Milan Randić
email: mrandic@msn.com

Marjana Novič

Marjan Vračko

Dejan Plavšić

Study of proteome maps using partial ordering

Keywords:

come

Study of proteome maps using partial ordering

Milan Randić, Marjana Novič, Marjan Vračko,

INTRODUCTION

When an experimental animal is exposed to drug or toxin, the protein content of cells will change and may change dramatically and in an unpredicted way. It is of interest to characterize variations of the relative abundance of proteins from that which characterizes a normal cell, and if possible try to related the perturbations in altered proteome map to variations in agents causing the change. Such work may eventually show the extent to which Emil Fishers "similar drugs have similar effects" extends also to the collection of proteins, cellular proteome. To facilitate characterization of changes of protein composition in cells one needs to characterize individual proteome maps by a set of map descriptors, which are mathematical invariants derived from proteome map. In this communication we will outline novel numerical map descriptors for characterization of proteome maps, which are conceptually and computationally not very involved.

Typical proteome map may display in 2D gel 2000 protein spots, proteins separated by charge and mass. The experimental information of proteome maps are usually represented as a list, the x-coordinate giving the charge and the y-coordinated giving the mass, followed by the relative abundance of individual spots. At left in Table 1 we have illustrated such results for twenty most intensive spots of normal rat liver cells, while the remaining part of Table 1 lists the abundances of the same 20 proteins for proteome maps obtained from liver cells obtained from animals exposed to four chemical agents. The experimental data listed in Table 1 are from the work of Anderson and coworkers (1996) who were studying the effect of several peroxisome proliferators on rats liver cells. Fig. 1 shows the locations of the twenty most abundant protein spots labeled 1 -20 in 2D gel, the coordinates x, y of which are listed in Table 1.

-----------------Table 1 around here -----------------------------------------Fig. 1 around here ------------------------Our interest is in developing a "tool" for comparative study of proteome maps. So for such work neither the number of selected points nor necessarily the criteria of selection is essential. However, because of unknown variability of experimental data we focused on the most abundant proteins, the relative experimental errors for which are expected to be the least.

The question on the number of protein spots that may suffice to represent a map or cellular proteome as a whole has been considered [START_REF] Randić | Dependence of a characterization of proteomics maps on the number of protein spots considered[END_REF][START_REF] Randić | Quantitative characterization of proteome: Dependence on the number of proteins considered[END_REF] and appears to be an order of magnitude larger, but as mentioned above for the purpose of developing novel approaches for quantitative characterization of proteome maps the number of proteins considered is not critical.

Proteome maps invariant are quantities that are independent of the labeling of the proteins in proteome maps or 2D gel. Such quantities characterize a map as a whole or more correctly they characterize simplified map consisting of proteins selected for numerical characterization of a map. Obviously if one is to capture salient feature of information-rich proteome maps one needs several map descriptors. This paper is a continuation of our efforts to develop suitable mathematical characterization of proteome maps [START_REF] Randić | On graphical representation of proteomics and their numerical characterization[END_REF]Randić et al., 2001;[START_REF] Randić | A graph theoretical characterization of proteomics maps[END_REF]Randić et al., 2002;Randić and Basak, 2004;Randić et al., 2005). In particular in this article we will introduce novel invariants constructed by using partial ordering of proteins with respect to their charge and mass and the partial ordering diagram embedded over proteome maps.

PARTIAL ORDERING BASED ON CHARGE AND MASS

While ordering of objects with respect to a single property is trivial, ordering of objects with respect to two or more properties is far from trivial. Scottish mathematician Muirhead around the year 1900 initiated such studies that have lead to rules for construction of partial ordering [START_REF] Muirhead | Some Methods Applicable to Identities and Inequalities of Symmetric Algebraic Functions of Letters[END_REF][START_REF] Hardy | Inequalities[END_REF]. Partial ordering has not been much used in chemistry and is not widely known methodology in chemistry despite that it offers a useful tool for comparison of objects characterized by sequences. Among few applications of partial ordering in chemistry we may mention study of variations in properties of isomers isomers (Randić and Wilkins, 1979), and the search for a characterization of pharmacophore in QSAR [START_REF] Randić | The search for active substructure in structure-activity studies[END_REF][START_REF] Randić | On characterization of pharmacophore[END_REF]. For recent developments of partial orderings in chemistry see the special issue of MATCH (Klein and Brickmann, 2000).

In this contribution we will use partial order to construct a graph to be associated with a proteome map. Having a graph one can construct its adjacency matrix, which is a binary matrix with zero and one as elements, the ones indicating adjacent vertices in the graph. Selected invariants of adjacency matrix can be used as map descriptors. An advantage of using partial ordering graph for construction of map descriptors is that minor variations in experimentally measured mass and charge of proteins spots will have no influence on our analysis. For earlier applications of partial ordering for characterization of proteomics maps see [START_REF] Randić | A graph theoretical characterization of proteomics maps[END_REF] and Randić and Basak (2004).

To obtain the diagram of partial ordering one order the twenty spots first of Table 1 with respect to their charge and then with respect to their mass. The units of charge and mass are those as reported by [START_REF] Anderson | The effect of peroxisome proliferators on protein abundances in mouse liver[END_REF]. By ordering protein spots with respect to mass we obtain the sequence: 1, 15, 20, 7, 5, 9 . . . This is the order as spots appear in the map when looking from the top of the map (proteins having the greatest mass) to the bottom (proteins having the smallest mass). When one orders the same protein spots with respect to the charge one obtains the sequence: 12, 8, 15, 2, 17, 5 . . . which reflects the order in which protein spots appear if we go from right (the largest charge) to left (the smallest charge).

Partial ordering is defined as a set of all orderings involving subsets of elements of the two sequences that are true for both sequences. To find all such partial orderings it is convenient to list, as shown in Fig. 2, proteins in order of descending mass and descending charge. Once the two sequences are obtained, one connects the same protein in both sequences. Any set of noncrossing lines that one can extract from such diagram represents one partial ordering, that is, the ordering is satisfied for so selected proteins for both sequences. For example, the lines 15 -15, 5 -5, 18 -18, 4 -4, 16 -16, 6 -6, and 14 -14 do not cross giving the sub-sequence 15 > 5> 18 > 4 > 16 > 6 > 14, which means that protein 15 has bigger mass and bigger charge than protein 5, and protein 5 has bigger mass and bigger charge than protein 18, and so on. In Table 2 we have listed only all such fragmentary orderings that start with protein spot 1. The totality of all fragmentary orderings makes the Partial Ordering for the selected twenty proteins of map of Fig. 1.

-----------------Table 2 around here - ----------------------------------------Fig. 2 around here ------------------------It is customary to graphically represent partial ordering by a directed graph or a hierarchical diagram, such as one illustrated in Fig. 3 in which points at right dominate points to the left. The dominance here means that an element at the left in Fig. 3 is also below in both sequences of Fig. 2. Fig. 3 facilitates visual inspection of the partial ordering and immediately shows which protein dominate (by the mass and by the charge) other proteins, or is dominated by others. In general when depicting partial ordering one can place vertices in arbitrary positions as long as the top-bottom and the left-right dominance relations are preserved, but for better visibility one tries to construct diagram that will have as few as possible crossings of lines.

-----------------Fig. 3 around here ------------------------In the approach considered here we depart from the usual practice outlined above and we insist on construction of the partial ordering diagram that is directly superimposed on the proteome map. This is in fact the diagram already illustrated in Fig. 3. This is a critical step, because we want to model underlying proteome maps, and partial ordering is used only to arrive at suitable geometrical object for our considerations. This approach has been for the first time introduced for representation of partial ordering diagrams of proteome maps in ref. [START_REF] Randić | On graphical representation of proteomics and their numerical characterization[END_REF]. Such embedding of graphs in 2-D space (plane) in Graph Theory implies a fixed geometry for the graph considered, which means that in addition to adjacency relationship between vertices in such graphs are also determined the distances between all vertices, whether adjacent or not.

