([nt]) C(j/[nt]) -δ (this is possible thanks to (1.6)), we get that

|l n (x)| C [nt] j=1 j [nt] -q(D+δ) 1 [ j-1 [nt] j [nt] ) (x) Cx -(D+δ)q for all x ∈ (0, 1).
The function in the bound is integrable on (0, 1). Hence, the dominated convergence theorem yields

1 L([nt]) q [nt] 1-qD [nt] j=1 j -qD L(j) q → 1 0 l ∞ (x)dx = 1 0 x -qD dx = 1 1 -qD ,
which is equivalent to (5.58).

Let us go back to the proof of the non-central limit theorem. We have (5.57). But, since L([nt])/L(n) → 1 (t is fixed), we actually get that

[nt] j=1 j -qD L(j) q ∼ n 1-qD t 1-qd L(n) q 1 -qD as n → ∞, (5.59)
so that, by combining (5.59) and (5.57), lim sup

n→∞ ϕ     1 n 1-qD/2 L(n) q/2 [nt] k=1 U s (X k )   2   t 2-qD ε 1 -qD .
Since ε > 0 is arbitrary, this implies that

1 n 1-qD/2 L(n) q/2 [nt] k=1 U s (X k ) L 2 → 0 as n → ∞
for all s q + 1. As a consequence, in the rest of the proof we can assume without loss of generality that Q = a q U q . Thus, set

F n (t) = a q n 1-qD/2 L(n) q/2 [nt] k=1 U q (X k ) = a q n 1-qD/2 L(n) q/2 G n (t), t 0,
where G n is given by (4.47). Using (4.48), we have that

ϕ(F n (t 1 ) . . . F n (t p )) = (a q ) p n p-pqD/2 L(n) pq/2 [nt 1 ] k 1 =1 . . . [ntp] kp=1 (r 1 ,...,r p-1 )∈B(qp) 1 i<j p ρ(k i -k j ) α ij (r) ,
(5.60) where q p = (q 1 , . . . , q p ) = (q, . . . , q).

To obtain the limit of (5.60) as n → ∞ and thus to conclude the proof of (1.10), we proceed in five steps.

Step 1 (Determination of the main term). We split the sum

[nt 1 ] k 1 =1 . . . [ntp]
kp=1 in the right-hand side of (5.60) into

k 1 =1,...,[nt 1 ] ... kp=1,...,[ntp] ∀i =j: |k i -k j | 3 + k 1 =1,...,[nt 1 ] ... kp=1,...,[ntp] ∃i =j: |k i -k j | 2 ,
(5.61) and we show that the second sum in (5.61) is asymptotically negligible as n → ∞.

Up to reordering, it is enough to show that, for any r = (r 1 , . . . , r p-1 ) ∈ B p ,

R n := 1 n p-pqD/2 L(n) pq/2 k 1 =1,...,[nt 1 ] ... kp=1,...,[ntp] |k p-1 -kp| 2 1 i<j p ρ(k i -k j ) α ij (r)
(5.62) tends to zero as n → ∞. In (5.62), let us bound |ρ(k ik p )| by 1 when i ∈ {1, . . . , p -1}. We get that

R n 5 n p-pqD/2 L(n) pq/2 [nt 1 ] k 1 =1 . . . [nt p-1 ] k p-1 =1 1 i<j p-1 ρ(k i -k j ) α ij (r) .
(5.63) Going back to Definition 4.1, there is q p-1 = ( q 1 , . . . , q p-1 ) and r = ( r 1 , . . . , r p-2 ) ∈ B( q p-1 ) such that α ij ( r) = α ij (r) for all i, j = 1, . . . , p -1. This is because the connexions between the remaining p -1 blocks are unchanged. Moreover, since we remove the q connexions associated to the block p (this involves 2q dots, see Figure 3), we have q 1 + . . . + q p-1 = (p -2)q.

(5.64) Hence, using Lemma 4.2 with T = Z, ν i =

[nt i ]
k=1 δ k (δ k being the Dirac mass at k) and e(k, x) = e k (x) as in (4.52), we get that

[nt 1 ] k 1 =1 . . . [nt p-1 ] k p-1 =1 1 i<j p-1 ρ(k i -k j ) α ij (r) = (. . . (( g 1,n (t 1 , •) r 1 ⌢ g 2,n (t 2 , •)) r 2 ⌢ . . .) r p-2 ⌢ g p-1,n (t p-1 , •) (5.65)
where, for any i = 1, . . . , p -1,

g i,n (t i , x 1 , . . . , x q i ) = [nt i ] k=1 e k (x 1 ) . . . e k (x q i ).
Iterative applications of (2.26) in (5.65) lead to

[nt 1 ] k 1 =1 . . . [nt p-1 ] k p-1 =1 1 i<j p-1 |ρ(k i -k j )| α ij (r)
g 1,n (t 1 , •) L 2 (R q 1 ) . . . g p-1,n (t p-1 , •) L 2 (R q p-2 ) .

But, for any i = 1, . . . , p -1,

g i,n (t i , •) 2 L 2 (R q i ) = [nt i ] k,l=1 ρ(k -l) q i = |j|<[nt i ] ρ(j) q i [nt i ] -|j| = [nt i ] + 2 1 j<[nt i ]
j -q i D L(j) q i [nt i ]j

[nt i ]   1 + 2 1 j<[nt i ]
j -q i D L(j) q i   Cn 2-q i D L(n) q i , for some C > 0, and where the last inequality holds because of (5.59). We deduce

[nt 1 ] k 1 =1
. . .

[nt p-1 ] k p-1 =1 1 i<j p-1 |ρ(k i -k j )| α ij (r)
Cn p-1-( q 1 +...+ q p-1 )D/2 L(n) ( q 1 +...+ q p-1 )/2

= Cn p-1-(p-2)qD/2 L(n) (p-2)q/2 , by (5.64). By putting all these bounds together, and because qD < 1,we conclude from (5.63) that R n given by (5.62) tends to zero as n → ∞.

Step 2 (Expressing sums as integrals). We now consider the first term in (5.61) and express it as an integral, so to apply the dominated convergence theorem.

