On Characterization of Dose Variations of 2-D Proteomics Maps by Matrix Invariants Milan Randic´ ,* Marjana Novicˇ , and Marjan Vracˇ ko National Institute of Chemistry, Ljubljana, Slovenia Received December 23, 2001
We explore the characterization of 2-D electrophoresis proteomics maps by certain structural invariants derived from matrixes constructed by considering for all pairs of spots in a proteomics maps the shortest (Euclidean) distances and distances measured along zigzag lines connecting protein spots of the neighboring abundance. This paper is a sequel to previous papers in which we outlined the idea of characterizing 2-D proteomics maps by graph-theoretical descriptors. To illustrate the approach, we selected data of Anderson et al. (Anderson, N. L.; Esquer-Blasco, R.; Richardson, F.; Foxworthy, P.; Eacho, P. The effects of peroxisome proliferators on protein abundances in mouse liver. Toxicol. Appl. Pharmacol. 1996, 137, 75-89) on protein abundance in mouse liver under a series of dose of peroxisome proliferator LY1711883. We found strong linear correlation between the experimentally applied doses and the leading eigenvalue of a D/D-type matrix (Randic´ , M.; Kleiner, A. F.; DeAlba, L. M. Distance/ distance matrices. J. Chem. Inf. Comput. Sci. 1994, 34, 277-286) constructed for the experimental proteomics maps. Keywords: map invariants • chemical graph theory • D/D matrix • proteomics maps • peroxisome proliferators
Introduction Proteomics became one of the most expanding fields of applied biochemistry because for the first time experimental data have been collected on an abundance of proteins in single animal organ cells. The growth in accumulation of experimental data, mostly in the form of 2-D proteomics maps in which proteins are separated by charge (electrophoresis) and by mass (chromatography), is not followed by the development of theoretical methodologies that would assist in “digestion” of the plethora of data. We have recently initiated one such theoretical approach that shows promise3-7 that is based on construction of map invariants, which are analogous to a degree to sequence invariants used for characterization of DNA8-12 and ultimately to graph invariants used in QSAR (quantitative structure-activity relationship).13-16 An invariant is a mathematical property of a system that may be a molecular graph, molecular structures in 3-D, DNA sequence, or a 2-D map, the special case of which are proteomics maps. The strategy that we have developed in order to arrive at map invariants is to associate with a map a graph or a zigzag line. After embedding such graph or zigzag line onto the map, one tries to associate a matrix with the so-embedded geometrical object. Matrix is a natural mathematical tool for characterization of such objects because one can associate with matrix element (i, j) information concerning pair of vertexes i and j. The elements of the particular matrix that we will consider are given as a quotient of distances. One of the distances consid* Professor Emeritus, Department of Mathematics and Computer Science, Des Moines, IA 50311. Present address: 3225 Kingman Road, Ames, IA 50014. Fax: (515) 292 8629. 10.1021/pr0100117 CCC: $22.00
2002 American Chemical Society
ered is the Euclidean distance between two points considered (protein spots in 2-D gel), and the other is obtained by measuring the distance along the edges of a graph or along the zigzag line. Hence, we succeeded to combine information on the adjacency of the spots along the zigzag line and on the mutual distances of all spots. In this way we obtain the socalled D/D matrix for the embedded zigzag line. The D/D matrix has been initially designed for characterization of conformations (i.e., the geometrical shape) of chainlike structures.2,17-21 Once the D/D matrix is obtained by use of standard matrix algebra, one can extract various matrix invariants to serve as the map descriptor and generate additional structurally related map matrixes. The intuitive argument why the D/D matrix could in principle characterize a map is based on the fact that such a matrix combines information on the distances between the spots with the information on the sequential ordering of spots. Both the graph theoretical distance matrix22 and the geometrybased distance matrix have been employed in chemistry for characterization of molecules. Although they are informationrich, they do not carry information on adjacencies, i.e., which atom is bonded to which. In the cases of molecules, however, the bonding pattern can be deduced by simply finding pairs of atoms associated with the shortest distances. The adjacency matrix, on the other hand, only tells which atoms in a molecule are bonded but is devoid of any information on molecular geometry. The D/D matrix combines information given by both matrixes in a single matrix, which is based on information that relates to through-space and through-bonding distances. This is essential for characterization of maps because the separations Journal of Proteome Research 2002, 1, 217-226
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Randic´ et al. Table 1. Coordinates of Protein Spots and Their Abundance for Different Concentrations of Peroxisome Proliferator LY171883 spot
x
y
0
0.003
0.01
0.03
0.1
0.3
0.6
7 13 14 19 22 23 26 29 31 33
78 103 78 84 120 73 90 63 107 107
147 161 128 144 130 129 144 181 143 123
24.6 15.3 54.1 17.6 17.2 26.4 13.9 9.8 10.7 40.1
24.6 18.4 54.1 21.1 25.8 26.4 16.7 6.9 8.6 45.0
32.0 18.4 59.5 21.1 22.4 26.4 16.7 6.9 11.8 28.6
34.4 16.8 59.5 19.4 24.1 26.4 15.3 5.9 12.8 28.6
41.8 16.8 59.5 19.4 37.8 29.0 15.3 5.9 12.8 36.8
46.7 15.3 70.3 19.4 49.9 29.0 15.3 5.9 15.0 28.6
56.6 16.8 75.7 21.1 55.0 31.7 18.0 5.9 20.3 20.5
Figure 1. Schematic representation of a two-dimensional protein pattern for 99 proteins of whole mouse liver homogenante considered in this paper and based on the map reported in ref 1.