Observe an important feature of the embedded graph of Fig. 3 in that all the lines in the graph have positive slope. This is a consequence of the dominance relation, and is a property that can be used for a direct construction of partial ordering diagram for a given map without a need for construction of Fig. 2 and the accompanying search for components of partial ordering. To obtain Fig. 3 directly from Fig. 1 one can start with the top vertex (spot 1 in Fig. 3) and connect it to the next lowest spot to left of 1 which is spot 7. One continues the same with vertex 7 and connects it to the next lowest vertex below it and to left, which is spot 3. One continues to connect 3 to 19 and finally 19 to 14. By exhausting this particular trail one return to vertex A and repeat the process: one connect 1 to 9 and then 9 to 3. In next step we connect 1 to 6 and finally 6 to 14. By backtracking one connects 6 to 11. Finally one connects 1 to 16 and 1 to 13, which are connected to 6 and 11, respectively. This has exhausted all the fragmentary orders starting with protein spot 1, which have been listed in Table 2. The process continues with spot 15, then 8 and 12, which completes construction of the embedded graph of partial order for the map considered. In passing we may mention that the partial ordering diagram depicted in ref. [START_REF] Randić | On graphical representation of proteomics and their numerical characterization[END_REF], which considers the same twenty proteins, has a missing link that should connect protein 2 (2804.3, 903.6) and protein 16 (2032.7, 902.8), which was apparently overlooked visually because of such a small difference in the two y-coordinates,

THE ADJACENCY MATRIX OF PROTEOME MAPS

Adjacency matrix is well known binary matrix of Graph Theory [START_REF] Harary | Graph Theory[END_REF], the matrix elements of which are defined as:

A ij = 1 if vertices i and j are adjacent;

A ij = 0 otherwise.
For the graph of partial ordering of the proteome map of Fig. 1, illustrated in Fig. 3, the 20x20 adjacency matrix is shown in Table 3.

-----------------Table 3 around here ------------------------Our numerical characterization of the five proteome maps of Table 1 will be based on augmented adjacency matrix, in which we will replace the diagonal zeros by the corresponding relative abundances for individual proteome maps. The x, y coordinate do not enter directly into analysis but determine the adjacency of the spots, while we will incorporate the abundance of proteins by augmenting the adjacency matrix by introduction of non-zero diagonal terms.

Whenever experimental quantities are measured in different units, as is the case with proteome maps, which involve mass, charge and abundance, if one is to use experimentally measure numerical values in a subsequent analysis one has to consider suitable normalization, in order that one of the properties does not dominate the other, because of use of different scales. [START_REF] Kowalski | A powerful approach to interpreting chemical data[END_REF] recommended that in such situations one should re-scale the units used for measuring these quantities and scale them both to a same interval, such as (-1, +1). In our case we have three quantities, the protein charge (coordinate x), the protein mass (coordinate y) and the protein abundance (coordinate z) that we would like to combine into our analysis.

In view that the coordinates x and y of proteome maps are of the same order of magnitude, even if they would directly enter into our analysis we could skip their rescaling.

However, we have to consider the suggestion of Kowalski and Bender about scaling of the third coordinate (abundance). Observe that the approach of Kowalski and Bender does not consider a possibility that the number of points considered may also have an effect on the analysis, which is the case with use of augmented matrices (to be introduced later). Their approach is adequate only in cases when analyses are using augmented matrices of the same size. This, however, is not the case in our analyses of proteome maps, because on different occasions, and even for the same proteome maps, one may change the number of proteins considered and thus change the size of matrices considered.

For augmented matrices there is a problem that the same (0, 1) scaling will give different weight to diagonal and off-diagonal matrix elements. If diagonal entries are normalized on (0, 1)

scale (assuming the we consider adjacency matrix, whose elements are 0 or 1) as the size of matrices increases the number of off-diagonal entries increases faster than the size of diagonal entries and off-diagonal entries will play more and more dominant role with respect to the role of diagonal entries. In general the number of off-diagonal entries will increase with n 2 while the number of diagonal entries will increases with n, where n is the size of adjacency matrix.

As outlined by Randić and Orel (2010) there is a way to arrive at a matrix in which both the off-diagonal entries and the diagonal entries will have balance roles. This is achieved by having the average matrix element (excluding diagonal elements) and the average trace element equal, or which is the same, having the sum of all matrix elements (excluding diagonal) equal to the sum of all diagonal elements. The sum of off-diagonal entries for the adjacency matrix of Table 3 is 72, so the abundance of Table 1 have to be multiplied by 72/(Abundance sum), which is for the proteome maps of the control group 72/ 27758.54 = 0.002594. The so normalized abundances for the five proteome maps of Table 1 are listed in Table 4.

-----------------Table 4 around here ------------------------

In the last row in Table 4 we have listed the so scaled abundance sums for the five maps.

As expected for the control group the sum equals 72, because the graph of Fig. 3 -----------------Fig. 4 around here ------------------------From Table 4 it is interesting to observe individual variations in proteins. Thus protein 15 in the case of PFOA and PFDA has decreased its abundance considerably but in the case of DEHP it has increased its abundance. In many cases there are considerable changes in abundances in comparison with that of the control group even if the changes are within the bounds of ± 50 %. In some cases abundances of proteins increased slightly after exposure to the four agents (like proteins 2 and 11) assuming that the changes are statistically significant.

Similarly some proteins diminished their abundance, although often not evenly (like proteins 7, 9, and 17). Protein 14 appears to be among the least affected by any of the four agents considered.

We are going to try to construct numerical characterization of the five proteome maps.

We have selected to consider the eigenvalues and eigenvectors of the augmented adjacency matrices. In Table 5 we have listed the coefficients of the eigenvector accompanying the leading eigenvalue, the largest eigenvalue, obtained from the five augmented adjacency matrices. We will refer to the columns of Table 5 as the leading eigenvectors. Observe how the coefficients of the leading eigenvectors for the five proteome maps, which are illustrated in the bottom part of Fig. 4, parallel the relative abundances of the proteins illustrated at the top of Fig. 4. This parallelism suggests that the leading eigenvectors may offer a fair numerical characterization of the proteome maps. While the leading eigenvalues have found use as molecular descriptors for study of structure-property relationship, this is the first time that the leading eigenvectors have found useful application in chemical graph theory and bioinformatics.

-----------------Table 5 around here ------------------------

In Table 6 we show the degree of similarity/dissimilarity for pair wise comparisons of the five proteome maps based on the degree of similarity/dissimilarity for the corresponding leading eigenvectors. The values in the table were computed by viewing columns in Table 4 as a 20component vector. The Euclidean distance (in 20-dimensional vector space) gives the distance between the corresponding end-points of vectors. The smaller is the distance the more similar are vectors (or alternatively the more similar are the corresponding proteome maps). The first row gives the similarity of the four perturbed maps with the map of the control group. As one see between the four peroxisome proliferators clofibrate and DEHP cause the least perturbation of liver cell proteome, while PFDA is causing the largest perturbation. Between all five proteome maps the most similar proteome maps are those of clofibrate and DEHP. However, above comparisons may obscure details on how each agent affects individual protein types, and the overall similarity does not imply that neither the two chemicals have necessarily similar affect on all proteins, nor that the magnitudes of the overall perturbation will manifest in similar biological behavior. That the former is the case is illustrated, for instance PFOA and PFDA, which chemically differ only somewhat in their size, in that PFOA makes little change on the abundance of protein 5, while PFDA has drastically reduced the abundance of protein 5 in liver cells.

-----------------Table 6 around here ------------------------Clearly a single descriptor, such as the 20-component vectors representing the leading eigenvector cannot capture complexity of variations occurring even for a relatively small maps such as map considered here that is based on information on only twenty proteins. It is therefore desirable to have additional descriptors, one such additional descriptor will be outlined in the next section.