From Definition 4.1, we deduce immediately that 1 i<j p α ij (r) = pq/2 when r = (r 1 , . . . , r p-1 ) ∈ B(q p ). This fact, combined with the specific form (1.4) of ρ, yields (a q ) p n p-pqD/2 L(n) pq

k i -k j n -Dα ij (r) L(|k i -k j |) L(n) α ij (r) ×1 [ k 1 -1 n , k 1
n ) (s 1 ) . . . 1 [ kp-1 n , kp n ) (s p ).
Step 3 (Pointwise convergence). We show the pointwise convergence of l n,r . Since, for fixed |k ik j |/n and as n → ∞, one has

L(|k i -k j |)/L(n) = L(n × |k i -k j |/n)/L(n) → 1,
one deduces that l n,r (s 1 , . . . , s p ) → l ∞,r (s 1 , . . . , s p ) for any s 1 , . . . , s p ∈ R + , where l ∞,r (s 1 , . . . , s p ) = 1 [0,t 1 ] (s 1 ) . . . 1 [0,tp] (s p ) 1 i<j p |s is j | -Dα ij (r) .

Step 4 (Domination). We show that l n,r is dominated by an integrable function. If k ik j 3 (the case where k jk i 3 is similar by symmetry), s i ∈ [ k i -1 n , k i n ) and s j ∈ [

k j -1 n , k j n ), then 3 n k i -k j n s i + 1 n -s j ,
so that s is j 2 n , implying in turn

k i -k j n s i -s j - 1 n s i -s j 2 .
(5.66)

Since qD < 1, choose a small enough δ so that q(D + δ) < 1 and L(k ik j )/L(n) C((k ik j )/n) -δ (this is possible thanks to (1.6)). We get that

1 [ k 1 -1 n , k 1
n ) (s 1 ) . . . 1 [ kp-1 n , kp n ) (s p )
C 2 (D+δ)pq/2 1 [0,t 1 ] (s 1 ) . . . 1 [0,tp] (s p ) r) , by (5.66). The function in the bound is integrable on R p + . Indeed, for any fixed r = (r 1 , . . . , r p-1 ) ∈ B(q p ), we can write, thanks to (4.54) and with f 1-q(D+δ)/2,q given by (3.40), = C (. . . ((f 1-q(D+δ)/2,q (t 1 , •) r 1 ⌢ f 1-q(D+δ)/2,q (t 2 , •)) r 2 ⌢ . . .) r p-1 ⌢ f 1-q(D+δ)/2,q (t p , •), where C > 0, so that by an iterative use of (2.26), f 1-q(D+δ)/2,q (t 1 , •) L 2 (R q ) . . . f 1-q(D+δ)/2,q (t p , •) L 2 (R q ) = (t 1 . . . t p ) 1-q(D+δ)/2 < ∞, where the last equality holds because of (3.42).

1 i<j p |s i -s j | -(D+δ)α ij (
Step 5 (Dominated convergence). By combining the results of Steps 2 to 4, we obtain that the dominated convergence theorem applies and yields ϕ(F n (t 1 ) . . . F n (t p )) → (a q ) p t 1 0 ds 1 . . . We recognize that, up to a multiplicative constant, this is the quantity in (4.49) with H = 1-qD/2. More precisely, we have ϕ(F n (t 1 ) . . . F n (t p )) → ϕ a q H(2H -1) R H,q (t 1 ) × . . . × a q H(2H -1) R H,q (t p ) , which concludes the proof of (1.10).

/ 2 k 1

 21 =1,...,[nt 1 ] ... kp=1,...,[ntp] ∀i =j:|k i -k j | 3 (r 1 ,...,r p-1 )∈B(qp) 1 i<j p ρ(k ik j ) α ij (r) = (a q ) p n p k 1 =1,...,[nt 1 ] ... kp=1,...,[ntp] ∀i =j: |k i -k j | 3 (r 1 ,...,r p-1 )∈B(qp) 1 i<j p k ik j n -Dα ij (r) L(|k ik j |) L(n) α ij (r) = (a q ) p (r 1 ,...,r p-1 )∈B(qp) r (s 1 , . . . , s p )ds 1 . . . ds p , where l n,r (s 1 , . . . , s p ) = k 1 =1,...,[nt 1 ] ... kp=1,...,[ntp] ∀i =j: |k i -k j | 3 1 i<j p

  |l n,r (s 1 , . . . , s p )| C k 1 =1,...,[nt 1 ] ... kp=1,...,[ntp] ∀i =j: |k i -k j | 3 1 i<j p k ik j n -(D+δ)α ij (r)

  |s is j | -(D+δ)α ij (r)

t 1 0

 1 ds 1 . . .

  |s is j | -(D+δ)α ij (r)

  r 1 ,...,r p-1 )∈B(qp) 1 i<j p|s is j | -Dα ij (r) .

  Definition 3.1. Let H ∈ (0, 1). A non-commutative fractional Brownian motion (ncfBm in short) of Hurst parameter H is a centered semicircular process S H = {S H (t) : t

					0} with covariance
	function	ϕ(S H (t)S H (s)) =	1 2	t 2H + s 2H -|t -s| 2H .	(3.29)

R f (x)dS(x) : f ∈ L 2 (R) forms a centered semicircular family in the sense of the forthcoming Section 2.9.

Acknowledgments. We would like to thank two anonymous referees for their careful reading of the manuscript and for their valuable suggestions and remarks. Also, I. Nourdin would like to warmly thank M. S. Taqqu for his hospitality during his stay at Boston University in October 2011, where part of this research was carried out.

* Ivan Nourdin was partially supported by the ANR Grants ANR-09-BLAN-0114 and ANR-10-BLAN-0121 at Université de Lorraine.

† Murad S. Taqqu was partially supported by the NSF Grant DMS-1007616 at Boston University.

1

Corollary 4.4. The non-commutative Tchebycheff process R H,q has stationary increments and is selfsimilar with parameter H.

Proof. Since the law is determined by the moments, it suffices to use expression (4.49). Let h > 0. Replacing R H,q (t i ) by R H,q (t i + h) -R H,q (h) in the left-hand side of (4.49) changes the integrals t i 0 in the right-hand side by integrals t i +h h . Since this does not modify the right-hand side, the process R H,q has stationary increments. To prove selfsimilarity, let a > 0, replace each t 1 , . . . , t p by at 1 , . . . , at p in (4.49) and note that the right-hand side is then multiplied by a factor a pH .

