C Gouttebroze

F Louf

L Champaney

Multiple model updating using the finite element method over a polynomial algebra

Vibration control is an important issue when it comes to preserving the structural integrity of mechanical structures, particularly in the case of lightweight structures found aboard flying vehicles, such as printed circuit boards. In order for a simulation to be effective, a suitable numerical model of the structure must be used. Such a model can be obtained using a model updating method, such as that which is presented in this work, based on the modified constitutive relation error principle.

However, like many other model updating strategies, this method can be very expensive. When the structural parameters change over time, it would be too timeconsuming to perform another full calculation. In order to circumvent this problem, we introduce a new technique called the 'finite element method over a polynomial algebra', which involves modules defined over a ring of truncated polynomials in multiple variables.

In this paper, we illustrate the method with the updating and reduction of a model of a beam instrumented with piezoelectric sensors and actuators, which are all taken into account in the numerical model. This is a simple problem, yet it is representative of the updating of printed circuit boards.

Introduction

In the field of mechanics applied to the development of intelligent systems, the main areas of investigation are the control of the health or integrity of structures, the control of their shape, and active vibration damping. In this paper, we consider the case of lightweight structures, such as the printed circuit boards found aboard flying vehicles (planes, rockets,. . . ). In the different flight configurations (liftoff, landing,. . . ), these cards are subjected to low-frequency vibrations which can damage the components and electrical connections, causing them to age prematurely. The classical remedy, based on passive damping, works in only a limited range of frequencies and increases the weight significantly. A solution based on active damping allows the vibrations to be reduced in a wide frequency range while keeping the increase in weight moderate. Generally speaking, the active control process can be divided into three steps. First, the displacements of the structure are measured by sensors and a data processing system: this is the observation phase, in which the most commonly used sensors are piezoelectric components. Then comes the thinking phase, which consists in the elaboration of an answer. Finally, in the actuation phase, the resulting command is delivered to the structure. Again, the most commonly used actuators in this phase are piezoelectric components. The thinking phase requires an adequate model in order to deliver a satisfactory answer. In the case of lightweight structures, this model must take into account the presence of the piezoelectric components. One possible method to obtain such a model consists in using a model updating strategy.

A state-of-the-art review of finite element model updating methods can be found in [START_REF] Ienny | Identification from measurements of mechanical fields by finite element model updating strategies. a review[END_REF]. They all rely on stiffness, damping an mass matrices obtained through changes in the physical parameters of the model. Their differences come from the objective function they use. A first group of techniques try to minimize input residuals like the Sensitivity-Based Element-by-Element Updating Procedure [START_REF] Farhat | Updating finite element dynamics models using an element-by-element sensitivity methodology[END_REF] or the Equilibrium Gap Method [START_REF] Claire | A finite element formulation to identify damage fields : the equilibrium gap method[END_REF]. The second group is based on output residuals like the Modal Assurance Criterion [START_REF] Allemang | Investigation of Some Multiple Input/Output Frequency Response Function Experimental Modal Analysis Techniques[END_REF] or the Reciprocity Gap Method [START_REF] Bui | Numerical identification of linear cracks in 2d elastodynamics using the instantaneous reciprocity gap[END_REF]. A third group relies on the residual over the constitutive equations like the Constitutive Gap Method [START_REF] Geymonat | Identification of mechanical properties by displacement field measurement: A variational approach[END_REF] or the Modified Constitutive Relation Error (MCRE) [START_REF] Ladevèze | Updating of finite element models using vibrations tests[END_REF], the one we use in this paper.

The first developments of this method demonstrated the effectiveness of such a model in structural dynamics, then in forced vibrations with updated models obtained from eigenmodes. This approach, which uses the concept of Drucker error, provides an effective means of updating mass and stiffness matrices. Its governing principle consists in partitioning the equations of the problem into two groups, a reliable group and an uncertain group. Then, among the mechanical fields which satisfy the equations of the reliable group exactly, one seeks to minimize a norm containing the residuals of the uncertain group and the distance between the observed experimental data and the numerical calculation. Like many other popular updating techniques, it is an optimization-based method whose objective function is very computation-intensive.

The structural parameters of the structure can change over time due to ageing, damage, relative movements in the joints,. . . As this happens, a previous vibration control can become ineffective. In order to avoid going through another expensive model updating process, we propose to use a multivariate analysis approach. In this work, we introduce a new multivariate analysis technique called the Finite Element Method over a Polynomial Algebra (FEMPA). In this method, the coefficients of the vectors of the finite element basis are polynomials in multiple variables of interest.

First, we present the MCRE, the model updating method used here. Then, we introduce the FEMPA and describe its application to the one-time updating of a model of a structure representing a printed circuit board. Finally, we illustrate how the FEMPA can be used in order to update a structure rapidly when its parameters change over time.

∂ 1 Ω ∂ 2 Ω ∂ 3 Ω Ω F d f d u d u exp

Presentation of the model updating method

The reference problem

For the sake of simplicity, we present the model updating method using a viscoelastic structure as an example (see Figure 1). The application of the method to a viscoelastic piezoelectric structure was studied in [START_REF] Formosa | Contribution à l'amélioration de la modélisation de structures légères en vue de leur contrôle actif[END_REF] The reference problem consists in finding a solution in terms of s(x, t) = (u(x, t), (x, t), Γ(x, t)) (the displacement, stress and acceleration quantities respectively), with x ∈ Ω, t ∈ [O, T ], which satisfies the kinematic constraints:

u |∂ 1 Ω = u d , (1)
the equilibrium equations:

∇ + f d = Γ, (2)
n |∂ 2 Ω = F d , (3)
and the constitutive equations:

= K ¯(u) + D ¯( u), (4)
Γ = ρ ü. (5)
K denotes Hooke's tensor, D the damping tensor and ρ the density. These quantities can be defined by a set of n p structural parameters. During an experimental test, one measures a displacement u exp over ∂ 3 Ω. In order to agree with the experiments, the solution s(x, t) must satisfy:

u |∂ 3 Ω = u exp . (6)

The Modified Constitutive Relation Error approach

The MCRE concept, like the CRE concept, is based on a partitioning of the equations into reliable equations and less reliable equations. The difference comes from the presence of equation ( 6), which introduces a distance between the numerical model and the experimental data. For a given set of structural parameters, a displacement field u ad is said to be kinematically admissible if it satisfies [START_REF] Ienny | Identification from measurements of mechanical fields by finite element model updating strategies. a review[END_REF]. Stress and acceleration fields ad , Γ ad are said to be dynamically admissible if they satisfy (2) and (3). The MCRE approach consists in finding an admissible solution s ad = (u ad , ad , Γ ad ) which minimizes the residuals of ( 4), ( 5) and ( 6):

e 2 (s) = γ T t=0 K ¯(u) + D ¯( u) -2 dt + (1 -γ) T t=0 ρ ü -Γ 2 dt + r 1 -r T t=0 u -u exp 2 dt, (7)
where γ ∈ [0, 1] and r ∈ [0, 1[ are weighting parameters. Parameter r represents one's degree of confidence in the experimental data. The term r 1-r ∈ [0, ∞[ is used to apply the desired weight to the experimental part of the error, even if its initial magnitude is very different from that of the other errors. Different versions of the norms involved in equation ( 7) can be found in [START_REF] Deraemaeker | Sur la maitrise des modèles en dynamique des structures à partir de résultats d'essais[END_REF]. The discrete form of the norms we are using in this paper will be presented in equation (15).