THE EIGENVALUE OF MAP MATRICES

We continue to consider the five proteome maps for control groups, PFOA, PFDA, clofibrate and DEHP of Table 1, and will examine use of eigenvalues of the augmented adjacency matrices as map descriptors. In Table 7 are listed (in the reversed order) eigenvalues of the augmented adjacency matrices for the normal (control) rat liver cells and four peroxisome proliferators. One can view the columns in Table 7 as vectors in 20-dimensional vector space and one can construct the similarity/dissimilarity matrix by calculating the Euclidean distances between these vectors. Before reporting the results let us mention that there are no many illustrations that the set of eigenvalues of structural matrices have been offering suitable descriptors for molecular structure-property studies. Even less this would be the case for characterization of complex chemical and biological systems [START_REF] Randić | On graphical representation of proteomics and their numerical characterization[END_REF], Randić et al., 2001;Randić and Basak, 2004). An exception are the leading eigenvalue of few matrices, which could be interpreted in term of structural concepts. Thus Lovász and Pelikan (1973) interpreted the leading eigenvalue of the adjacency matrix as an index of branching in acyclic graphs, and the leading eigenvalue of the D/D matrix (the elements of which are given as the quotient of the through space and through bonds distances) as an index of bending of chain structures [START_REF] Randić | Distance/Distance matrices[END_REF], and more generally an index of compactness for molecular graphs in general Randić and Plavšić, 2010). Also the leading eigenvalue of path matrices [START_REF] Randić | On structural ordering and branching of acyclic saturated hydrocarbons[END_REF]Randić et al., 2000), which are special case of graphical matrices [START_REF] Randić | Double invariants[END_REF], Randić et al., 2004) is very good index of molecular branching.

-----------------Table 7 around here ------------------------When the eigenvalues of Table 7 are used as components of vectors in 209D space one obtains the entries in Table 8 as their Euclidean distances. This table can be used as an alternative measure of the degree of similarity or dissimilarity between the five proteome maps.

Some parallelism but also some differences can be observed when Table 8 is compared to Table 6 in which the similarity/dissimilarity between the five proteome maps was based on the leading eigenvectors, rather than the set of eigenvalues. This time one finds as the most similar proteome maps those of PFOA and clofibrate. Other small entries in Table 8 are between the control and DEHP, PFOA and PFDA and control and clofibratre.

The parallelism between the similarity/dissimilarity of the five proteome maps based on leading eigenvector and all eigenvalues is illustrated in Fig. 5.

-----------------Table 8 around here -----------------------------------------Fig. 5 around here ------------------------

DISCUSSION

That there is parallelism between the characterization of proteome maps by the leading eigenvector and the set of all eigenvalues is interesting but need not be fully surprising. The eigenvalues if placed on the diagonal of a 20x20 null-matrix represent a transformation of the initial augmented adjacency matrix into diagonal form. On the other hand the experimental abundances, which we have included as the diagonal entries, apparently have played the dominant role in the matrix eigenvalue problem. One can view the augmented adjacency matrix as a diagonal matrix that is perturbed, the perturbation being present in the form of the offdiagonal elements. This suggests that the agreement between the two characterizations of the proteomics maps will agree even more if we are further to emphasize the role of diagonal elements. The limiting case of completely negligible role of the off-diagonal matrix elements correspond to transforming the problem of characterization of proteomics maps to that of characterization of cellular proteome, because the role of the x, y) coordinates has been eliminated. That the analysis of proteome map and proteome, though related, is distinct is well illustrated by a paper of [START_REF] Randić | Order from chaos: Observing hormesis at the proteome level[END_REF], who examined the reported variations of protein abundances as concentration of particular peroxisome proliferator was changes over two orders of magnitude. Thus, by ignoring the (x, y) coordinates they were able to detect a nonlinear dose response at the cellular level, the phenomenon that has been known previously to holds for an organism (animal, plant) as a whole [START_REF] Calabrese | Toxicology rethinks its central belief[END_REF].

The similarity/dissimilarity tables such as Table 6 and Tables 8 do show some differences which illustrate the fact, that even though two approaches are based on similar methodologies, they capture somewhat different map information. As already mentioned the smallest entry in Table 6 belongs to the pair (clofibrate, DEHP), which is not the case with Table 8 where the smallest entry is associated with the pair (PFOA, clofibrate). Similar results have been found with use of "higher order" matrices and 10-D and 15-D vectors, respectively in [START_REF] Randić | A comparative study of proteomics maps using graph theoretical biodescriptors[END_REF]. A relatively small value of the Euclidean distance entry in a similarity/dissimilarity table does not necessarily imply that the effects of the corresponding chemicals are the very similar, but it signifies that the effect of those chemicals on proteome on average may be similar. Even though we used a small number of proteins we have already seen that in number of instances the same proteins have been affected differently so it would be premature to speculate that higher similarity between molecules will necessarily be reflected to high the degree of parallelism, but such conclusion have to wait on experimental data derived from use of more similar compounds.

It remains to be seen modification of this or other approaches may be more suitable when comparing different proteome maps. At this stage it is important to recognize that the presented approach has some flexibilities. One should recall that here we chose to consider a particular set of invariants, the leading eigenvector and the set of eigenvalues of augmented adjacency but other matrix and graph invariants ought to be examined. Important novelty of the current approach is that we found a scheme that allows experimental data to compact into matrices that thus become the source for further exploration of comparative proteome.

As already hinted to relate similarity /dissimilarity among proteome maps with similarities/dissimilarities of agents causing perturbations would be facilitated by having data on a set of chemical compounds showing closer structural features. If there is some parallelism between the biological descriptors of proteome maps and physico-chemical descriptors inducing changes in proteome one could test the degree to which the paradigm that similar chemical structures induce similar properties, that holds in QSPR (Quantitative Structure Property Relationship) and QSAR (Quantitative Structure Activity Relationship), also applies to proteome, even if proteome data may point to drastic changes of the cell proteome under minor perturbations of biological systems. Another testing ground for outlined numerical characterizations of proteome maps is examination of perturbations of proteome induced by gradual increase of concentrations of chemicals that produce visible perturbations of proteome, which is currently underway in our laboratory. We should add that all the calculations were made using Excel and the standard matrix manipulations available in MATLAB software (MathWorks, 2010).

CONCLUDING REMARKS

Here we have tested two measures of "proteome similarity" of the chemicals based on augmented adjacency matrices that include the information on the abundances of protein spots.

One approach considers the "leading" eigenvectors (eigenvectors belonging to the leading eigenvalue which have all coefficient positive) and the other consider eigenvalues as descriptors.

These two measures give similar but different pictures of the similarity of the effects of four peroxisome proliferators on rat liver cell proteome. This brings us close to the well-known problem of similarity methods for molecules: Similarity methods are not uniquely defined, they are rather user-defined and context specific [START_REF] Randić | Design of molecules with desired properties. Molecular similarity approach to property optimization[END_REF][START_REF] Randić | Similarity methods of interest in chemistry[END_REF]. In the area of molecular similarity for relatively small molecules, The adjacency matrix for the partial ordering graph of Fig. 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 3 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 5 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 6 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 7 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 9 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 10 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 11 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 13 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 15 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 16 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 17 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 18 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 19 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 20 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0

  has 36 edges, each being counted twice in the adjacency matrix. The last row in Table4immediately shows the overall effect of peroxisome proliferators on the liver cells of experimental animals. We see an overall decrease of most abundant twenty proteins for three peroxisome proliferators (PFOA [perfluorooctanoic acid], PFDA [perfluorodecanoic acid], and clofibrate) and an increase for the peroxisome proliferator DEHP [di(2-ethylhexyl)phthalate]. Observe that proteome maps having the same proteins (the same list of the (x, y) coordinates) are scaled using the same normalization factor (based on the map of the control group).AUGMENTED MATRICESWhat makes the five proteome maps different is the information on abundance of proteins as measured by the intensity of gel spots. This information differentiates maps and will reflect the role of drugs and other xenobiotics agents that induce changes in the proteome maps. We will incorporate the information on abundance as diagonal entries of the matrix in analogy with differentiation of heteroatoms in construction of variable connectivity indices(Randić. 1991;[START_REF] Randić | Optimal molecular connectivity descriptors for nitrogen-containing molecules[END_REF] Randić et al., 2000). This approach has been outlined in the introductory papers on the mathematical characterization of proteome maps based on 2D zigzag pattern of proteome map and its generalization to 3-D zigzag pattern[START_REF] Randić | On graphical representation of proteomics and their numerical characterization[END_REF], Randić et al., 2001; Randić and Basak, 2004), however normalization used there were not chosen to parallel matrix size-dependence.The selected twenty most abundant spots of the control group determine selection of proteins for the remaining proteome maps of PFOA, PFDA, clofibrate and DEHP. While the relative abundance to the control group steadily decrease the relative abundances of the remaining four maps oscillate at an irregular pattern illustrated at the top of Fig.4. The most abundant appear to be protein having label 6 for liver cells exposed to PFDA (2.217), and the least abundant among proteins considered is protein 3 for liver cells exposed to DEHP (0.081).