Proof of Theorem 1.2

In the proof of the central limit theorem (1.9), we shall use the following Wiener-Wigner transfer principle, established in [START_REF] Nourdin | Multidimensional semicircular limits on the free Wigner chaos[END_REF]Theorem 1.6]. It provides an equivalence between multidimensional limit theorems involving multiple Wiener integrals and multiple Wigner integrals respectively, whenever the limits of the multiple Wiener integrals are normal.

Proposition 5.1. (Statement of [13, Theorem 1.6]) Let d 1 and q 1 , . . . , q d be some fixed integers, and consider a positive definite symmetric matrix c = {c(i, j) : i, j = 1, ..., d}. Let (G 1 , . . . , G d ) be a d-dimensional Gaussian vector and (Σ 1 , . . . , Σ d ) be a semicircular vector, both with covariance c as defined in Section 2.9. For each i = 1, . . . , d, we consider a sequence {f i,n } n 1 of symmetric functions in L 2 (R q i + ). Let B be a classical Brownian motion and let I B q (•) stand for the qth multiple Wiener integral. Let S be a free Brownian motion and let I S q (•) stand for the qth multiple Wigner integral. Then:

1. For all i, j = 1, . . . , d and as

2. If the asymptotic relations in (1) are verified then, as n → ∞,

if and only if

5.1 Proof of the central limit theorem (1.9)

Recall that X = {X k : k ∈ Z} is a stationary semicircular sequence with ϕ(X k ) = 0, ϕ(X 2 k ) = 1 and correlation ρ. Consider first its Gaussian counterpart (in the usual probabilistic sense), namely Y = {Y k : k ∈ Z} where Y is a stationary Gaussian sequence with mean 0 and same correlation ρ. When dealing with Y , the important polynomials are not the Tchebycheff polynomials but the Hermite polynomials, defined as H 0 (y) = 1, H 1 (y) = y, H 2 (y) = y 2 -1, H 3 (y) = y 3 -3y, . . . , and determined by the recursion yH k = H k+1 + kH k-1 .

We assume in this proof that k∈Z |ρ(k)| q < ∞; this implies k∈Z |ρ(k)| s < ∞ for all s q. Since Q given by (1.7) is a polynomial, we can choose N large enough so that a s = 0 for all s N . Set

H s (Y k ), t 0, s = q, . . . , N.


Juraj Bolyó

Thomas Mair

Gabriela Kuncová

Marcus J B Hauser
email: marcus.hauser@ovgu.de

Spatio-temporal dynamics of glycolytic waves provide new insights into the interactions between immobilized yeast cells and gels

Keywords: reaction-diffusion-coupling, pattern formation, metabolic synchronization, open reactor, NAD(P)H-fluorescence

The immobilization of cells or enzymes is a promising tool for the development of biosensors, yet the interactions between the fixative materials and the cells are not fully understood, especially with respect to their impact on both the cell metabolism and cell-to-cell signaling.

We show that the spatiotemporal dynamics of waves of metabolic synchronization of yeast cells provide a new criterion to distinguish the effect of different gels on the cellular metabolism, which otherwise could not be detected. Cells from the yeast Saccharomyces carlsbergensis were immobilised into agarose gel, silica gel (TMOS), or a mixture of TMOS and alginate. We compared these immobilized cells with respect to their ability to generate temporal, intracellular oscillations in glycolysis as well as propagating, extracellular synchronization waves. While the temporal dynamics, as measured by the period and the number of oscillatory cycles, were similar for all three immobilized cell populations, significant differences have been observed with respect to the shape of the waves, wave propagation direction and velocity in the three gel matrices used.

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

Introduction

The immobilization of microorganisms has attracted increased scientific interest, since it provides a promising tool for biotechnological applications [START_REF] Avnir | Recent bio-applications of sol-gel materials[END_REF][START_REF] Martynenko | Immobilization of champagne yeasts by inclusion into cryogels of polyvinyl alcohol: Means of preventing cell release from the carrier matrix[END_REF][START_REF] Livage | Living cells in oxide glasses[END_REF][START_REF] Champagne | Cream fermentation by immobilized lactic acid bacteria[END_REF]. In addition, immobilized living cells may be used for the construction of biosensors, e.g. in the food industry or in the monitoring of environmental pollution. Several immobilization procedures have been reported [START_REF] Gupta | Entrapment of biomolecules in sol-gel matrix for applications in biosensors: Problems and future prospects[END_REF][START_REF] Pierre | The sol-gel encapsulation of enzymes[END_REF][START_REF] Preininger | Optical Fiber Sensor for Biological Oxygen Demand[END_REF][START_REF] Naessens | Fiber Optic Biosensor Using Chlorella vulgaris for Determination of Toxic Compounds[END_REF][START_REF] Trogl | Selectivity of whole cell optical biosensor with immobilized bioreporter Pseudomonas fluorescens HK44[END_REF][START_REF] Gavlasova | Whole cell biosensor for polychlorinated biphenyl analysis based on optical detection[END_REF][START_REF] Doran | Effects of Immobilization on the Nature of Glycolytic Oscillations in Yeast[END_REF][START_REF] Poulsen | Single cell studies and simulation of cellcell interactions using oscillating glycolysis in yeast cells[END_REF], each having its advantages and drawbacks. One problem concerns the dynamics and regulation of the metabolism, which seems to be affected in the immobilized microorganisms [START_REF] Doran | Effects of Immobilization on the Nature of Glycolytic Oscillations in Yeast[END_REF][START_REF] Poulsen | Single cell studies and simulation of cellcell interactions using oscillating glycolysis in yeast cells[END_REF][START_REF] Podrazky | Determination of living immobilized yeast cells by fluorescence spectroscopy[END_REF][START_REF] Callone | Immobilization of yeast and bacteria cells in alginate microbeads coated with silica membranes: Procedures, physico-chemical features and bioactivity[END_REF]. A fast and easy method to test for such perturbations is related to glycolytic oscillations. They are easy to detect by measuring the NAD(P)H fluorescence of the immobilized cells and information about the metabolic state of the cells can be derived from characteristic parameters as for example the period and amplitude of the oscillations [START_REF] Goldbeter | Biochemical oscillators and cellular Rhythms[END_REF][START_REF] Richard | The rhythm of yeast[END_REF].