In the case of a forced vibration problem at an angular frequency ω, the calculation of the objective function over a duration T ′ which is a multiple of the period leads to:

e 2 ω (s) = γ 2 (K + T ′ ω 2 D)¯(u) - 2 + 1 -γ 2 ρω 2 u -Γ 2 + r 1 -r u -u exp 2 . (8)

Discretized form of the modified error

For numerical convenience, the problem can be described in terms of displacements by introducing fields v ad and w ad such that the associated stress and acceleration fields are dynamically admissible:

= K ¯(v ad ) + D ¯( vad ), (9)
Γ = ρ ẅad . (10)
Now, let us move on and discretize the problem. We introduce a finite element basis {¨} = {φ i } i=1,...,n dof such that the displacement fields are approximated in the form:

u ad = n dof i=1 u i φ i , v ad = n dof i=1 v i φ i , w ad = n dof i=1 w i φ i , (11)
where u i , v i and w i denote the i th coefficients of the vectors of the nodal displacements {U}, {V } and {W } respectively. We build the discretized stiffness matrix [K], damping matrix [D] and mass matrix [M] such that their i th -line and j th -column coefficients are:

[K ij ] = Ω Tr K ¯(φ i )¯(φ j ) dV , (12)
[D ij ] = Ω Tr D ¯(φ i )¯(φ j ) dV , (13)
[M ij ] = Ω ρ φ i .φ j dV. ( 14
)
With these notations, the discrete form of e 2 ω (s) is given by:

e 2 ω ({U}, {V }, {W }) = γ 2 {U -V } ⋆ ([K] + T ′ ω 2 [D]){U -V } + 1 -γ 2 ω 2 {U -W } ⋆ [M]{U -W } + r 1 -r {Π 3 U -U exp } ⋆ [G]{Π 3 U -U exp }, ( 15
)
where ⋆ is the conjugate transpose operator, and Π 3 a projection operator onto ∂ 3 Ω. Matrix [G] quantifies the error in the measurements and can be expressed as:

[G] = γ 2 Π 3 ([K] + T ′ ω 2 [D])Π ⋆ 3 + 1 -γ 2 ω 2 Π 3 [M]Π ⋆ 3 . (16)
In addition, the solution ({U}, {V }, {W }) must be admissible, i.e. it must satisfy:

Π 1 {U} = {U d }, (17)
([K] + iω[D]){V } -ω 2 [M]{W } = {F }, (18)
where Π 1 is a projection operator onto ∂ 1 Ω and {F } is the excitation force vector. The expression of e 2 ω (s) under the admissibility constraints is obtained through the introduction of Lagrange multipliers. Then, for a given a set of the structural parameters, one can calculate {U},{V } and {W }. During these calculations, which are not described in detail here, the Lagrange multiplier corresponding to equation ( 18) is eliminated and the change of variables from ({U}, {V }, {W }) to ({U}, {U -V }, {U -W }) comes naturally. We end up with the resolution of a system of linear equations [A]{X} = {B} with:

A =      γ 2 ([K] + T ′ ω 2 [D]) 1-γ 2 ω 2 [M] r 1-r [G] 0 γ 2 ([K] + T ′ ω 2 [D]) 1-γ 2 ([K] + iω[D]) 0 0 -([K] + iω[D]) ω 2 [M] [K] + iω[D] -ω 2 [M] Π T 1 0 0 Π 1 0      (19) X =          {U -V } {U -W } {U} {λ}          , B =          r 1-r [G]{U exp } 0 {F } {U d }          . ( 20
)
Once the fields {U},{V } and {W } have been determined, the value of the objective function is calculated using equation (15).

The finite element method over a polynomial algebra

The two families of model reduction techniques

Let us consider a problem discretized spatially into n dof degrees of freedoms in a frequency range discretized with n ω points. For any set of the structural parameters which define the stiffness, damping and mass matrices, the calculation of the objective function requires the resolution of n ω linear systems of size 3n dof (see equation ( 20) ).

If n dof and/or n ω are large, the objective function

e 2 (s ad ) = ωn ω ω=ω 1 e 2 ω (s ad ) (21)
must be calculated approximately. There are two families of techniques, leading to different kinds of approximations.

The first family, based on a mechanical approach to the problem, consists in reducing n dof , which can be done by reducing the structure to its eigenmodes in a frequency range of interest [START_REF] Allemang | Investigation of Some Multiple Input/Output Frequency Response Function Experimental Modal Analysis Techniques[END_REF]. Alternatively, one can use a condensation method. The second family, based on a mathematical approach to the problem, consists in replacing the objective function by a simpler function. The most natural way to do that is to compute a Taylor series expansion of the objective function in the structural parameters, which can be done by three differents approaches.

The easiest maner to perform it is to have a sampling approach [START_REF] Kim | Second-order variance estimation in poststratified two-stage sampling[END_REF][START_REF] Helton | Survey of sampling based methods for uncertainty and sensitivity analysis[END_REF].

It is based on the evalution of the objective function for n samp sets of the structural parameters. Thus is requires the resolution of n samp n ω linear systems of size 3n dof . It is easy to implement but the choice of the sampling points can be a sentitive issue for the quality of the Taylor series expansion. This problem is avoided by the techniques based on the perturbation theory, like the perturbation finite element method [START_REF] Xie | The perturbation finite element method for solving problems with nonlinear materials[END_REF][START_REF] Huang | Perturbation finite element method of structural analysis under fuzzy environments[END_REF]. They introduce the parametric variabilities into the numerical model under variational form. It leads to the resolution of n omega linear systems of size 3n dof n ′ samp , where n ′ samp is the dimension of the Taylor series expansion. The main drawback is the needed of heavy software development in order to change the basis where the solution is seeked. We introduce here a new method to obtain the Taylor series expansion of the objective function, called the Finite Element Method over a Polynomial Algebra (FEMPA), that keeps the dependence over the parameters without changing the finite element basis. It is done by changing the ring over which the finite element method is build, using a multivariate polynomial ring instead of the real field or the complex field. It leads to the resolution of n omega linear systems of size 3n dof which coefficients are polynomials.

The polynomial algebra

Let P(n d , n X , Ã) denote the set of the polynomials in n X variables with maximum degree n d and coefficients in Ã. In the rest of the paper, for the sake of simplicity, this set will be denoted P when there is no ambiguity. The n X variables will be denoted X = (X 1 , . . . , X n X ). A polynomial p ∈ P can be expressed as:

p = i n d p i X i = i 1 +...+in X n d p i 1 ,...,in X X i 1 1 . . . X in X n X . (22)
For k ∈ Ã and (p, q) ∈ P 2 ,

p = i n d p i X i , q = i n d q i X i , (23)
let + denote the internal operation

p + q = i n d (p i + q i )X i , (24)
let . denote the external operation

k.p = i n d kp i X i , (25)
and let × denote the internal operation

p × q = i n d   (α,β)∈Σ i p α q β   X i , (26)
where Σ i is the set:

Σ i = {(α, β)|∀s ∈ [1, . . . , n X ], α s + β s = i s }. (27)
With these definitions, it is very easy to prove that (P, +, ., ×) has the required properties to be an algebra over the field à and that (P, +, ×) as the required properties to be a ring. It is not a field because an element has an inverse for operation × if, and only if, its constant term is nonzero. Let P ÷ denote the subset of P whose elements have an inverse for operation ×. We will show that P ÷ = P -{0 P }, where 0 P is the neutral element of operation +. For a given p ∈ P, denoting 1 P is the neutral element of operation ×, one can find q ∈ P such that:

p × q = i n d p i X i × i n d q i X i = 1 P (28)
when

q i =          1 p (0,...,0)
for i = (0, . . . , 0)

-1 p (0,...,0) (α,β)∈Σ ′ i p α q β for i = (0, . . . , 0)

, ( 29
)
k 1 k 2 u 1 u 2 f Figure 2. Two springs
where Σ ′ i is the set:

Σ i = {(α, β)|(α, β) ∈ Σ i and α = (0, . . . , 0)}. ( 30
)
In other words, such a polynomial q exists if, and only if, p (0,...,0) = 0. In such a case, we will denote q = 1 P ÷ p.