Fig. 1 Fig. 2

 12 Fig. 1 Location of twenty most abundant proteins for proteomics maps of the control group

Fig. 3

 3 Fig. 3 Partial ordering diagram for twenty proteins of Fig. 1

Fig. 4 Fig. 5

 45 Fig.[START_REF] Randić | On graphical representation of proteomics and their numerical characterization[END_REF] The experimental and calculated abundances for twenty most abundant proteins of the

Fig. 1

 1 Fig.1

Fig. 2

 2 Fig.2

Fig. 3

 3 Fig.3

Fig. 5

 5 Fig.5

Table 1

 1 [START_REF] Basak | Determining structural similarity of chemicals using graph-theoretic indices[END_REF];Basak and Grundwald (1994); and[START_REF] Basak | A hierarchical approach to the development of QSAR models using topological, geometrical, and quantum chemical parameters[END_REF] calculated and used various molecular similarity methods based on graph invariants to select analogs, cluster large libraries of chemicals[START_REF] Basak | Applications of topological indices in property/bioactivity/toxicity prediction of chemicals[END_REF], and estimate properties of chemicals from their selected neighbors[START_REF] Basak | A comparative study of molecular similarity, statistical, and neural network methods for modes of action of chemicals[END_REF] Basak et al., 2001) in the selected spaces. It was concluded that no universal similarity method can handle all situations. It would be interesting to see how far map invariants are able to characterize various aspects of proteome maps and their perturbation derived from the exposure of organisms to drugs Gel coordinates and abundance for the twenty most intensive protein spots of rat liver cells of normal cells and the corresponding abundance for cells exposed to four proliferators

	and toxins.

Table 2

 2 Fragmentary orderings of proteins spots holding with respect to charge and mass respectively that start with protein 1.

	1	1 ⇒	7 ⇒	3 ⇒	19 ⇒ 14
	2	1 ⇒	9 ⇒	3 ⇒	19 ⇒ 14
	3	1 ⇒	13 ⇒ 11
	4	1 ⇒	13 ⇒ 14
	5	1 ⇒	16 ⇒ 6 ⇒	11
	6	1 ⇒	16 ⇒ 6 ⇒	14
	7	1 ⇒	16 ⇒ 6 ⇒	14
	8	1 ⇒	16 ⇒ 19 ⇒ 14

Table 3

 3

Table 4

 4

		Scaled Abundance values
	Spot Control	PFOA	PFDA	Clofibrate	DEHP
	1	5.200	3.916	3.423	5.299	5.976
	2	5.174	5.604	6.794	5.760	5.586
	3	4.923	4.102	5.413	5.895	0.292
	4	4.582	3.572	2.632	2.761	4.038
	5	4.272	4.063	1.793	3.956	5.000
	6	4.140	6.933	7.982	5.983	6.506
	7	4.044	2.114	1.402	2.636	2.777
	8	3.923	0.940	1.828	1.654	4.209
	9	3.539	3.284	2.989	3.033	3.348
	10	3.372	2.996	2.267	2.877	3.940
	11	3.242	4.659	4.048	4.058	4.301
	12	3.124	2.549	2.835	3.810	4.189
	13	3.056	2.659	1.638	2.591	3.510
	14	2.972	2.665	2.683	3.051	3.190
	15	2.953	0.581	0.594	2.164	5.367
	16	2.883	2.785	2.885	2.739	3.633
	17	2.876	0.749	0.472	1.398	1.939
	18	2.622	2.751	1.900	2.003	2.789
	19	2.600	2.809	2.175	1.686	2.814
	20	2.502	1.363	0.581	2.059	2.568
	Sum	72	61.095	56.334	65.417	75.974

Table 5

 5 Leading eigenvector for the Twenty Most Intensive Protein Spots of Rat Liver Cells of the Normal Cells and Cells Exposed to Four Chemicals

	Protein	Control	PFOA	PFDA	Clofibrate	DEHP
	1	0.5314	0.3195	0.2374	0.5065	0.5287
	2	0.3059	0.1973	0.1853	0.2823	0.2471
	3	0.2157	0.0608	0.0447	0.1918	0.0417
	4	0.2134	0.1052	0.0573	0.1073	0.1358
	5	0.1629	0.0431	0.0138	0.0800	0.1067
	6	0.3039	0.7737	0.8699	0.5646	0.5716
	7	0.2404	0.0675	0.0396	0.1429	0.1119
	8	0.0231	0.0041	0.0033	0.0080	0.0109
	9	0.2501	0.0919	0.0524	0.1701	0.1443
	10	0.0919	0.0207	0.0081	0.0404	0.0507
	11	0.1284	0.2663	0.2012	0.1987	0.1904
	12	0.0192	0.0053	0.0039	0.0121	0.0108
	13	0.2802	0.1915	0.1162	0.2326	0.2490
	14	0.1478	0.1887	0.1679	0.1796	0.1681
	15	0.1287	0.0359	0.0258	0.0746	0.1344
	16	0.3021	0.2711	0.2335	0.2935	0.3162
	17	0.0896	0.0302	0.0237	0.0514	0.0479
	18	0.1026	0.0333	0.0124	0.0438	0.0558
	19	0.1282	0.0951	0.0659	0.1042	0.0907
	20	0.5314	0.0360	0.0165	0.0785	0.0823

Table 6

 6 Similarity/Dissimilarity Among Perturbations of Abundances of Proteome Rat Liver Cells for the Control and the Four Peroxisome Proliferators based on 20component leading eigenvectors vectors of Table 4

		Control	PFOA	PFDA	Clofibrate	DEHP
	Control	0	2.2793	2.7583	1.1820	1.2639
	PFOA		0	0.6508	1.0973	1.1020
	PFDA			0	1.5997	1.5404
	Clofibrate				0	0.4925
	DEHP					0

Table 7

 7 Eigenvalues for the Twenty Augmented Adjacency Matrices Belonging to the Most Intensive Protein Spots of Normal Cells Rat Liver Cells and Cells Exposed

		to Four Chemicals
	Eigenvalue	Control	PFOA	PFDA	Clofibrate	DEHP
	20	-0.0830	-1.1439	-1.7941	-0.6431	-0.8055
	19	0.6326	-0.6731	-0.7304	-0.0715	0.4254
	18	0.7767	0.4575	-0.0998	0.3382	0.7267
	17	1.4262	0.6936	0.3285	0.6511	1.3338
	16	2.2079	0.8606	0.5743	1.5114	1.7088
	15	2.3826	1.3571	0.9712	1.7600	2.3483
	14	2.5278	1.6779	1.4737	2.0168	2.8501
	13	3.0976	2.4354	1.8390	2.4977	3.4341
	12	3.1487	2.5309	2.1415	2.5601	3.6500
	11	3.4520	2.8756	2.4984	3.0014	4.0585
	10	3.5987	3.0562	2.6893	3.3902	4.1709
	9	3.9179	3.4333	3.2123	3.4929	4.1991
	8	4.1029	3.5074	3.3452	4.0193	4.3161
	7	4.6763	4.3467	3.8255	4.2165	4.7448
	6	4.8083	4.6362	4.2210	4.3761	5.1266
	5	5.2633	4.8190	4.2995	5.2386	5.2619
	4	5.5240	5.2513	4.7327	5.6414	6.0093
	3	6.3278	6.0646	6.3835	6.5875	6.4306
	2	6.4204	6.6249	7.4747	6.7621	7.3740
	1	7.7914	8.2840	8.9477	8.0701	8.6109

Table 8 The

 8 Similarity/DissimilarityTable Between the Control Data and Data for the Four Chemicals Based on the eigenvalues

		Control	PFOA	PFDA	Clofibrate	DEHP
	Control	0	12.2991	20.1990	8.5783	7.1881
	PFOA		0	8.4263	5.8563	14.8792
	PFDA			0	12.2635	20.5157
	Clofibrate				0	11.1962
	DEHP					0

Acknowledgments

MR thanks the laboratory for Chemometrics, National Institute of Chemistry, Ljubljana, Slovenia, for warm hospitality. This work has been supported in part by the Ministry of Science and Higher Education of Republic of Slovenia, under research grant P1-017 and the Ministry of Science, Education and Sports of the Republic of Croatia under Project "NMR Spectroscopy and Modeling of Bioactive Molecules." We also thank Professor A. T. Balaban (Texas A&M University at Galveston) who was kind to examine the manuscript and suggested improvements that resulted in better presentations of our work.