Monitoring of fluorescence intensities of NAD(P)H during aerobic-anaerobic transition has also been used to determine the viability of yeast cells immobilized in transparent matrices.

Immobilized cells from the exponential and the stationary growth phases were monitored [START_REF] Podrazky | Determination of living immobilized yeast cells by fluorescence spectroscopy[END_REF].

Macroscopic glycolytic oscillations of yeast cell populations result from metabolic synchronization of the individual cells [START_REF] Richard | The rhythm of yeast[END_REF][START_REF] Ghosh | Metabolic Coupling and Synchronization of NADH Oscillations in Yeast Cell Populations[END_REF][START_REF] Richard | Acetaldehyde mediates the synchronization of sustained glycolytic oscillations in populations of yeast cells[END_REF]. These oscillations have a short period of about 0.5-1 min and occur when starved cells of Saccharomyces cerevisiae are supplemented with glucose pulses, either after switching to anaerobiosis [START_REF] Hess | Mechanism of Glycolytic Oscillation in Yeast, I. Aerobic and anaerobic growth conditions for obtaining glycolytic oscillation[END_REF] or after addition of cyanide, i.e. at energy limiting conditions. Acetaldehyde has been found to act as the signaling compound for this type of cell-cell synchronization. Sustained glycolytic oscillations occur only when the yeast cells are harvested during the diauxic shift [START_REF] Richard | Around the growth phase transition S. cerevisiae´s make-up favours sustained oscillations of intracellular metabolites[END_REF].

These experiments have been performed with stirred cell suspensions. However, oscillatory systems may generate spatio-temporal patterns when the oscillatory reaction is coupled to mass transport processes, such as diffusion [START_REF] Prigogine | Biological order, structure and instabilities[END_REF][START_REF] Mikhailov | Control of waves, patterns and turbulence in chemical systems[END_REF]. As this is the case in gel-fixed yeast cell layers, the glycolysis in entrapped yeast cells may give rise to a series of waves and patterns, that are well known from self-organized reaction-diffusion systems [START_REF] Mikhailov | Control of waves, patterns and turbulence in chemical systems[END_REF][START_REF] Kapral | Chemical waves and patterns[END_REF][START_REF] Epstein | An introduction to nonlinear chemical dynamics -Oscillations, waves, patterns, and chaos[END_REF][START_REF] Steinbock | Control of chemical waves in excitable media by external perturbation[END_REF]. Metabolic synchronization of gel-fixed cell populations may thus occur via traveling waves of the signaling compound, which cannot form in stirred cell suspensions. Results from chemical systems demonstrate that the propagation dynamics as well as the shape of such traveling reaction-diffusion waves contain information about the state of the system [START_REF] Steinbock | Chemical wave logic gates[END_REF][START_REF] Gorecki | Sensing the distance to a source of periodic oscillations in a nonlinear chemical medium with the output information coded in frequency of excitation pulses[END_REF]. Hence,

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

these waves may be of relevance for biological information processing [START_REF] Mair | Propagating waves of biological activity[END_REF][START_REF] Loose | Spatial regulators for bacterial cell division self-organize into surface waves in vitro[END_REF], and as such also for biotechnological applications.

The first experimental evidence for the generation of reaction-diffusion waves in glycolysis came from experiments with yeast extract [START_REF] Mair | Traveling NADH and proton waves during oscillatory glycolysis in vitro[END_REF][START_REF] Mair | Spatio-temporal dynamics in glycolysis[END_REF]. There, traveling NADH and proton waves could be observed during oscillatory glycolysis. In these experiments, the necessary coupling between reaction and diffusion was achieved by performing the glycolytic reaction in a gel matrix. A drawback of these investigations is that they have been conducted under batch conditions. Since the generation of oscillations and reaction-diffusion waves requires nonequilibrium conditions, such experiments are best run in open reactors [START_REF] Horvath | An experimental design method leading to chemical Turing patterns[END_REF].

Recently, Bagyan et al. monitored reaction-diffusion waves in the yeast extract immobilized in an agarose gel mounted into an open spatial reactor [START_REF] Bagyan | Glycolytic oscillations and waves in an open spatial reactor: Impact of feedback regulation of phosphofructokinase[END_REF]. They could show that in case of the yeast extract local desynchronization plays an important role for the wave dynamics [START_REF] Bagyan | Spatial desynchronization of glycolytic waves as revealed by Karhunen-Loève analysis[END_REF].

Lavrova et al have simulated glycolytic wave propagation in this open spatial reactor in order

to explain some of the phenomena observed at low protein concentrations [START_REF] Lavrova | Modeling of glycolytic wave propagation in an open spatial reactor with inhomogeneous influx[END_REF]. Due to the possibility of maintaining the reaction under stationary non-equilibrium conditions,, this reactor should also be useful for the measurement of oscillations and waves in yeast cell suspensions.

In the present work, we show first results about the application of this type of reactor for cell suspensions immobilized in gels with a focus of the impact of various gel matrices on the dynamics of glycolytic patterns. We aimed at investigating (i) the effect of the immobilizing gel medium on the generation of glycolytic oscillations in yeast cells and on (ii) the spatiotemporal patterns that may arise in immobilized cell layers, i.e. when there is a coupling between reaction and diffusion. Three different gel matrices were used: agarose, prepolymerized tetramethoxysilane (TMOS) and compositions of pre-polymerized tetramethoxysilane with alginate (alginate/TMOS).

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

Materials and methods

Growth of yeast

The yeast Saccharomyces carlsbergensis ATCC 9080 was cultivated aerobically in a rotary shaker at 180 rpm in liquid semi-synthetic minimal media [START_REF] Hess | Mechanism of glycolytic oscillation in yeast, 1. Aerobic and anaerobic growth conditions for obtaining glycolytic oscillations[END_REF] at 28 °C. After complete consumption of glucose the cells were harvested by centrifugation at 5000 g at 21°C for 5 min and were washed twice with distilled water. The cells were resuspended in potassium phosphate buffer (0.1 M KH 2 PO 4 , pH 6.5) yielding cell suspensions in concentrations of 10, 20 or 30% (wet weight per volume). Subsequently, these cell suspensions were starved at 21°C for at least 6 h until further use.