An example of application

Let us consider a series of two springs (see Figure 2) whose stiffnesses k 1 and k 2 vary around values k (0)

1 and k (0) 2 respectivly. In order to apply the FEMPA, we express the stiffnesses as :

k 1 (X) = k (0) 1 (1 + X 1 ), k 2 (X) = k (0) 2 (1 + X 2 ). (31)
If additional information is available concerning the variation of these stiffnesses, for example following two functions g 1 and g 2 , one can use the Taylor series expansions of functions g 1 and g 2 around k Let us now apply a force f to the second spring. The problem can be described using the equations:

k 1 u 1 -k 2 (u 2 -u 1 ) = 0 k 2 (u 2 -u 1 ) = f . (32)
This problem can be rewritten in matrix form. In the framework of the FEMPA, this leads to matrices and vectors whose coefficients belong to P:

k (0) 1 (1 + X 1 ) + k (0) 2 (1 + X 2 ) -k (0) 2 (1 + X 2 ) -k (0) 2 (1 + X 2 ) k (0) 2 (1 + X 2 ) u 1 (X 1 , X 2 ) u 2 (X 1 , X 2 ) = 0 f . ( 33
)
The analytic solution of this system is:

       u 1 (θ) = f 1 k 1 u 2 (θ) = f k 1 + k 2 k 1 .k 2 , ( 34
)
which can be rewritten, using the FEMPA notations, in the form: During the resolution of the linear system (33) using a specific algorithm, the elementary operations leading to the solution (35) are carried out with the definitions given in equations ( 23), ( 26) and (29). When the degree n d is equal to 2, we obtain the solution:

   u 1 (X) = f ÷ k (0) 1 (1 + X 1 ) u 2 (X) = f × k (0) 1 (1 + X 1 ) + k (0) 2 (1 + X 2 ) ÷ k (0) 1 (1 + X 1 ) × k (0) 2 (1 + X 2 ) . (
       u 1 (X 1 , X 2 ) = f k (0) 1 - f k (0) 1 X 1 + f k (0) 1 X 2 1 u 2 (X 1 , X 2 ) = f k (0) 1 + f k (0) 2 - f k (0) 1 X 1 + f k (0) 1 X 2 1 - f k (0) 2 X 2 + f k (0) 2 X 2 2 . (36)
If one needs to calculate a quantity of interest using the vectors of the nodal displacements [u 1 , u 2 ] T , one can also use the operations defined in Subsection 3.2.

Application of the FEMPA to the one-time updating of a model

The strategy

In order to apply the FEMPA to the one-time updating of a model, we write the n p structural parameters under polynomial form as in equation (31). Thus, the i-th quantity we want to update z i is written:

z i (X) = z (0) i (1 + X i ) ( 37
)
where z (0) i is an initial guess of this quantity. In order to compute a polynomial form of the displacement {U(X)}, {V (X)}, {W (X)} we solve the linear system (20) whose coefficients lie now in P(n d , n p , Ê), using the definitions given in equations ( 23), ( 26) and ( 29). We also use them and equation ( 15) to compute a first Taylor series expansion of the error :

e ≈ e (0)

P (X) = e (0) P (X 1 , . . . , X np ). (38)
Then, we try to find a local minimum which is close to (0, . . . , 0), denoted

a (1) = (a (1)
1 , . . . , a

np ) when it exists. The minimization algorithm choice is given to the reader, but a variable-step BFGS algorithm is recommended, with the identity matrix as the initial guess for the Hessian matrix. Once the minimum is found, we have a new guess of the quantity we want to update, the i-th being:

z (1) i = z (0) i (1 + a (1) i ), (39)
which can be rewritten under polynomial form

z i (X) = z (1) i (1 + X i ), (40)
in order to repeat all the steps described above until convergence as described in figure 3.

Presentation of the example

Let us consider an aluminum isotropic beam instrumented with one piezoelectric sensor and one piezoelectric actuator. Both piezoelectric components are assumed to be transverse isotropic. All the dimensions are shown in Figure 4. The material properties are given in Table 1 for the beam and in Table 2 for the piezoelectric components. With this structure, we tested our model updating method by modifying the Young's modulus and the structural damping of the beam. The experimental measurements were simulated by perturbing the structural parameters by:

E (exp) = 0.7 E, D (exp) = 1.4 D. (41)
We obtained a Frequency Response Function (FRF). In order to limit the number of frequencies used to calculate the error (see equation ( 21)), we retained only the measured frequencies with high FRF amplitude: this is justified by the fact that in actual experiments it is at these frequencies that there is the least measured noise. In our example, the 4 first frequencies were 6, 33, 88 and 178 Hz. In addition, we chose r = γ = 0.5. The calculations of the successive minima of the error (see equations 38) was performed with a variable-step BFGS algorithm, with the identity matrix as the initial guess for the Hessian matrix.

Results of the model updating

The initial guess of the structural parameters was E (0) = E and D (0) = D. Figure 5 shows the exact objective function (whose minimum is (E (exp) , D (exp) )) and its successive approximations with polynomials of degree 4. Table 3 shows that the minimum was reached after three iterations. The degree nd of the polynomials has an influence on both the computation time and the quality of the approximation. As shown in Table 4, on the one hand, the lower the degree, the better the computation time; on the other hand, increasing the degree improves the quality of the approximation and, therefore, decreases the number of iterations needed to achieve convergence. For our problem, for which we sought an error in the final guess of the parameters of less than 10 -6 for each polynomial degree, nd = 4 was the best choice.

In some cases which are not addressed here, it may be useful to know how the computation time varies with the number of variables n X . This variation is the same as the variation with the degree nd because they have symmetrical roles in the expression of the dimension of the algebra (P(n d , n X , Ã), +, ., ×), which is (n d +n X )! n d !n X ! .

Application of the FEMPA to multiple model updating

General principle

In order to update a structure whose behavior changes due to ageing effects, we proceed in two steps.

The first step consists in constructing an objective function which represents the evolution of the structure's behavior. In addition to expressing how the error varies with the structural parameters, we are interested in its dependence on the experimental data u exp . In order to express this dependence, we rely on the measurments before ageing.

The second step consists in updating the model for each ageing stage using its specific measured FRF. In order to do that, we take the restriction of the objective function to the experimental data of the ageing stage we are interested in. Thus, we get a function whose only variables are the structural parameters and whose minimization cost is very small.