Darius Seyedi

Guy Nicolas

François Hild

Sébastien Meunier

S Granet

A non-local damage model for heterogeneous rocksapplication to rock fracturing evaluation under gas injection conditions

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A non-local damage model for heterogeneous rocksapplication to rock fracturing evaluation under gas injection conditions

Seyedi Darius, BRGM 3 avenue Claude Guillemin, 45060, Orléans, Franced.seyedi@brgm.fr Guy Nicolas, BRGM, Orléans & LMT-Cachan, France (now at IFPEN) Hild François, LMT-Cachan, Cachan, France Meunier Sébastien, EDF R&D, Clamart, France Granet Sylvie, EDF R&D, Clamart, France Keywords: Initiation, Non-local damage model, Propagation, Probabilistic approach, hydromechanical coupling

Abstract:

All underground engineering works have to deal with heterogeneous rock mass (e.g., nuclear waste disposal in geological media, deep underground injection, geothermal energy extraction, enhanced recovery from oil and gas reservoirs, and underground storage of natural gas). More recently, geological storage of carbon dioxide (CO 2 ) into permeable deep aquifers has been suggested as an important potential method to reduce the emission of greenhouse gases into the atmosphere. Deep well injection changes the stress state in the geological formation. A change in the stress field may affect the hydraulic response of the medium. The main goal of the present work is to introduce a coupled hydromechanical damage model that takes into account the effect of mechanical heterogeneities on the hydromechanical response of rock masses.

A probabilistic damage model is developed to study possible cracking of the rock due the effective stress changes. Two different thresholds are considered for crack initiation and propagation. A Weibull model is used to account for the stochastic nature of crack initiation(s) and then a fracture mechanics based threshold is considered to model crack propagation. The model is integrated in the finite element package (Code_Aster) as a non-local damage model. Both damage thresholds are checked using a "regularised" stress field to avoid mesh dependence and localization phenomena. The use of the regularized stress enables us to calculate correctly the stress intensity factors for the propagating cracks modelled as a set of completely damaged elements. The interaction between propagating cracks and initiation sites is accounted for naturally through the stress relaxation induced around the existing cracks.

A robust and efficient sequential approach is proposed for hydromechanical coupling. The coupled phenomena of fluid flow in complex geological systems are solved with a thermo-hydraulic module, and the mechanical response of the geological medium is evaluated with the mechanical module of Code_Aster®. These processes are linked using a sequential algorithm. The use of a damage variable to represent the induced cracks permits a straightforward damage-permeability relationship to be considered.

Following the above methodology, a typical gas injection example is considered. The effects in terms of effective stress changes for different injection scenarios are assessed. The effect of different in situ parameters such as material heterogeneity, mechanical parameters and initial stress field is studied through parametric analyses and the damage risk is evaluated.

Alert Workshop 2011 -Session number 2. -Abstract form

For other authors:

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Figures …………….


Jean-Baptiste Gossé

Dominique Plihon
email: plihon@univ-paris13.fr.

The future of financial markets and regulation: What Strategy for Europe? 1 The Future of Financial Markets and Regulation: What Strategy for Europe?

Keywords: Financial Stability, Supervision and Regulation, Financial Integration JEL Classification: E44, F36, G18, G28

This article provides insight into the future of financial markets and regulation in order to define what would be the best strategy for Europe. To preserve financial stability, Europe has to choose between financial opening and independently determining how to regulate finance.

Among the five scenarios we defined, three achieve financial stability both inside and outside Europe. In terms of market efficiency, the multi-polar scenario is the best and the fragmentation scenario is the worst, since gains of integration depend on the size of the new capital market. Regarding sovereignty of regulation, fragmentation is the best scenario and the multi-polar scenario is the worst because it necessitates coordination at the global level which implies moving further away from respective national preferences. However, the more realistic option seems to be the regionalisation scenario: (i) this level of coordination seems much more realistic than the global one; (ii) the market should be of sufficient size to enjoy substantial benefits of integration. Nevertheless, the "European government" might gradually increase the degree of financial integration outside Europe in line with the degree of cooperation with the rest of the world.

Introduction

The recent financial crisis, which is not over yet, has shown that finance matters a lot and exerts a powerful influence on the world economy. The increasing importance of finance, both private and public finance, is particularly true in Europe. Finance plays an essential role for the (mis)allocation of capital among sectors and countries. It is also a major factor of instability, both for the financial and the real sectors of the economy, at all levels: national, regional and international.

A priori, it seems difficult to forecast what will be the evolution of financial markets in the coming decades, not least because finance is fundamentally unstable! Nevertheless, it is crucial to define what could be the future of finance due to its central importance for the world and European economies.

Forecasting the evolution of finance is needed for regulation purposes: there is a need to adapt the rules to the transformation of financial markets in the future. This study is an attempt to analyze the wide array of possibilities in the future, as they range: (i) from a reform based on micro-prudential regulation to a strong state-controlled macro-prudential regulation; (ii) from regulation at a national level to regulation at a global level; and (iii) from strong capital controls between economic areas to strong financial integration.

The methodology of this study will consist in building contrasted scenarios in order to allow for strategic thinking about the evolution of finance up to 2030. The scenarios will be built around the strategies of the major actors, public and private, of the world economy and financial markets, including the interrelated role of financial regulation and financial innovation. This study will also focus on the role of Europe as concerns financial markets and the impact of finance on Europe in 2030.

The study is composed of three sections. The first section provides a description of the spontaneous trends of financial markets, with a focus on major players. Section 2 builds on the evolution of financial system depending on the different types of regulations implemented in the future decades. Finally, section 3 provides a panorama of five alternative scenarios based on different assumptions as to the strategies and the role of major players. These scenarios provide a framework for the analysis of the future of Europe in the world economy.

I-The spontaneous tendency of financial markets

In this part, we define the spontaneous tendency of finance in order to pinpoint what could be the future strategies of operators, the future locations of financial centres, the most promising markets, and the new major players.

Future strategies of financial operators

The financial operators' view is determinant in shaping future finance, since they will take strategic decisions inside banks and financial institutions. According to European bank managers and financial executives3 , the main challenges are (i) the improvement of risk management, (ii) the adaptation to changing regulation, (iii) the intensification of price competition, (iv) the concentration of financial markets, and (v) the standardization of financial operations to facilitate automation, replacing workers with machines. Thus, financial institutions will try to increase their size and to reduce costs of financial transactions to deal with stronger competition in more globalised financial markets. Furthermore, while financial operators seem to have realized at least partially the shortcomings of their own risk management in the past, they worry now more about new regulatory constraints. To put it in a different way, they would prefer to deal with financial instability at the firm level rather than be closely controlled by state supervisors and regulatory authorities.

Future locations of financial centres

The future locations of financial centres will probably be rebalanced with the emergence of deeper markets in the emerging market economies. First, the development of Islamic finance could help centres to thrive in the Middle East and in Asia4 . Secondly, if commodities became major safe havens for investors, some new centres in emerging countrieswhich produce themcould play a growing role5 . Thirdly, the opening of financial markets in the emerging countries would undoubtedly change the situation greatly. For example, in 2010 China decided to make the Renminbi convertible for non-residents, paving the way for Honk Kong to become the most important financial centre in Asia.