Gel preparation and immobilization procedure

After at least 6 h of starvation, the yeast cells were immobilized in one of the following three gel matrices that were prepared as follows:

1% agarose gel: Agarose (Type VII; Sigma-Aldrich) (0.03 g) in potassium phosphate buffer (1.5 mL of 0.1 M KH 2 PO 4 , pH 6.5) was heated for 5 min at 70°C. The solution was placed in a water bath at 28°C. After 2 min, the cell suspension (1.5 mL; incubation at 28°C) was carefully mixed with the agarose and subsequently this mixture was placed between two glass plates (20 cm x 15 cm) separated by 1.5 mm thick spacers. The gelation process took place during 15 min at ca. 0°C. A flat surface of the gel was obtained by placing a 5 kg weight on the upper glass plate during the gelation process. Thereafter a circular piece (diameter 24 mm) of the gel slide was cut out and placed on a black nitrocellulose membrane (Millipore) of 0.8 µm pore size [START_REF] Bagyan | Glycolytic oscillations and waves in an open spatial reactor: Impact of feedback regulation of phosphofructokinase[END_REF].

Silica gel: Tetramethoxysilane (TMOS; Fluka) was mixed with distilled water and HCl (0.1 M) in TMOS:H 2 O:HCl molar ratio = 1:5:10 -2 , to form a clear solution. This solution was left to pre-polymerize for 24 h at 4°C [START_REF] Podrazky | Determination of living immobilized yeast cells by fluorescence spectroscopy[END_REF][START_REF] Kuncova | Monitoring of the viability of cells immobilized by sol-gel process[END_REF]. The pre-polymerized TMOS was used for immobilization in the following way: The cell suspension (1 mL) was mixed with 0.05 M NaOH (0.5 mL) and with pre-polymerized TMOS (0.5 mL). This mixture was vortexed and poured on the black nitrocellulose membrane which was covered with an appropriate spacer ring (diameter 24 mm, teflon). After gelation (within ca. 1-2 min, at room temperature),

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

planar film disks (of 24 mm diameter and 15 mm thickness) of immobilized cells were very quickly cut out and the spacer ring was removed.

Alginate/TMOS matrix: At first a stock solution of sterile 3.2% (w/v) sodium alginate (Sigma-Aldrich, type IV, practical grade) was prepared as follows: the alginate and distilled water were sterilized separately (20 min, 120°C, 0.1 kPa). Thereafter, the sterilized alginate was dissolved in the sterilized distilled water under sterile conditions and stored at 4°C until further use. For preparation of the gel the cell suspension (1 mL) was mixed with 0.05 M NaOH (0.5 mL), dissolved alginate (0.167 mL) and pre-polymerized TMOS (0.333 mL).

Then this mixture was very quickly vortexed and poured on the black nitrocellulose membrane covered with an appropriate spacer ring (diameter 24 mm, teflon). After gelation (within ca. 1-2 min at room temperature) the biofilm was immersed into a CaCl 2 solution (7 g/L) for 30 min [START_REF] Podrazky | Determination of living immobilized yeast cells by fluorescence spectroscopy[END_REF][START_REF] Kuncova | Monitoring of the viability of cells immobilized by sol-gel process[END_REF]. Subsequently, planar (disk) films of immobilized cells (24 mm diameter and 15 mm thickness) were cut out and the spacer ring was removed.

Monitoring of glycolytic activity

The glycolytic activity of yeast cells was measured under two different conditions: (i) in stirred suspensions in a conventional spectrophotometer and (ii) in gel-fixed suspensions in an open spatial reactor, i.e. at unstirred conditions. For both systems the glycolytic activity was measured by the NAD(P)H fluorescence at excitation and emission wave lengths λ = 366 and 460 nm, respectively.

(i) Glycolytic oscillations in stirred cell suspensions. 1.9 ml of 10 % (w/v) yeast cells were placed in a spectrofluorimeter (AB2 Luminescence spectrophotometer, SLM AMINCO, Milton Roy Co., USA). The cells were stirred in a thermostatically regulated cuvette at 21°C (Hellma, Germany). Glycolytic oscillations were initiated by addition of glucose (0.1 mL; 1 M) into the cell suspension.

(ii) Glycolytic oscillations and waves from immobilized cells: The immobilized yeast cell discs were placed in an open spatial reactor (Fig. 1). The open spatial reactor consists of diffusive gel layer in contact with the content of a continuous-flow stirred tank reactor (CSTR). This CSTR acts as a feeding reservoir for the immobilized cell layer. The immobilized yeast cells were separated from the CSTR by an Anodisc membrane, with 20 nm

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pore size (Whatman). The black nitrocellulose membrane and the immobilized yeast cell disc were placed on top of this Anodisc membrane. Glycolytic oscillations of the immobilized yeast cells were initiated by feeding a solution containing 60 mM glucose, 0.1 M KH 2 PO 4 , pH 6.5, and 0.1 mM acetaldehyde into the open spatial reactor. Spatiotemporal concentrations changes of NAD(P)H (i.e. NAD(P)H waves) were monitored by means of an image intensified CCD camera (Corail, Optronics) and the resulting movies digitized at a temporal resolution of 2 images per second with a frame grabber card using a standard personal computer. For further details on the experimental set-up, see Bagyan et al [START_REF] Bagyan | Glycolytic oscillations and waves in an open spatial reactor: Impact of feedback regulation of phosphofructokinase[END_REF].

For analysis of the local temporal dynamics, the intensity of the NAD(P)H-fluorescence (grey levels) of a selected area (20 x 20 pixels) from each image of the movie was averaged and plotted as a function of time. For analysis of the spatiotemporal dynamics of the glycolytic waves a time-space plot of propagating NAD(P)H waves was constructed from the recorded movies along selected lines. The intensity along the selected line from each image of the movie was stacked on top of each other, resulting in the time-space-plot, which displays the spatiotemporal dynamics of propagating waves along this line. The reciprocal of this line corresponds to the wave velocity.