Construction of the objective function

During n ω tests at several frequencies, the real and imaginary parts s (0) = (s (0) 1 , . . . , s (0) 2ncnω ) of the measurements are collected and expressed in polynomial form:

s (0) i (X) = s (0) i (1 + X np+i ), (42)
for i = 1, . . . , 2n c n ω . (The shifted indices are due to the fact that the n p unknowns are retained for the quantities we want to update.) In order to get a Taylor series expansion of the error e (0) P (X 1 , . . . , X np+2ncnω ), we solve the linear system (20) with coefficients in P(n d , n p + 2n c n ω , ), then use equation (15).

Use of the objective function

For the k-th ageing stage of the structure, a test leads to 2n s n ω measurements s

(k) 1 , . . . , s (k)
2ncnω which are linked to the initial measurements through a set of parameters η (k) = {η i |i = 1, . . . , 2n c n ω } such that:

s (k) i = s (0) i (1 + η (k) i ), (43)
for i = 1, . . . , 2n c n ω . This calculation cannot be carried out if s 0 i = 0, which occurs when the sensor is on a node of the structure. In this case, it is necessary to move the sensor or to change the frequency of the experimental test. If neither is possible, the data from the particular sensor must be omitted in the calculation of the multi-structure error.

Then, we proceed with the calculation of the restriction of that error to the k-th ageing stage: (44)

We seek the local minimum of ε (k)

Iω which is closest to (0, . . . , 0) and denote θ (k) when it exists. This enables us to find the values z (k) of the quantities of the k-th ageing stage:

z (k) i = z (0) i (1 + θ (k) i ), (45)
for i = 1, . . . , n p .

Results

We took the same example as in the previous section. The computation of the Taylor series expansion of the error e (0) P (X 1 , . . . , X np+2ncnω ) took 14 seconds. Several additional experiments were carried out with different values of the structural parameters. Table 5 summarizes the quality of the multiple model updating results.

Model reduction for vibration control

Vibration control requires a reduced mechanical model of the structure. The reduced stiffness, damping and mass matrices [K r ], [D r ] and [M r ] can be calculated using

K r ] = [Φ] T [K][Φ], (46)
[D r ] = [Φ] T [D][Φ], (47)
[M r ] = [Φ] T [M][Φ], (48)
where [Φ] is the n ω × n dof matrix of the n ω eigenmodes of the structure which lie within some frequency range of interest. The FEMPA can be used to calculate these reduced matrices. The coefficients of these matrices lie in P(n d , n p , Ê) and can be updated rapidly when {α (j) } is known.

Conclusion

In previous works, the MCRE was used successfully to update models of large industrial structures. In order to do that, however, the objective function had to be approximated very often through mechanical reduction of the model. In this work, we introduced the finite element method over a polynomial algebra in order to calculate an approximate value of the error mathematically. This enables one to update the model of the structure rapidly when the structural parameters change, and also to calculate the reduced mechanical model rapidly. This technique lends itself to multiple application fields which should be explored in the future.

Figure 1 .

 1 Figure 1. The domain being studied

  . The structure vibrates over a time interval [0, T ]. It is subjected to body forces f d , prescribed displacements u d over ∂ 1 Ω and distributed forces F d over the boundary ∂ 1 Ω.

Figure 3 .

 3 Figure 3. Algorithme de recalage suivant une stratgie algbrique

Figure 4 . 1 .Table 2 .

 412 Figure 4. The instrumented beam

  e

P

  (X 1 , . . . , X np ) = e (0) P (X 1 , . . . , X np , η

Table 4 .

 4 Computation times in relation to polynomial degree

	Polynomial Time for one	Number of	Total
	degree	approximation approximations time
	2	15 s	∞	∞
	3	21 s	5	105 s
	4	31 s	3	93 s
	5	46 s	3	138 s
	6	73 s	2	146 s
	4.4. The influence of the degree of the polynomials

Table 5 .

 5 The model updating results E, 0.9 D} {1.1002 E, 0.9061 D} 0.04 s {0.7 E, 1.2 D} {0.7012 E, 1.1935 D} 0.04 s {0.8 E, 1.3 D} {0.7997 E, 1.2922 D} 0.04 s {1.6 E, 0.6 D} {1.5973 E, 0.6114 D} 0.04 s

	Given	Result	Computation
	perturbation	perturbation	time
	{1.1 equations
	[

Each calculation of an approximation of e 2 (s) took 31 seconds on a 2 GHz dualcore processor with 512 Mb RAM, using a code written in C++. The total time of 93 seconds was 2.5 times less than with the usual scalar field method, which required 232 seconds to obtain results of similar quality, i.e. an error in the final guess of the parameters of less than 10 -6 .


Jurga ; Bernatoniene

Savickas

S Trumbeckaite

D Majiene

R Baniene

G Baliutyte

A Savickas

A Toleikis
email: adolfas.toleikis@kmu.lt

The effect of Crataegus fruit extract and some of its flavonoids on mitochondrial oxidative phosphorylation in the heart The effect of Crataegus extract on the heart mitochondrial function

Keywords: Crataegus fruit extract, heart mitochondria, oxidative phosphorylation, flavonoids Crataegus fruit extract, heart mitochondria, oxidative phosphorylation, flavonoids Crataegus fruit extract, PC -Crataegus phenolic compounds, ROS -reactive oxygen species, BSA -bovine serum albumin, TMPD -N,N,N',N

Crataegus (Hawthorn) fruit extracts (CE) are widely used for the treatment of various cardiovascular diseases (arrhythmias, heart failure, myocardial weakness, etc). Despite the fact that many of these diseases are associated with disturbances of mitochondria, no data have been found on the effect of CE on their function. The aim of this study was to perform an oxygraphic investigation of the effect of CE (in concentration range from 70 ng/ml to 13.9 µg/ml of Crataegus phenolic compounds (PC)) and its several pure flavonoids on isolated rat heart mitochondria respiring on pyruvate+malate, succinate and palmitoyl-L-carnitine+malate. CE at doses under 278 ng/ml of PC had no effect on mitochondrial functions. At concentrations from 278 ng/ml to 13.9 µg/ml of PC, CE stimulated State 2 respiration by 11% -34% with all used substrates, and decreased the mitochondrial membrane potential by 1.2-4.4 mV measured with a tetraphenylphosphonium-selective electrode and H 2 O 2 production measured fluorimetrically. Similar uncoupling effects on mitochondrial respiration were observed with several pure CE flavonoids. The highest CE concentration also slightly reduced the maximal ADP-stimulated and uncoupled respiration, which might be due to inhibition of mitochondrial respiratory chain between flavoprotein and cytochrome c. Whether or not the uncoupling and other effects of CE on mitochondria may be realized in vivo remains to be determined.

INTRODUCTION

Cardiovascular diseases remain the principal cause of mortality in both developed and developing countries. Many of them are associated with the structural and functional disturbances in heart mitochondria. Since mitochondria make up 35% of the cell volume in the cardiac muscle cells and produce 95% of energy necessary for the heart functioning, their role in the cell is obvious. Therefore, the therapeutic agents that could influence mitochondrial dysfunction are of special importance, and this leads to an increased attention paid to the effect of various medicinal preparations on the mitochondrial structure and function.