Promising markets

It is rather difficult to identify promising markets for the next 20 years. There are some emerging new markets such as ethical and green finance which could be destined to play an important role in the future, but it is too early to assess their future importance. This depends crucially on future innovations, which are closely linked to regulation. According to both specialists and practitioners, the strategic markets should be derivatives. Otherwise, over-thecounter markets could grow fast to avoid organized markets which could be much more regulated in the future. But this evolution may be stopped by new regulation, under study today, about the obligation of all actors on OTC to register with clearinghouses.

Players

The major players of financial markets will be both public and private entities. In the first place, the growth in the number of sovereign wealth funds6 , which prefer riskier investments, could increase the exposure of government-controlled investments. Whereas central bank purchases of safe assets reduced the global interest rate in the 2000s, the growth of risky investments made by sovereign funds may lead to an increase in asset prices. On At the same time, private actors such as hedge funds and private equity are called upon to play an increasing role. It is difficult to define what would be their activities in the future, because those depend on the regulatory reforms that will be put in place. Nevertheless, speculative activities of hedge funds imply important systemic risks and may create contagion effects in the banking system. Public actorsregulators, states, central banks, and international organizationsare also bound to play an important role if the instability of financial markets is to be persistent. It may be that in some emerging countriessuch as Chinapublic authorities will play a dominant role, while the influence of private players is more important in countries where the market economy is more developed, as is the case in the US and in Europe. At the international level, civil society (meaning non-governmental organizations defending the general interest) is also bound to play an increasing role. International organizations, such as the IMF, may also have a growing influence, depending on the nature of the likely reforms of the international monetary system.

Financial innovation vs. regulation

The development of financial innovation has been motivated for a long time by the desire to avoid taxes or government regulation. We can therefore expect that new regulatory constraints will be circumvented by financial innovations to come. We can distinguish two ways of avoiding regulatory constraints. On the one hand, financial operators can escape regulators by transferring their capital to offshore financial centres with a high degree of financial secrecy.

On the other hand, institutions can create some new financial products and develop a "shadow banking system" or a "shadow financial market" in order to steer clear of regulation and supervision.

II-Tools to regulate financial markets

The two dimensions of financial regulation: micro-and macro-prudential regulations

The role of prudential regulation is to prevent bank failures and financial crises. The reason for preventing financial crises is that the costs to society are more important than crises in other sectors and exceed the private cost to individual financial institutions. Prudential regulation aims at internalizing these externalities in the behaviour of such institutions.

Prudential regulation has two dimensions: micro-prudential and macro-prudential. Microprudential regulation concerns itself with the stability of individual entities and the protection of clients of these institutions. Macro-prudential regulation concerns itself with the stability of the financial system as a whole. Micro-prudential regulation consists of such measures as (i) the certification of those working in the financial sector; (ii) rules on what assets can be held by whom; (iii) how instruments are listed, traded, sold and reported; (iv) measures of the value and riskiness of assets. The Basel Committee on bank supervision has played a major role in defining the rules and instruments of micro-prudential regulation. One of the main tools put forward by the Basel Committee is that of capital adequacy requirements.

Until recently, the Basel approach rested on the principle that the purpose of regulation is to ensure the soundness of individual institutions against the risk of loss on their assets. The Basel Committee doctrine was based on the false assumption that actions enhancing the soundness of a particular institution should also promote overall stability. However, ensuring the safety of each individual institution is not a sufficient condition for the soundness of the system as a whole. It is possible, indeed often likely, that attempts by individual institutions to remain solvent can push the system to collapse. The recent crisis was caused by banks transferring risky toxic assets to other financial unregulated financial institutions such as hedge funds. One of the major causes of the crisis has been the deficit of macro-prudential regulation.

Micro-prudential regulation examines the responses of an individual bank to exogenous risks.

By definition, it does not incorporate endogenous risk. It also ignores the systemic importance of individual institutions as defined by their size, the degree of leverage they use, and their interconnectedness with the rest of the system. This is why we need to complement microprudential regulation with macro-prudential regulation. The macro-prudential approach to regulation considers the systemic implications of the collective behaviour of banks.

A critical feature of macro-prudence and systemic stability is the heterogeneity of the financial system. Homogeneityeveryone selling or buying at the same timeundermines the system. Invariably, market participants start off being heterogeneous but a number of factorssuch as the use of similar techniques of risk measurement by banksdrive them to homogeneity. In this regard systemic risk is endogenous, and macro-prudential regulation is about identifying those endogenous processes that turn heterogeneity into homogeneity.

There is a growing consensus that the most important manifestation of market failure in banking and financial markets is pro-cyclicality. According to this view, the purpose of macro-regulation is to act as countervailing force to the pro-cyclical behaviour of banks which is based on their underestimation of risks in a boom and their overestimation of risks in the subsequent collapse. This shift in risk perception from "too low" to "too high" is an essential problem. The purpose of macro-prudential regulation is to moderate financial cycles by narrowing this gap in forcing banks to improve their measurement of risks in boom and bust.

A critical part of micro-prudential regulation in the last decade was the increasing use of market prices in valuation and risk assessment. This was done in the name of transparency, risk-sensitivity and prudence, but what it achieved was increasing homogeneity and cyclicality of market behaviour, hence also increased systemic fragility. Micro-prudential behaviour can thus endogenously create macro-prudential risks.

Counter-cyclical bank regulation can be introduced through banks' dynamic provisions systems, linking provisioning to the credit cycle. This technique has already been used in Spain and Portugal. Such a system requires higher provisions when credit grows more than historical average and lower provisioning in slumps. An alternative approach for counter-cyclical bank regulation is via capital requirements. Basel III guidelines presented in September 2010 propose making bank capital charges counter-cyclical.

Micro and macro-prudential regulation differ in their needed professionalism. Microprudential regulation should be carried out by banking and financial market supervisors, whereas macro-prudential regulation should be put under the responsibility of central banks.

Central banks should have to monitor credit expansion which is a major channel of bubbles and financial crises. Such instruments as loan-to-value ratios and progressive compulsory reserves on bank credit may be used by central banks to reduce credit cycles. Not all banks are alike. Regulation should acknowledge that some banks are systemically important. Tighter supervision of the latter banks needs to be implemented. By the same token, international cooperation is required for the regulation and supervision of banks which operate in several countries.

In short, the subprime crisis has shown that the existing framework of banking regulation was insufficiently macro-prudential. A reform of financial and banking regulation is under way to introduce new instruments, and to achieve a new balance between micro-and macroprudential regulations. This is illustrated by the new Basel III guidelines presented in September 2010 or the reforms implemented in 2011 in the European Union along the lines proposed by the Larosière Report.

Financial regulation at the international level

At the international level, the regulator has three main objectives which are not necessarily compatible: (i) financial stability, (ii) independence of regulatory policy, (iii) financial integration.

Financial stability seems to be the most obvious target. Since the last crisis a fairly clear consensus has emerged among the G20 central bankers (Banque de France, 2011). They agree on the need for enhanced supervision of the financial sectorespecially for systemically important financial institution (SIFI)and greater coordination of national policies in order to ensure financial stability internationally. One can note two interesting nuances among central bankers of emerging countries. They expect a change in the International Monetary System (IMS), which would imply more competition against the dollar, and better regulation of speculative capital flows, by nature more volatile, since those have a particularly destabilizing effect on emerging markets.

The independence of regulatory policy may appear desirable, because it allows each country to regulate its financial market according to national preferences. For example, following the example of Germany, other countries might wish to ban short sales or opt for more stringent prudential standards for systemically important actors (e.g. hedge funds). Indeed, preferences in the regulation of financial markets are quite heterogeneous in the world. On one side, Europeans seem willing to put in place a regulatory regime based on a strong coercive power of the authorities. On the other side, the United States appear more directed towards a system focused on market-based insurance mechanisms [START_REF] Goodhart | The Role of Macro-Prudential Supervision[END_REF].

For many years much of the economic literature has been arguing in favour of a greater degree of financial integration which is defined as better access to foreign capital markets for investors. Financial integration is supposed to have several advantages. Firstly, it enables a more efficient capital allocation, because savers benefit from a wider choice in their investment decisions and potentially have access to more lucrative investments. Secondly, the deregulation of national markets allows investors to diversify the risk of their portfolios more effectively by holding assets that are less correlated due to the fact that they are issued in countries with different economic characteristics.