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Results

Induction of glycolytic oscillations in stirred yeast cell suspensions

Addition of glucose to starved yeast cells induces glycolytic oscillations, provided the cells are under energy limiting conditions. At the chosen growth conditions of the yeast, a starvation period of about 3-5 h is required, when stirred yeast cells are investigated (Fig. 2).

During the starvation period the amplitude of the oscillations increases from 0.12 to 0.31 a. u.

until it reaches a plateau after about 6-7 h (Fig. 3). On the other hand, the oscillatory period remains relatively constant at 30 ±4 s. These values serve as a reference for the experiments with the immobilized cells. For all immobilization experiments we used yeast cells that had been starved for 9 h.

Glycolytic oscillations of immobilized yeast cells in gel matrices

Gel-fixed yeast cells represent a diffusive layer and thus open the possibility to investigate their ability to generate spatiotemporal patterns of the reaction-diffusion type in an open system. To this purpose, the gel layer of the fixed cells is coupled via a membrane to a continuous stirred tank reactor (CSTR), which acts as a feeding reservoir [START_REF] Bagyan | Glycolytic oscillations and waves in an open spatial reactor: Impact of feedback regulation of phosphofructokinase[END_REF] (Fig. 1). The generation of glycolytic patterns in yeast cell layers is expected to occur via extracellular acetaldehyde waves [START_REF] Schütze | Spatio-temporal dynamics of glycolysis in cell layers. A mathematical model[END_REF]. These extracellular waves, however, can be visualized only indirectly. e. g. via intracellular NAD(P)H. Changes in the intracellular NAD(P)H fluorescence can be seen as an indicator for the concentration changes in the extracellular acetaldehyde, because the oscillations of these two compounds exhibit a fixed phase relation [START_REF] Richard | Acetaldehyde mediates the synchronization of sustained glycolytic oscillations in populations of yeast cells[END_REF]. Thus, we have taken the spatiotemporal dynamics of intracellular NAD(P)H fluorescence as an indirect measure of extracellular acetaldehyde waves.

Concomitant application of glucose and acetaldehyde to the gel-fixed yeast cells induces glycolytic oscillations in these cells. Acetaldehyde has been applied together with glucose in order to avoid a drop of its concentration below a critical value [START_REF] Richard | Acetaldehyde mediates the synchronization of sustained glycolytic oscillations in populations of yeast cells[END_REF]. The analysis of local fluorescence changes reveals the generation of glycolytic oscillations, irrespective of the type of gel used (Fig. 4). The period of the glycolytic oscillations (53 s) as well as the number of oscillatory cycles [START_REF] Podrazky | Determination of living immobilized yeast cells by fluorescence spectroscopy[END_REF][START_REF] Callone | Immobilization of yeast and bacteria cells in alginate microbeads coated with silica membranes: Procedures, physico-chemical features and bioactivity[END_REF][START_REF] Goldbeter | Biochemical oscillators and cellular Rhythms[END_REF] is nearly the same in the three gels studied (Fig. 5). However, a comparison of gel-fixed cells with stirred cell suspensions demonstrates a 50% drop in the period for the stirred cells, whereas the number of cycles remains nearly unaffected (13-15 vs.

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. Since the period is also related to temporal events we may take this change as an indicator for the slower diffusive processes in the gels, which are overcome by stirring in experiments with cell suspensions.

Spatio-temporal dynamics

In the extended gel-immobilised cell system, we also followed the spatiotemporal distribution of NAD(P)H. The signals are very weak so that the signal to noise ratio of the images is quite poor. However, the evaluation of the movies after image processing (background subtraction, A further difference between yeast cells immobilized in silica (TMOS) and agarose gel at the one side and cells entrapped in alginate/TMOS gel is the direction of propagation of the glycolytic waves (Fig. 6). While waves in agarose and silica gels were found to propagate from the centre of the reactor outwards, waves of cells in alginate/TMOS gels propagate in the opposite direction, i.e. inwardly.

The time-space-plots (Fig. 7) constructed from the movies at three different spatial locations (marked as white lines in Fig. 6B) demonstrate the impact of the type of gel on the wave propagation dynamics. On these time-space plots horizontal lines are due to oscillations in the medium, while inclined white lines originate from propagating NAD(P)H waves. Whereas in agarose the spatial patterns are quite complex even showing several wave dislocations (see arrows Fig. 7A), the patterns in the alginate/TMOS gel are very regular with only minor spatial differences (Fig. 7C). In TMOS alone we can see regular propagating waves mainly in the middle of the gel (Fig. 7B, 1/2), whereas at the borders the waves propagate at lower velocity.

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The impact of the gel composition on the propagation velocity of the waves can be also deduced from the time traces taken from the time-space-plots of figure 7, which are plotted in figure 8. On average we find a phase shift between the centre and the border of the gel of 7.5 s for agarose (Fig. 8A), 25.9 s for TMOS (Fig. 8B) and 0.5 s for alginate/TMOS (Fig. 8C).

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Discussion

The results presented here demonstrate for the first time the generation of propagating synchronization waves in yeast cell immobilized into layers of gels. The results also show that the open spatial reactor, used to perform these experiments, is well suited for the investigation of spatiotemporal pattern formation, not only in yeast extract [START_REF] Bagyan | Glycolytic oscillations and waves in an open spatial reactor: Impact of feedback regulation of phosphofructokinase[END_REF][START_REF] Bagyan | Spatial desynchronization of glycolytic waves as revealed by Karhunen-Loève analysis[END_REF][START_REF] Lavrova | Modeling of glycolytic wave propagation in an open spatial reactor with inhomogeneous influx[END_REF] but also in yeast cell layers.

Although the use of the open reactor is intended at maintaining the cell population in a sustained oscillatory state, we find only transient oscillations (Fig. 4). This indicates that the cells do not settle on an oscillatory state because some drifts in the reaction conditions cannot be fully controlled. Possibly, either substances required to maintain the glycolytic NAD(P)H oscillations leak out of the gel layer, or substances that inhibit the oscillations accumulate in the gel. Acetaldehyde itself may be a potential candidate for both of these possibilities since a drop below or an increase above the oscillatory concentration window abolishes the oscillations [START_REF] Richard | Acetaldehyde mediates the synchronization of sustained glycolytic oscillations in populations of yeast cells[END_REF].