The range of effective plant products for the target phytotherapy of cardiovascular system diseases is not wide. Crataegus monogyna Jacq. (Hawthorn) extract is among the most popular herbal medicinal products and has been used around the world since ancient times [START_REF] Fong | Hawthorn[END_REF]. Nowadays extracts of Crataegus fruit, leaves and flowers are used for the prevention and treatment of various cardiovascular diseases (hypertension, arrhythmias, congestive heart failure, coronary artery disease, myocardial weakness etc.) due to their positive inotropic and negative chronotropic effects, reduction in coronary spasm, blood pressure and total plasma cholesterol, and for their antioxidant and antiinflammatory properties [START_REF] Chang | Hawthorn: potential roles in cardiovascular disease[END_REF][START_REF] Rigelsky | Hawthorn: pharmacology and therapeutic uses[END_REF]. Such a wide pharmacological effect is due to polyphenol substances in the extract, consisting of flavonoids, triterpenes, phenolic acids, etc. Despite a number of preclinical and clinical studies, the mechanisms of the protective effects of Crataegus extract remain poorly understood. Moreover, there are no F o r P e e r R e v i e w 5 findings on its action on energetic metabolism of the cardiac muscle cells, which is essentially determined by the activity of mitochondria, ensuring the main physiological function of cardiac muscles, i.e. continuous mechanical contraction. Much more is known on the effects of polyphenols on mitochondrial function. Although their ability to protect cells from the "oxidative stress" (antioxidant activity) has been demonstrated, there is increasing evidence that the same polyphenol compounds could behave as both antioxidants and prooxidants, depending on the concentration and the source of free radicals [START_REF] Nemeikaite-Ceniene | Quantitative structure-activity relationships in prooxidant cytotoxicity of polyphenols: role of potential of phenoxyl radical/phenol redox couple[END_REF][START_REF] Laughton | Antioxidant and pro-oxidant actions of the plant phenolics quercetin, gossypol and myricetin. Effects on lipid peroxidation, hydroxyl radical generation and bleomycin-dependent damage to DNA[END_REF]. Because some of Crataegus flavonoids are the same as in Ginkgo biloba, one can assume that CE, like Ginkgo biloba extract [START_REF] Trumbeckaite | Effect of Ginkgo biloba extract on the rat heart mitochondrial function[END_REF], could possess the uncoupling activity on heart mitochondria, which was also demonstrated in our study (Trumbeckaite et al., 2006) for some pure flavonoids. However, it is difficult to predict the behavior of the same flavonoids in the natural extracts containing many different components, including those of unknown nature.

In this regard, it is interesting to note that moderate mitochondrial depolarization (uncoupling of oxidative phosphorylation) is known to attenuate the generation of mitochondrial reactive oxygen species (ROS) and could be beneficial in protection against cardiac ischemia-reperfusion injury [START_REF] Skulachev | Uncoupling: new approaches to an old problem of bioenergetics[END_REF][START_REF] Sack | Mitochondrial depolarization and the role of uncoupling proteins in ischemia tolerance[END_REF][START_REF] Halestrap | The role of mitochondria in protection of the heart by preconditioning[END_REF] and, possibly, against other pathological situations. ROS are formed within mitochondria under physiological and pathological conditions, particularly under ischemia/reperfusion. Therefore, the aim of this study was to investigate in vitro the direct influence of Crataegus fruit extract and some of its flavonoids on the respiration of isolated rat heart

MATERIAL AND METHODS

Preparation of CE and quantitative analysis. The dry fruits of Crataegus monogyna Jacq. were extracted (1:10) with 70% ethanol, the particle size was 2-3 mm, the production method was percolation, and the flow speed of the extract was 0.5 ml/min [START_REF] Bernatoniene | Production technology and analysis of blood circulation improving tincture[END_REF].

The content of total phenolic compounds in CE was determined spectrophotometrically (Hitachi 557) using the Folin-Ciocalteu method. 10 ml of appropriate dilutions of CE were added to a 50 ml volumetric flask containing 10 ml of distilled water, 4 ml of Folin-Ciocalteu reagent, and 6 ml of a 20% sodium carbonate.

Distilled water was added to increase the volume up to 50 ml. The color was developed for 2 hours at room temperature, the suspension was centrifuged at 5000 rpm, and the absorption of the obtained solution was measured at 760 nm wavelength. The measurement was compared to a standard curve of prepared gallic acid solution. Total phenolic contents are expressed as Gallic Acid Equivalent.

HPLC analysis was carried out using the Waters 2690 Alliance HPLC system (Waters Corporation, Milford, MA, USA), equipped with a Waters 2487 UV/Vis detector, an on-line degasser and an auto sampler, and a Waters XTerra RP18 150×3.9 mm column. UV detection was achieved at 360 nm on the detector. The chromatographic elution was accomplished by a gradient solvent system consisting of water containing 0.1% TFA (A) and acetonitrile containing 0.1% TFA (B). The gradient conditions were the following: 0 min, 95% A, 5% B; 45 min, 55% A, 55% B, kept to 50 min; 55 min, 95% A, 5% B at the flow rate of 400 µl/min. The injection volume was 10 µl. Buffer solutions were filtered through 0.2 µm disposable membrane filter (Roth, Karlsruhe, Germany) and degassed prior to use. Data were collected and analyzed using the Waters Millennium 2000® chromatographic manager system. The eluted constituents were identified by comparison of the retention time. Regression of the calibration curves of the reference standards was linear, correlation coefficient (R 2 ) of all curves was >0.9999, and the resolution (R s ) of standards peaks was >1.5.

Isolation of rat heart mitochondria. Male Wistar rats weighing 250-300 g were used for our study. The animals were killed according to the rules defined by the European Convention for the protection of Vertebrate Animals Used for Experimental and Other Purposes (License No.0006). Hearts of rats were excised and rinsed in ice-cold 0.9% KCl solution. Heart mitochondria were isolated in the medium containing 220 mM mannitol, 70 mM sucrose, 5 mM N-tris[Hydroxymethyl]methyl-2-aminoethane-sulfonic acid and 0.5 mM EGTA (pH 7.4, adjusted with Trizma base; 2 °C) and 2 mg/ml bovine serum albumin (BSA; fraction V, A4503, Sigma). The homogenate was centrifuged at 750 x g for 5 min, then the supernatant was re-centrifuged at 10,000 x g for 10 min, and the pellet was washed once (10 min 10,000 x g) in the isolation medium without BSA, suspended in it and kept on ice. The mitochondrial protein concentration was determined by applying the biuret method with BSA used as standard. 3) 9 µM palmitoyl-L-carnitine + 0.24 mM malate as substrates. The solubility of oxygen was estimated to be 422 nmol O 2 /ml. The activity of cytochrome c oxidase was measured with 10 mM ascorbate + 0.5 mM N,N,N',N'-tetramethyl-pphenylenediamine (TMPD) as electron donors. Mitochondrial respiration rates were expressed as nmol O 2 /min/mg protein. The final mitochondrial protein concentration in all experiments was 0.5 mg/ml. All stock solutions of pure flavonoids (Carl Roth GmbH, Karlsruhe, Germany) that were used in our study were prepared in 70% ethanol. Control measurements showed that the maximal used ethanol concentration (10 µl of 70% ethanol per 1 ml of the mitochondrial incubation medium) did not affect any functional parameters of mitochondria.