The second point, however, raises two questions. The first question concerns the impact of globalization on national circumstances. In recent decades traditional barriers to trade in goods and services and to capital flows have eroded and economies have become increasingly interdependent. When a shock occurs, it is felt throughout the system via fewer commercial opportunities, lower investment incomes, and devaluations of foreign assets. The second question is that of the consequences of greater financial integration on the resilience of financial markets. Indeed, there are good grounds for considering that breaking down barriers between national financial markets increases the global systemic risk. Thus, the reduction of portfolio risk expected from the international diversification is not so obvious, since financial integration would cause an increase in the risk to which all assets are exposed. Thus, the deepening of financial integration is beneficial only if systemic risk is reduced by an appropriate regulation at the global level.

To sum up, the efficiency of micro-prudential and macro-prudential measures depends on the capacity to control international financial markets. We can identify three main objectives of regulation at the international level. First, authorities try to reach a high degree of financial integration to allow a better allocation of capital and to permit a better diversification of risks.

Second, governments want to improve financial stability, including the need to avoid international arbitrage between financial centres and their regulations (e.g. tax havens). Third, decision makers wish to maintain the independence of the national regulatory system in order to decide how financial activities should be regulated (e.g. limitations on short selling). These three objectives cannot be combined. The trilemma in financial regulation at the international level can be represented by a triangle with one of the objectives at each vertex. The regulator has to leave aside one of them. As financial stability is currently the first priority, authorities must either reduce financial integration or accept to reinforce international governance 7 . Thus, it is necessary to adapt the scale of international regulation to the scale of risk caused by greater financial integration at the global level.

In the context of open capital markets, it is impossible for a single nation-state to define an effective regulation of finance. Financial institutions will seek the least restrictive supervision system so as to avoid compliance with the standards set by the regulator 8 . Thus, even the most powerful prudential arsenal at the national level will be rendered inefficient by the opening-up of financial markets which permits investors to direct capital flows to the less regulated financial centres. G20 measures targeting tax havens seem inadequate. The list provided by the OECD is based primarily on criteria of transparency and exchange of information that are neither able to prevent offshore financial centres operating nor even neutralize the off-balance sheet investment vehicles 9 .

III-Five scenarios

After describing the spontaneous tendencies of finance and the tools available to regulate it, we will now discuss the effect of different regulatory policies on the natural evolution of markets. The issue is to describe the impact they could have on the world, according to the configuration chosen, and to determine what would be the best strategy for Europe. To do this, we define five scenarios 10 that are distinguished by the leading actors who have the power to impose their strategies to the world. Alternatively, we define five groups distinguished by decision makers: Chimerica (United States and China), multinational corporations, nation-states, regional blocs, and supranational authorities.

Chimerica: United States-China

In this scenario, current trends continue, assuming that on the economic and financial levels the United States and China are in a position to impose their decisions, since none of the major players opposes them. On the one hand, the United States pursues a domestic demandled growth strategy. The Fed conducts an accommodative monetary policy, and fiscal policy 7 See Aglietta (2011). 8 For further details about the inadequacy between international financial markets and institutions, and national supervision and crisis management, see [START_REF] Goodhart | Border Problems[END_REF]. 9 See on this point, the protest of Josef Pröll, the Austrian Minister of Finance (Vanessa Houlder, « Ports in a storm », Financial Times, November 17, 2009. 10 See Table 1 for details on scenarios. is occasionally used to cover shortfalls in demand. On the other hand, China favours an export-led growth and continues to accumulate foreign exchange reservesin dollars and, increasingly, in other currenciesto perform a slow and gradual revaluation of the Renminbi.

In this way, the price competitiveness of China is maintained and the transition to domestic demand-led growth takes place smoothly.

In this context, commodity export revenues will continue to grow. Sovereign wealth funds in commodity-exporting countries acquire an increasing amount of assets in developed countries, and very soon a steady income replaces commodity revenues. China and commodity-producing countries, which hold a large share of their assets in U.S. currency, maintain their peg to the US dollar which remains the international reference currency. Europe and the rest of the world are subjected to the growth strategies of other actors. The euro appreciates, because the U.S. and Chinese growth strategies cause a sharp increase in the supply of dollar assets relative to other currencies. European growth is slowed by the rise of prices expressed in euros. As debt sustainability depends on interest rates and on the growth rates of the GNP, GIIPS 11 should find more and more difficult to meet interest payments. The euro area experiences increasingly strong tensions. Germany pursues its strategy of wage moderation, while the relative unit labour cost continues to grow in GIIPS (except in Spain and Ireland where it has been decreasing since 2008 12 ). Since the Eurozone becomes weaker, the European influence on the global economy decreases rapidly. Moreover, as China and commodity-exporting countries diversify their exchange reserves in favour of Euros, Europe should pay increasingly high interests to these countries, which reduces GNP.

Consolidation: multinational corporations (minimum state)

We continue another important trend in recent decades: the declining power of states, matched at the same time by the growing influence of multinational corporations. We assume that large firms are becoming increasingly powerful. Thus, they can take advantage of competition between states that seek to become more attractive, particularly in terms of taxation, regulation, or confidentiality. Thus, multinationals have power to bypass regulation and supervision that governments try to establish. Investors will flock to financial markets 11 Greece, Italy, Ireland, Portugal and Spain. 12 See [START_REF] Grauwe | The Financial Crisis and the Future of the Eurozone[END_REF], Figure 4.

with soft regulation at the expense of those trying to impose measures that reduce the competitiveness of firms, but are essential to the establishment of financial stability.

Financial institutions are given free rein to develop their business globally. Financial industry becomes more concentrated, because players seek to reach the minimum size needed to be internationally competitive. Meanwhile, stock exchange markets continue to merge. After the creation of Euronext 13 in 2000 and its merger with the New York Stock Exchange to create the NYSE Euronext group, a new merger could take place with Deutsche Börse. Thus, if the integration process taking place today in Europe and North America continues, it could well lead to the creation of a single global financial market that would improve liquidity.

Governments tend to establish the least stringent regulatory standards possiblewhat is often termed the "race to the bottom"and prefer self-regulation to attract large financial institutions that generate higher incomes. Within this oligopolistic global financial market, the trend towards automation of financial transactions continues without being too heavily constrained by regulators. The development of high-frequency trading generates an increase in gross trading volume, while net trading volume stagnates. We observe recurrent financial panics which can result from decisions of large operators and from the spread of automated trade execution 14 . Finally, the reinforcement of global financial integration increases systemic risk, but there is no supranational authority to regulate this supranational market.

The few measures that have been taken to improve macro-prudential regulation are rapidly circumvented, because governments feel the need to woo financial institutions which make high profits. At the same time, the concentration of finance implies an increase in the number of systemically important financial institutions (SIFI) and hence aggravates financial instability. In such a context, it is likely that financial crises and stock-market panics will become more frequent. This scenario could also lead to the fragmentation of the eurozone, which would result from a sudden stop in capital inflows to GIIPS. 13 The group was founded in 2000 with originally Amsterdam Stock Exchange, Brussels Stock Exchange and Paris Bourse. Then, LIFFE joined them in 2001 and the Bolsa de Valores de Lisboa e Porto (BVLP) in 2002. 14 The stock market crash of 6 May 2010 provides an interesting example of such a financial panic. On this day, the main US financial indexes fell by 10% within 15 minutes, before returning to their previous level. For further information on the role played by automated trading strategies, see the report of the CFTC and SEC (2010) on that so-called "flash crash."

Fragmentation: Nation-state (economic nationalism)

The negative effects of globalization experienced by workers create a feeling of economic patriotism. Most economies in Europe and elsewhere prefer to adopt purely national economic governance and strictly control international capital flows. The globalized financial and economic space is divided to correspond to the social and political space. Fragmentation is a major concern of the private sector which fears a return to a strong state. [START_REF] Ernst | The Future of Finance: Driving business value through the performance of the Finance function[END_REF] notes, moreover, a buzz on the word « protectionism » at the 2009 Davos Conference and that the fight against it is a priority for most decision makers.