Supply of glucose to the immobilized cells not only induces temporal glycolytic oscillations but also traveling waves (Fig. 6). This means that there is a coupling between the intracellular oscillatory production of acetaldehyde, which is released into the extracellular medium, and the extracellular transport of the messenger acetaldehyde by diffusion. The propagation dynamics of the travelling waves consequently depend on both, the oscillatory reaction and the transport step. Thus, possible interactions of the gels with the metabolism of the cells (oscillatory glycolysis) as well as with the extracellular signaling compounds (extracellular diffusion of acetaldehyde) may be encoded in these waves.

When comparing the characteristics of the temporal glycolytic NAD(P)H oscillations observed in gel-immobilized yeast cells to those of cells in suspension, we observe that the period of oscillations is twice as large in the gel-immobilized cells than in the suspended ones (Fig. 5). Since the glycolytic oscillations are the result of an acetaldehyde-mediated synchronization between the cells [START_REF] Richard | Acetaldehyde mediates the synchronization of sustained glycolytic oscillations in populations of yeast cells[END_REF], stirring of the cell suspensions provides an immediate distribution of this compound into the entire reactor, whereas in the gel the transport of this messenger is provided exclusively by molecular diffusion, which is much slower. Thus, the increased oscillatory period in the gels is caused by the slower diffusion of extracellular

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Saccharomyces cerevisiae, suspended in phosphate buffer or immobilized in gelatin [START_REF] Doran | Effects of Immobilization on the Nature of Glycolytic Oscillations in Yeast[END_REF].

Although the immobilization procedure and the type of gel differed markedly from ours, a pronounced increase of the period of glucose induced oscillations in intracellular NAD(P)H for the cells immobilized in gelatin was also found. When measured at 25 o C, the period rose from 54 s for suspended cells to about 240 s for the immobilized cells [START_REF] Doran | Effects of Immobilization on the Nature of Glycolytic Oscillations in Yeast[END_REF].

In the following, we discuss the influence of the type of gel on both, the temporal and spatiotemporal dynamics of entrapped yeast cells. To this purpose the (i) periods of the temporal oscillations as well as (ii) the direction and (iii) speed of the propagating waves will be discussed. Interestingly the temporal dynamics show no difference among the different gels, although the three gels studied in the present article have different pore sizes. Obviously, the differences in extracellular diffusion in the gel types studied are not sufficiently large to significantly alter the period of oscillations.

The picture changes, when it comes to comparing the spatiotemporal wave patterns of NAD(P)H concentration in the three gels studied (agarose, TMOS, and alginate/TMOS). The wave patterns are relatively similar to each other, thus demonstrating the robustness of oscillatory glycolysis arising even from yeast cells immobilized in gels. Nevertheless, significant differences were found when it comes to the wave propagation direction and velocity, which depend sensitively on the identity of the immobilizing gel. In fact, glycolytic waves from cells entrapped in silica (TMOS) and agarose gels always propagated outwards, while the propagation direction in alginate/TMOS gels was always inward. The difference in wave propagation direction correlates with the behaviour of the wave speed. Again, the wave speeds are similar in agarose and silica gel, while it is twice as high in alginate/TMOS. These observations suggest that the availability of acetaldehyde for the cells entrapped in agarose or silica gels on the one hand and in alginate/TMOS at the other hand differ. This may be due either to subtle differences in the transport efficiency of acetaldehyde, since such changes in the transport process have recently been shown to affect the direction of wave propagation [START_REF] Bagyan | Spatial desynchronization of glycolytic waves as revealed by Karhunen-Loève analysis[END_REF][START_REF] Lavrova | Modeling of glycolytic wave propagation in an open spatial reactor with inhomogeneous influx[END_REF], or to a different way of entrapping the cells in the gel matrices.

To better understand how the yeast cells are entrapped into the gel matrices, we have performed a preliminary screening electron microscopic (SEM) study, which indicates that the yeast cells are monodispersely entrapped in both agarose and silica gels. This contrasts

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with the alginate/TMOS gel, where the cells are found to form small clusters comprising a small number of cells. The formation of cell clusters in alginate/TMOS matrices may explain the different direction of glycolytic wave propagation in this matrix: In small clusters, the cells are packed tightly together, so that the acetaldehyde released from one cell immediately triggers the release of acetaldehyde of the other cells in the cluster. The clusters are randomly distributed in the gel, and thus act as randomly distributed nucleation sites for wave initiation.

Unevenly, but randomly distributed clusters consequently favour the generation of waves that propagate inwardly.

The wave propagation speed is similar for cells entrapped in agarose and silica gels, while the wave velocity is significantly higher in alginate/TMOS gel. This is somewhat surprising, since this composite gel contains weak chemical bonds between the hydroxyl groups of the polymerized aqueous silica and the polysaccharide chains of alginate [START_REF] Coradin | Synthesis and characterization of alginate/silica biocomposites[END_REF][START_REF] Xu | Preparation and catalytic properties of novel alginate-silica-dehydrogenase hybrid biocoposite beads[END_REF]. Therefore the alginate/TMOS gel would have been expected to have a considerably smaller pore sizes than TMOS alone or the agarose gel. In turn, this would than have led to a decreased diffusion and to a lower wave propagation velocity.

However, the more effective transport in alginate/TMOS gels seems to be linked to the cluster formation of yeast cells, which may locally lead to the release of a higher concentration of acetaldehyde than it is the case of monodispersely distributed cells (as in agarose or silica gels). Since the excitation of a yeast cell requires a superthreshold concentration of the messenger acetaldehyde, this threshold is reached sooner when higher concentrations of the messenger are present. This leads to higher effective wave propagation velocity. The more homogeneous appearance of the wave propagation in the alginate/TMOS gel (Fig. 7) is a further indication for a more effective transport of acetaldehyde between the entrapped cells.

These findings suggest that there is a stronger mutual coupling of the cells in the alginate/TMOS gel than in the other two gels. Additional support for a more effective acetaldehyde transport in the alginate/TMOS gel is provided by the phase shift between the center and the borders of the gel (Fig. 8). In alginate/TMOS there is nearly no phase shift (0.5 s), whereas in agarose we find a phase shift of 7.5 s and in TMOS of 26 s.