Measurement

Mitochondrial membrane potential was measured in a closed, stirred and thermostatically controlled 1.5 ml vessel fitted with a tetraphenylphosphonium (TPP + )selective electrode using a TPP + -binding correction factor of 0.162 (µl/mg) [START_REF] Brown | Proton/electron stoichiometry of mitochondrial complex I estimated from the equilibrium thermodynamic force ratio[END_REF]. The binding correction factor was determined from the ratio of Rb + to TPP + accumulation as a function of mitochondrial volume. The experiments were performed at 37 °C using 6 mM pyruvate + 6 mM malate as oxidizable substrate, and 0.5 mg/ml mitochondria in the same medium as mitochondrial respiration.

ROS generation was estimated as the release of H 2 O 2 from isolated rat heart mitochondria (0,25 mg/ml) fluorimetrically (Thermo Scientific). Mitochondria were F o r P e e r R e v i e w 9 incubated for 7 min at 37 0 C in the same medium as for mitochondrial respiration supplemented with pyruvate (5 mM), malate (5 mM) and antimycin A (0.016 µg/ml).

After incubation, Amplex Red (5 µM) and horseradish peroxidase (2U/ml) were added, and fluorescence (excitation at 544 nm, emission at 590 nm) was measured. Amplex Red fluorescence response was calibrated by adding known amounts of H 2 O 2.

Statistical analysis. Data are presented as means ± S.E.M. Nonparametric methods were applied for making inferences about the data. Differences between mean values in dependent groups were tested using Wilcoxon matched pairs test. Differences between mean values in independent groups were tested using nonparametric Kruskal-Wallis test with Dunns post-hoc evaluation. P < 0.05 was taken as the level of significance.

Statistical analysis was performed using the software package Statistica 1999, 5.5, StatSoft Inc., USA.

RESULTS

In this study we used Crataegus fruit extract that contained 139 ± 8 mg/100 ml of total phenolic compounds. Using the HPLC analysis method, we identified several classes of flavonoids in our investigated CE: flavonols (quercetin -1.26 ± 0.1 µg/ml, hyperoside -29.32± 0.5 µg/ml, rutin -2.64 ± 0.2 µg/ml, quercitrin -0.49 ± 0.01 µg/ml), flavanols (epicatechin -44.92 ± 0.9 µg/ml, procyanidin B 2 -0.71 ± 0.06 µg/ml), and phenolic acid (chlorogenic acid -3.46±0.03 µg/ml).

The effect of CE on the integrity and function of mitochondrial membranes was determined oxygraphically using isolated rat heart mitochondria. The State 2 respiration In the first series of experiments, we investigated the effect `of various concentrations of CE (70 ng/ml -13.9 µg/ml of PC) on the mitochondrial respiration with complex-I dependent substrates pyruvate+malate. CE was added into incubation medium after 1 min registration of State 2 or State 3 respiration. The percentage change in the mitochondrial respiration parameters is presented in Fig. 1. Our findings showed that CE at the lowest investigated concentration (70 ng/ml of PC) had no effect on any mitochondrial respiratory parameters. Higher concentrations of CE (278 ng/ml -6.95 µg/ml of PC) increased Vo (11-24%) but did not affect any other mitochondrial respiration rates (V ADP , V ADP+C ). The highest CE concentration used in our experiments (13.9 µg/ml of PC) induced a 34% increase in State 2 respiration rate and a decrease in 11 these results showed that the decrease in State 3 respiration rate at highest used CE concentration was not due to the loss of cytochrome c. Further data demonstrated that CE at the concentration 13.9 µg/ml of PC also suppressed the CCCP-uncoupled respiration rate with pyruvate+malate as substrates by 25-53% (data not shown). Thus, this demonstrates that the respiration rate at the highest CE concentration was decreased due to the inhibition of mitochondrial respiratory chain. Our experiments also showed that the activity of the terminal respiratory chain complex IV (cytochrome c oxidase) measured by recording ascorbate + TMPD oxidation was not affected by the highest CE concentration (control, 586 ± 41 nmol O 2 /min/mg protein; in the presence of CE, 627 ± 42 nmol O 2 /min/mg protein, n=4 measurements).

Mitochondrial membrane potential in State 2 was the same (163.2 ± 1.3 mV) in both control mitochondria oxidizing pyruvate + malate and at the lowest used CE concentration (70 ng/ml). At higher CE concentrations in the medium (278 ng/ml, 6.95 µg/ml and 13.9 µg/ml of PC), the mitochondrial membrane potential decreased by 1.2 mV, 2.5 mV and 4.4 mV (P<0.05, all versus control, respectively; n = 3).

Our fluorimetric measurements demonstrated that CE at the maximal (13.9 µg/ml of PC) and at the half-maximal (6.95 µg/ml of PC) concentrations decreased H 2 O 2 production in mitochondria by 62.8±12% and 35.3±7%, respectively (n=5, p<0.05; control rate of H 2 O 2 production was 0.31 ± 0.03 nmol/min/mg protein).

In the next series of studies, we investigated the effect of CE on the oxidation of FAD-specific substrate succinate and palmitoyl-L-carnitine, the main respiratory substrate of the heart (Table 1). CE extract at the concentration of 70 ng/ml of PC had no effect on V 0 . At higher CE concentrations (6.95 and 13.9 µg/ml of PC), V 0 increased by 12 27% and 30% with succinate, and by 14% and 27% -with palmitoyl-L-carnitine. Only at the highest used concentration of CE (13.9 µg/ml of PC), the maximal respiration rate (V ADP ) decreased by 12% and 19% with substrate succinate and palmitoyl-L-carnitine, respectively. Lower concentrations of CE had no effect on the State 3 respiration rate.

In further experiments we investigated the effect of some pure active substances present in the composition of CE on mitochondrial respiration with pyruvate+malate.

Their concentrations were identical to those found in CE at the maximal used amounts (13.9 µg/ml of PC). Rutin had the greatest effect (Fig. 2) -mitochondrial State 2 respiration rate was stimulated by 70%; epicatechin, procyanidin, hyperoside, quercetin, and quercitrin increased it by 17-30%, whereas the chlorogenic acid had no effect on State 2. Rutin and quercitrin had no effect on State 3 respiration rate whereas it was slightly inhibited (-11%) by hyperoside and quercetin (data not shown).

DISCUSSION

One of the main functions of mitochondria is to supply a cell with energy, which is required for maintaining the regular functioning of heart. Therefore damage to these organelles may be the key factor for myocardial dysfunction. For this reason, the role of mitochondria in physiological and pathological conditions and the effect of various medical preparations on the mitochondrial processes are intensively investigated. In this study we prepared the ethanolic extract from Crataegus monogyna fruit and determined its effect in vitro on the function of mitochondria isolated from rat hearts.

Our investigation showed that at concentrations of 278 ng/ml -13.9 µg/ml of PC, CE stimulated State 2 respiration by 11 % -34 %, similarly to all used Complex I and Complex II substrates, and statistically significantly (by 1.2-4.4 mV) decreased mitochondrial membrane potential. Thus, the decrease in mitochondrial membrane potential and the increase in V 0 indicates that CE in vitro possesses a slight direct uncoupling effect in isolated cardiac mitochondria. This effect was dose-dependent, yet it was practically independent of the respiratory substrate. It is interesting to note that the CE-induced uncoupling was clearly lower than that induced by Ginkgo biloba extract even at much lower concentrations of flavonoids, as it had been shown in our previous study [START_REF] Trumbeckaite | Effect of Ginkgo biloba extract on the rat heart mitochondrial function[END_REF]. It could be that certain unknown components affected (potentiated or inhibited) the flavonoid action.