The financial markets are becoming more localized, and the regulation is country-specific. In some countries, the banking system may even be nationalized to closely regulate market activities. The retreat into protectionism also involves the establishment of exchange controls and the setting up of currency-undervaluation strategies which may further destabilize the foreign exchange market. In such a context, there exist major arbitrage opportunities for those with the ability to execute trades across borders.

The euro-zone is fragmented, and each member-state returns to its previous national currency. This scenario is far from being impossible, if one looks at the nicknames used to refer to the indebted economies. The group of countries consisting of Greece, Ireland, Italy, Portugal and Spain are often referred as PIIGS, GIPSI or "club med" by the economic press, in academia, and among bond analysts. These quite offensive nicknames stigmatize the lack of seriousness of indebted countries and reflect a deep resentment among North European countries against their Southern neighbors.

The abandonment of the euro would have serious consequences on Europe. The intra-EU trade would be reduced because of the instability of prices in foreign European currencies and increase in transaction costs. European currencies become vulnerable again to speculative attacks and to currency crisis. The borrowing costs rise in South European countries, whereas German growth substantially slows down because of the appreciation of the Deutsche mark which becomes the main safe haven for investors. Finally, neither the North nor the South European countries seems to have an interest in returning to national monetary and exchangerate policies.

Regionalisation: Regional integration

In this case, the negative effects of financial globalization generate a different reaction of the national states: instead of reducing the economic space at the national level, they decide to create regional economic spaces. At the same time, political power is organized at the regional scale to meet the size of the economic space. Regional-states put in place a common financial regulation at the bloc level. We can also imagine the emergence of regional initiatives to counterbalance the power of governments and the influence of firms. Thus, a group of European elected officials launched a call 15 to "organize the creation of a nongovernmental organization capable of developing a counter-expertise on activities carried out on financial markets by the major operators". The goal of the "finance watch" project is to compete with the private lobbies which contributed to the engagement in riskier lending activities [START_REF] Igan | A fistful of dollars: lobbying and the financial crisis[END_REF].

In this scenario financial regulation is region-specific. The regional governments control capital flows to avoid regulatory arbitrage between regions. The blocs can be constituted on the basis of existing agreements. Beside the European Union, we could imagine that NAFTA, ASEAN, MERCOSUR, the CIS or the Arab League would move towards a stronger financial integration. The fragmentation of financial markets requires the financial institutions to adopt a different strategy in each bloc in order to adapt to region-specific characteristics.

The eurozone enlarges to increase its weight in the world economy and the size of its domestic market. In fact, the wider the financial integration area and the more unified the regulation system are, the better the capital allocation and the diversification of portfolio risk will be. The full range of economic policy is put in place at the European level, beyond just monetary or fiscal policy. Europe benefits from advantages of financial integration at the continental level and can establish a financial regulation compatible with its systemic risk aversion, perhaps even creating a currency transaction tax. Afterwards, the "European government" might increase financial integration with the rest of the world in line with the degree of cooperation of regional blocs on prudential supervision and mutual surveillance of their banking systems.

Multi-polar: global integration and supranational regulation

The multi-polar scenario implies both a strong intervention capacity of governments and massive participation of civil society in the functioning of financial markets. First, governments of major world economies reach an agreement (e.g. at the level of G20, or an enlarged G20) on financial regulation and create the necessary institutions to enforce these common rules. Thus, political power is established at the scale of financial globalization with suitable supranational institutions capable of addressing the need for proactive global systemic risk management. Indeed, the cooperation of governments on a sufficiently binding regulation and the capability of supranational institutions to prevent the creation of offshore financial centresas well as to limit other shadow transactionsare prerequisites for stabilizing financial markets.

Second, this scenario necessitates the emergence of a counterbalancing force besides governments' and multinational corporations' powers. If there is already a world social forum for civil society -which is the answer to Davos' World economic forum for corporations and to G20 summits for governmentsthis third force is not yet sufficiently organized to respond to the challenges of globalization, especially at the financial level.

There are, however, some interesting citizens' initiatives aimed at rebalancing globalization.

On the one hand, Obstfeld and Rogoff (2009) claim that "the interaction among the Fed's monetary stance, global real interest rates, credit market distortions, and financial innovation created the toxic mix of conditions making the U.S. the epicentre of the global financial crisis." Thus, it seems more than necessary to create a civil society organizationi.e. the equivalent of the European "Finance Watch" at the global scalethat would keep an eye on financial authorities and institutions in order to avoid the repetition of such a disaster.

On the other hand, civil society could also put in place payment systems in addition to traditional banking. [START_REF] Lietaer | Monnaies régionales : des nouvelles voies vers une prospérité durable[END_REF] argues that the development of local monies would deal with the problem of crises in increasing the resilience of the economic system, even if it would reduce the efficiency of the monetary system. Therefore, the question of monetary creation is sensitive, since the excess of liquiditywhich means a creation of liquidity beyond the credit absorption capacity of the US economycontributed to the subprime crisis.

Ethical finance is an interesting alternative for investors wishing to devote their savings to progressively managed firms which respect the environment, comply with the international labour code, and refrain from speculative activities. The development of such finance could have stabilizing effects on markets.

To summarize, in this scenario we assume a combined success of governments and civil society to rebalance the financial globalization process. On the government side, success implies renouncing national preferences. Nation-states must adopt a similar regime of financial regulation with common rules in order to avoid regulatory arbitrage. This means the end of tax havens that would render ineffective the attempts to regulate at the global level. On the civil-society side, counterbalancing powers must be created to oppose the lobbies of financial institutions and to watch the developments of financial markets. Under these conditions Europe and the rest of the world could fully enjoy the benefits of financial globalization, yielding a more efficient allocation of capital and better distribution of risk.

Conclusions

The aim of this article was to provide insight into the future of financial markets and regulation in order to define what would be the best strategy for Europe. The main objectives of financial regulators should be stability and market efficiency. However, favourite methods to achieve stability can be rather different from country to country and may depend on national preferences or institutional tradition.

We have shown that it was difficult to combine financial stability with both financial integration and sovereign financial regulation. This implies that Europe has to choose between financial opening and independently determining how to regulate finance. The options are summed up in Table 2. Three of the five scenarios achieve financial stability both inside and outside Europe. In terms of market efficiency, the multi-polar scenario is the best and the fragmentation scenario is the worst, since gains of integration depend on the size of the new capital market. Regarding sovereignty of regulation, fragmentation is the best scenario and the multi-polar scenario is the worst because it necessitates coordination at the global level which implies moving further away from respective national preferences.

Finally, even if the multi-polar scenario is the first best solution, the more realistic option seems to be the regionalisation scenario. On the one hand, the regional level of coordination seems much more realistic than the global one, since the preferences are much more similar on that scale. On the other hand, it should be of sufficient size in order to enjoy substantial benefits of integration. However, it may be appropriate for the "European government" to gradually increase the degree of financial integration outside Europe in line with the degree of cooperation with the rest of the world. Obstfeld M. and K. Rogoff, (2009), "Global Imbalances and the Financial Crisis : Products of Common Causes", CEPR discussion paper, n 7606. World Economic Forum, (2009), "Financial System A Near-Term Outlook and Long-Term Scenarios", World Economic Forum USA Inc.

Figure 1 .

 1 Figure 1. The incompatibility triangle of financial objectives

For more details, see[START_REF] Engstler | Banks & Future preparing for the Scenario 2015[END_REF][START_REF] Kpmg | Finance of the Future: Looking forward to 2020[END_REF][START_REF] Ernst | The Future of Finance: Driving business value through the performance of the Finance function[END_REF].

According to International Financial Services London (2010), the countries which could benefit from the development of Islamic finance are: Iran, Saudi Arabia, Malaysia, Kuwait, UAE and Bahrain.

The BM&F Bovespa (Sao Paolo) or the National Commodity and Derivatives Exchange (Mumbai), which grew rapidly in the last years, may have a stronger role in the future. We can also mention the brand new Dubai Multi Commodities Centre.

Most of the sovereign wealth funds are in China, Russia and the OPEC countries which are major exporters of, respectively, manufactured goods and oil.

Domination of big banks and accommodation by central banks

Three international moneys: dollar, euro and yuan.

A world money for international transactions and reserves (SDR or Bancor)