Bagyan et al. [START_REF] Bagyan | Glycolytic oscillations and waves in an open spatial reactor: Impact of feedback regulation of phosphofructokinase[END_REF] found also phase shifts for glycolytic waves, however in immobilized yeast extract, measured with the same open spatial reactor as the one used in the present study.

Later, Lavrova et al simulated these results and attributed the phase shifts to a slight

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inhomogeneous substrate inflow [START_REF] Lavrova | Modeling of glycolytic wave propagation in an open spatial reactor with inhomogeneous influx[END_REF]. An inhomogeneous distribution of the cells, however, might have a similar effect as an inhomogeneous substrate inflow. Spatial desynchronization phenomena might be taken into consideration as well [START_REF] Bagyan | Spatial desynchronization of glycolytic waves as revealed by Karhunen-Loève analysis[END_REF]. Figure 8 also reveals a strong decrease of the amplitude in cells immobilized in alginate/TMOS. However, fluorescence is a relative measure and thus should be taken with care when different experimental conditions are involved.

In conclusion, our results demonstrate that spatiotemporal glycolytic oscillations can be generated in yeast cells that are immobilized in gel layers. This is in good agreement with recent studies where such waves were also seen in sedimented (but not fixated) cells layers, and also with studies where a gel-immobilized cell extracts were used. The spatiotemporal patterns of metabolic synchronization in gel-immobilized yeast cell layers provide a promising tool to investigate the interaction between the cells and the surrounding gel.

Especially the wave velocity, direction, and the wave patterns seem to be suitable parameters for this purpose, since these characteristic features were found to depend sensitively from both, the identity of and the physical parameters describing the embedding, fixating gel.

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  contrast enhancement) revealed the spontaneous formation of propagating patterns. They can be seen as white stripes or spots (increased NAD(P)H) on a dark background. To improve visualization of the waves we have marked them by dashed lines (Fig,6A). The direction of wave propagation is indicated by an arrow. With agarose as fixative (Fig6 A, B) we observe multiple wave sources which periodically emit waves that eventually combine to one large wave. Similar wave patterns are observed when the yeast cells were immobilized in TMOS (Fig.6C, D) or in alginate/TMOS. No significant differences in the wave velocity were found in agarose and TMOS (both at 130 ± 20 μm/s) gels, whereas in alginate/TMOS the propagation velocity it is about twice as large (240 ± 10 μm/s).

Figure 1 :

 1 Figure 1: Sketch of the open spatial reactor used to perform the experiments with gelimmobilized yeast cell layers.

Figure 2 :

 2 Figure 2: Glycolytic oscillations in stirred suspensions of S. carlsbergensis (cell concentration: 10 % wet (w/v)) after different times of starvation. Glycolysis was measured by NAD(P)H fluorescence in a conventional spectrofluorimeter. The oscillations were initiated by addition of glucose to the cell suspension (marked by arrow). The yeast cells had been starved for 0-11 h.

Figure 3 :

 3 Figure 3: Increase of the amplitude of glycolytic oscillations in stirred suspensions of S. carlsbergensis (cell concentration: 10% wet (w/v)) as a function of the starvation time. Oscillations have been measured by NAD(P)H fluorescence. The mean value is calculated from 10-12 amplitudes. Error bars show ± Standard deviation.

Figure 4 :

 4 Figure 4: Glycolytic oscillations of immobilized cells of S. carlsbergensis (10% w/v) in gel matrices. The glycolytic oscillations of the gels were measured in the open spatial reactor(stirring rate: 500 rpm, feeding rate: 74 mL/h and temperature 21°C). The cells were immobilized after 9 h of starvation and the oscillations were initiated by addition of glucose and acetaldehyde (at time 180 s, marked by the arrow).

Figure 5 :

 5 Figure 5: Comparison of the kinetic values of the glycolytic oscillations from the cells shown in figure 2 (after 9 h starvation) and figure 4. Period is given in seconds.

Figure 6 :

 6 Figure 6: Propagating waves of glycolytic activity in immobilized yeast cell layers after glucose addition, as measured by intracellular NAD(P)H-fluorescence. Immobilization was performed in agarose (A, B), TMOS (C, D) and alginate/TMOS (E, D). For improved visualization the wave fronts are marked by dashed lines (in C with a circle) and the propagation direction by an arrow. In B the three different locations (1/3, 1/2, 2/3 from top to bottom) at which time space plots have been constructed are marked by horizontal full lines. The time interval is 88 s for A to B, 78 s for C to D and 86 s for E to F. The x-axis of each image corresponds to 12 mm.

Figure 7 :

 7 Figure 7: Time-space plots of propagating waves in immobilized yeast cell layers (cell concentration: 10% (w/v)) in A) agarose, B) silica (TMOS) gel and C) alginate/TMOS gels. The different locations at which the time space plots were taken (1/3, 1/2 and 2/3) are shown in figure 6B. The white arrows in A indicate wave interactions due to annihilation and wave break up. The rectangles show the positions at which the time traces, shown in figure 8, have been taken.

Figure 8 :Figure 1 A

 81 Figure 8: Temporal dynamics of the local oscillations as taken from the time-space-plots in figure 7 in (A) agarose, (B) silica (TMOS) and (C) alginate/TMOS gels. The variable locations at which these time traces have been taken are marked by solid black lines (center), solid grey lines (left border) and dashed lines (right border).

Figure 2 A

 2 Figure 2

Figure 3 A

 3 Figure 3

Figure 4 A

 4 Figure 4

Figure 5 A

 5 Figure 5

Figure 6 A

 6 Figure 6

Figure 7 A

 7 Figure 7

Figure 8 A

 8 Figure 8

Acknowledgement

This study was financially supported by the Ministry of Education, Youth and Sport of the Czech Republic (grant no. OC121) and by the Deutsche Forschungsgemeinschaft (grant no. MU5939/19-1). Exchange grant and lab visit support was provided by the ESF research networking program "Functional Dynamics in Complex Chemical and Biological Systems" (FUNCDYN) and by COST action 865 "Bioencapsulation multiscale interaction analysis".

Note added

We dedicate this article to Dr. Thomas Mair, who sadly passed any during the publication process of this article.

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