It is known that "mild" uncoupling of isolated mitochondria, which leads to a slight decrease in mitochondrial membrane potential, declines ROS production [START_REF] Skulachev | Uncoupling: new approaches to an old problem of bioenergetics[END_REF][START_REF] Sack | Mitochondrial depolarization and the role of uncoupling proteins in ischemia tolerance[END_REF][START_REF] Halestrap | The role of mitochondria in protection of the heart by preconditioning[END_REF]. Therefore, it was proposed that such uncoupling is involved in the cellular defense system against oxidative stress. In good agreement with the data obtained using classical uncouplers of oxidative phosphorylation [START_REF] Skulachev | Uncoupling: new approaches to an old problem of bioenergetics[END_REF], our fluorimetrical measurements demonstrated that CE at concentrations causing mild uncoupling of oxidative phosphorylation also significantly decreased H 2 O 2 production in mitochondria. However, [START_REF] Brennan | Mitochondrial uncoupling, with low concentration FCCP, induces ROSdependent cardioprotection independent of KATP channel activation[END_REF] showed that adding low doses of an uncoupler (100 nM FCCP) prior to ischemia of an isolated perfused rat heart, increasing oxygen consumption (mild uncoupling) and ROS production but having no effect on the mitochondrial membrane potential, can also protect the heart against subsequent ischemia-reperfusion injury.

Several other studies showed that CE can be beneficial due to its antioxidant effects.

It was shown that 15-day oral administration of Crataegus flavonoids extracted from [START_REF] Zhang | Oral administration of Crataegus flavonoids protects against ischemia/reperfusion brain damage in gerbils[END_REF]. Oral pretreatment of rats for 7 days with Crataegus extract from leaves with flowers (10 mg/kg or 100 mg/kg) improved cardiac function and reduced infarct size in a rat model of prolonged coronary ischemia and reperfusion [START_REF] Veveris | Crataegus special extract WS 1442 improves cardiac function and reduces infarct size in a rat model of prolonged coronary ischemia and reperfusion[END_REF]. [START_REF] Jayalakshmi | Pretreatment with alcoholic extract of Crataegus oxycantha (AEC) activates mitochondrial protection during isoprotere[END_REF] showed that oral pretreatment of rats for 30 days with the ethanolic extract of Crataegus oxycantha fruit at a dosage of 0.5 ml/100 g body weight/day maintained mitochondrial antioxidant status, and prevented mitochondrial lipid peroxidative damage and decrease in Krebs cycle enzymes induced by isoproterenol in rat heart. Considerable antioxidant potential of water extracts of Crataegus aronia leaves and fruit (1 µg/mL -1 mg/mL) was also demonstrated on cultured Hep G2 cells [START_REF] Ljubuncic | Antioxidant activity of Crataegus aronia aqueous extract used in traditional Arab medicine in Israel[END_REF]. Noteworthy, during Crataegus extract administration by above mentioned research groups, the concentrations of total flavonoids varied widely, approximately between 20-170 µg/g body weight/day, thus it is difficult to compare these data to our in vitro study.

The much more drastic differences between doses were noted for pure flavonoids.

For example, the amount of quercetin administered to rats per os differed by about 24,000 times, i.e. from 0.033 mg/kg [START_REF] Brookes | Mitochondrial function in response to cardiac ischemia-reperfusion after oral treatment with quercetin[END_REF] to 100 mg/kg [START_REF] Guzy | Effect of quercetin on daunorubicin-induced heart mitochondria changes in rats[END_REF] and even 800 mg/kg per day (de Boer et al., 2005). Surprisingly, despite a 3,000fold difference of the amount of quercetin administered (i.e. at concentrations ranging from antioxidant to prooxidant) in the first two studies, cardioprotective effects in ischemia-reperfusion and daunorubicin-induced injury, respectively, were noted.

It should be also noted that the concentrations of CE used in our in vitro study were higher than therapeutic doses of ethanolic CE in human studies. However, they were 15 comparable to those used with rats in vivo [START_REF] Jayalakshmi | Pretreatment with alcoholic extract of Crataegus oxycantha (AEC) activates mitochondrial protection during isoprotere[END_REF]. The highest CE concentration used in our investigations (13.9 µg/ml of PC) not only increased State 2 respiration rate (similarly to all respiratory substrates), but also reduced the maximal ADP-stimulated mitochondrial respiration rate by 12-19% and the CCCP-uncoupled respiration rate with pyruvate+malate as substrates. Thus, this clearly demonstrates that higher concentrations of CE suppress some mitochondrial respiratory chain complexes.

Since the cytochrome c oxidase (complex IV) activity was not affected by this CE concentration, these data also suggest that the inhibition of electron transport in mitochondria is localized between flavoprotein and cytochrome c, i.e. between Complex I and III. However, we cannot exclude that ADP/ATP translocase might be inhibited to some extent as well. Concerning ATP synthase (complex V), it was revealed that some polyphenolic phytochemicals bind to and inhibit mitochondrial ATP synthase [START_REF] Gledhill | Mechanism of inhibition of bovine F1-ATPase by resveratrol and related polyphenols[END_REF][START_REF] Zheng | Inhibition of mitochondrial proton F0F1-ATPase/ATP synthase by polyphenolic phytochemicals[END_REF]; however its inhibition in our experiments seems unlikely due to much lower flavonoid concentrations used.

Since some of CE polyphenols may damage membranes and cause cytochrome c release from mitochondria, we used cytochrome c test in order to check this possibility and elucidate the reason for the inhibition of mitochondrial respiration. Our findings showing that the degree of external cytochrome c-induced stimulation of mitochondrial respiration (V ADP+C /V ADP ) was similar in the control group and after the addition of CE, denied such possibility.

It is known that inhibition of mitochondrial respiration by various respiratory chain inhibitors augments ROS production and oxidative stress that occurs under the conditions of inhibitor-diminished membrane potential [START_REF] Byun | Mitochondrial dysfunction by complex II inhibition delays overall cell cycle progression via reactive oxygen species production[END_REF], and could also protect 16 the cells at ischemia and reperfusion [START_REF] Halestrap | The role of mitochondria in protection of the heart by preconditioning[END_REF][START_REF] Kabir | Cardioprotection initiated by reactive oxygen species is dependent on activation of PKCepsilon[END_REF]. In contrast, the treatment of isolated, perfused rabbit heart with rotenone (mitochondrial complex I inhibitor) decreased the production of ROS in mitochondria and preserved the contents of cardiolipin and cytochrome c measured after 45 min of ischemia [START_REF] Chen | Modulation of electron transport protects cardiac mitochondria and decreases myocardial injury during ischemia and reperfusion[END_REF]. Interestingly, in isolated rat cardiocytes, antimycin A-caused ROS production was abolished by the mitochondrial uncoupler [START_REF] Kabir | Cardioprotection initiated by reactive oxygen species is dependent on activation of PKCepsilon[END_REF]. Thus, some inhibition of mitochondrial respiration observed at the highest CE concentration can be beneficial in some circumstances and may complement the effect of uncoupling. The blockade of electron transport and the partial uncoupling of respiration were regarded as two mechanisms whereby manipulation of mitochondrial metabolism during ischemia decreases cardiac injury [START_REF] Chen | Modulation of electron transport protects cardiac mitochondria and decreases myocardial injury during ischemia and reperfusion[END_REF].

Investigations concerning the action of individual classes of flavonoids, namely chalcones, flavones, and flavonols, showed that at higher concentrations (100 µM ) they inhibit beef heart mitochondrial succinatoxidase, NADH-oxidase, ATPases, and can have a pro-oxidant effect [START_REF] Hodnick | Inhibition of mitochondrial respiration and cyanide-stimulated generation of reactive oxygen species by selected flavonoids[END_REF]. Flavonoids at lower concentrations (IC 50 ≤1.5 µM) possess antioxidant activity and do not inhibit cellular enzymes [START_REF] Laughton | Antioxidant and pro-oxidant actions of the plant phenolics quercetin, gossypol and myricetin. Effects on lipid peroxidation, hydroxyl radical generation and bleomycin-dependent damage to DNA[END_REF].

In our study, we also tested the effect of some pure flavonoids present in the CE at the maximal concentration (13.9 µg/ml of PC) on the heart mitochondrial respiration with pyruvate+malate. We found (Fig. 2) that epicatechin (1.5 µM), procyanidin (0.012 nM), hyperoside (0.96 µM), quercetin (45 nM), and quercitrin (5.3 nM) stimulated State 2 respiration by 18-30%, i.e. similarly to the CE. Rutin (47 nM) increased it more effectively -by 70%. Thus, it seems likely that the effect of CE is mainly determined by [START_REF] Hodnick | Inhibition of mitochondrial respiration and cyanide-stimulated generation of reactive oxygen species by selected flavonoids[END_REF], and not optimal to achieve the maximal increase in State 2 respiration rate (Trumbeckaite et al., 2006).

It still remains to be determined whether or not the effects of CE may be realized in vivo -through uncoupling of oxidative phosphorylation in mitochondria observed in our in vitro studies or via other mechanisms. This issue is complicated due to several circumstances. It is known [START_REF] Brookes | Mitochondrial function in response to cardiac ischemia-reperfusion after oral treatment with quercetin[END_REF] that polyphenols, after intake, undergo various conversions, but the effects of most metabolites on mitochondria are not known.

Moreover, because of the long half-lives of elimination and the hydrophobic nature of polyphenols, repeated consumption of polyphenol-containing products (food or CE) should cause their accumulation in blood and membranes, including those of mitochondria. Since the daily dietary flavonoid intake (1) varies considerably in different populations (65-250 mg/day -for Western populations) [START_REF] Erdman | Flavonoids and heart health: proceedings of the ILSI North America Flavonoids Workshop[END_REF] and, in addition, (2) there exist very large blood, tissue and species differences with regard to the distribution and concentrations of flavonoids and its metabolites (de Boer et al., 2005), including differences in plasma half-life for these compounds in various species, differences in the background concentrations of flavonoids in the cells should be expected. Accordingly, it should entail the differences in the effect of CE on various subjects and tissues. Thus, to facilitate the solution of a problem/interpretation of the data, it seems reasonable that the concentrations of flavonoids in the plasma, cells, and mitochondria should be determined before and after treatment with CE. Based on the effects of chronic respiratory uncoupling of Hep G2 cells by classical uncoupler dinitrophenol (10 and 50 µM, 3 days) -an increase in cellular oxygen consumption, 18 oxidative capacity and cytochrome c oxidase activity -which was associated with an upregulation of its subunit IV and adenine nucleotide translocase 3 gene expression [START_REF] Desquiret | Dinitrophenol-induced mitochondrial uncoupling in vivo triggers respiratory adaptation in HepG2 cells[END_REF] -one may assume that long-term treatment with CE may induce mitochondrial biogenesis (for rev. see Tapia, 2006), which would be beneficial in heart failure treatment. On the other hand, large numbers of protein kinases (that have been reported as being potential targets for flavonoids due to their ability to bind to ATP sites on enzymes) as well as other accumulating evidence suggest that the cellular effects of flavonoids may be mediated by their interactions with specific proteins central to intracellular signaling cascades [START_REF] Williams | Flavonoids: antioxidants or signalling molecules?[END_REF].

In sum, our in vitro findings clearly demonstrated that CE and its main biologically active components cause partial uncoupling of oxidative phosphorylation and suppression of H 2 O 2 production in the isolated rat heart mitochondria. It remains to be determined whether these effects may be realized in vivo and thus could reduce the generation of free radicals within mitochondria, as well as whether slight mitochondrial respiratory chain inhibition plays any significant role in cardioprotection. The reagents were added in the following order: V 0 -6 mM pyruvate + 6 mM malate;

F

V ADP -1 mM ADP; V ADP+C -respiration rate in the presence of 32 µM cytochrome c; RCI -respiratory control index; number of experiments -3; *p<0.05 vs control.

  to test for their uncoupling potential and get more insight into the mechanism of their action.

  0 ) with different substrates was monitored for 1 min and then various amounts of CE were added in order to assess their effect on the State 2 respiration rate. Subsequently, ADP was added, and the maximal respiration in State 3 (V ADP ) was recorded. It reflects the maximal capacity of respiratory chain and ATP production rate. In order to check the intactness of the outer mitochondrial membrane, we used cytochrome c test and for this we added exogenous cytochrome c in State 3. The degree of respiration stimulation by cytochrome c (V ADP+C /V ADP ) allows for assessing the integrity of the outer mitochondrial membrane. Changes in V 0 show changes in the permeability of the inner mitochondrial membrane to protons or ions, and, together with calculated respiratory control index (RCI = V ADP /Vo), allows for determining the effectiveness of ADP phosphorylation in mitochondria.

  State 3 respiration rate by 18%, leading to the most pronounced decrease in RCI. The decrease in State 3 was not restored after the addition of exogenous cytochrome c. Thus,

  production and protected the brain against delayed cell death caused by ischemia/reperfusion injury

  Noteworthy, these concentrations were far lesser than those mentioned above

Fig. 1 .

 1 Fig. 1. The effect of different concentrations of Crataegus fruit extract on the

Fig. 2 .

 2 Fig. 2. The effect of the main active compounds of Crataegus fruit extract on

  of mitochondrial respiration, membrane potential and H 2 O 2 generation. Oxygen uptake rates were recorded at 37 o C by means of the Clark-type

	electrode system in a solution containing 20 mM imidazole, 20 mM taurine, 0.5 mM
	dithiothreitol, 1.6 mM MgCl 2 , 100 mM MES, 3 mM KH 2 PO 4 , 3.0 mM CaK 2 EGTA, and
	7.1 mM K 2 EGTA (free Ca 2+ concentration -0.1 µM) (pH 7.1 adjusted with KOH at
	37 o C) separately (1) with 6 mM pyruvate + 6 mM malate, (2) 12 mM succinate (+2 mM
	amytal), and (
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Table 1 .

 1 Dihal AA, van der Woude H, Arts IC, Wolffram S, Alink GM, Rietjens IM, The effect of Crataegus fruit extract on mitochondrial respiration Measurements were performed in the presence of 12 mM succinate (+2 mM amytal) or 9 µM palmitoyl-L-carnitine + 0,24 mM malate as substrates (V 0 ); Further additions : 1 mM ADP (V ADP ); 32 µM cytochrome c (V ADP+C ); RCI -respiratory control index; number of experiments -3; * -P<0.05 vs control.

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Acknowledgements

This work was supported by the Foundation of Kaunas University of Medicine and by the Agency for International Science and Technology Development Program in Lithuania.