Figure 2. Schematic chemical structure of LY171883.
between the spots along the zigzag line may vary considerably and thus one cannot deduce from the information on distances between spots which pairs of spots are connected by zigzag line. D/D Matrix. In Figure 1 we show schematically the proteomics map with 99 spots that has been reconstructed from Figure 2 of the paper of Anderson et al.1 It illustrates distribution of proteins in the control group of a study after the effects of peroxisome proliferators on protein abundance in mouse liver. First we need the x and y coordinates (that were not reported in ref 1) for 107 protein spots that have been identified and labeled in ref 1. We superimposed a 200 × 200 grid over the map in order to estimate the x, y coordinates and were able to obtain coordinates for 99 individual spots. In Table 1 we have listed coordinates (not reported in ref 1) and abundances (reported in ref 1) for the first 10 points of 99 that we selected from ref 1. To facilitate comparison with the original data, we used the same labels as given by Anderson et al. for the protein spots (shown in the first column of Table 1). The abundance in the column “0” is that of the control group. The remaining six columns give the abundance for the following six concentrations, respectively, of LY171883 in mouse diet: 0.003, 0.01, 0.03, 0.1, 0.3, and 0.6. The schematic chemical structure of LY171883 is shown in Figure 2. Because charge, mass, and abundance are measured each in their own units, they may result in numerical values of widely 218
Journal of Proteome Research • Vol. 1, No. 3, 2002
Figure 3. (x, y) projection of a zigzag line connecting the 20 most abundant proteins of Figure 1 spots having neighboring abundance values.
different magnitude. In order to ensure that the three coordinates x, y, and z (charge, mass, and abundance, respectively) play approximately the same roles, the first step in the construction of the D/D matrix is to scale the input x, y, z coordinates (Table 1). Following the recommendation of Kowalski and Bender,23 we re-scaled coordinates and abundances, all of which are now in the interval (-1, +1). In Table 2 we show for the first 10 spots listed in Table 1 the new re-scaled coordinates and abundances. For example, fot the case of the relative abundance the simple regression scaled ) 0.00609 (nonscaled) - 0.07809 relates the scaled and the nonscaled abundance (the correlation coefficient r ) 1.00000; the standard error s ) 0.051; the Fisher ration F ) 2,736,440). The next step is to construct a zigzag line, which connects spots of adjacent values of the abundance. In preparation for construction of the D/D matrix for the proteomics maps we first have to order spots according to relative abundance in the control group. We start with the most intensive spot no. 51, which is then connected to the next most intensive spot no. 129, which is connected to the next most intensive spot no. 162, etc. In Figure 3, we have depicted the projection of a so constructed zigzag line in the (x, y) plane. The line for which we compute the D/D matrix is in 3-D, the third dimension being given by normalized abundances (listed in Tables 2 and 3).
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2-D Proteomics Maps by Matrix Invariants
Table 2. Scaled Coordinates and Abundances of Protein Spots for Different Concentrations of Peroxisome Proliferator LY171883 spot
x
y
0
0.003
0.01
0.03
0.1
0.3
0.6
7 13 14 19 22 23 26 29 31 33
-0.103 -0.037 -0.103 -0.087 0.008 -0.116 -0.071 -0.143 -0.026 -0.026
0.118 0.159 0.063 0.110 0.069 0.066 0.110 0.218 0.107 0.048
0.072 0.015 0.251 0.029 0.027 0.083 0.007 -0.019 -0.013 0.171
0.054 0.022 0.203 0.036 0.060 0.063 0.014 -0.036 -0.027 0.157
0.099 0.026 0.248 0.040 0.047 0.069 0.016 -0.037 -0.010 0.081
0.127 0.023 0.275 0.038 0.066 0.079 0.014 -0.042 -0.001 0.093
0.168 0.021 0.272 0.036 0.145 0.093 0.012 -0.043 -0.002 0.139
0.208 0.015 0.353 0.040 0.227 0.099 0.015 -0.043 0.013 0.097
0.295 0.025 0.424 0.054 0.284 0.126 0.034 -0.049 0.049 0.050
Table 3. Ordered Spots by Their Abundance in the Control Group spot
x
y
0
0.003
0.01
0.03
0.1
0.3
0.6
51 129 162 14 111 94 33 34 79 73 262 219 127 217 23 77 7 85 91 19
-0.198 -0.198 0.184 -0.103 0.143 0.124 -0.026 -0.127 0.037 0.106 0.129 0.103 0.066 0.132 -0.116 -0.161 -0.103 0.164 0.119 -0.087
-0.192 -0.063 -0.163 0.063 -0.031 -0.025 0.048 -0.002 -0.043 0.030 -0.166 -0.166 -0.090 -0.222 0.066 -0.151 0.118 0.001 0.004 0.110
0.464 0.438 0.353 0.251 0.190 0.182 0.171 0.138 0.134 0.126 0.100 0.090 0.088 0.083 0.083 0.078 0.072 0.069 0.068 0.029
0.515 0.402 0.467 0.203 0.108 0.168 0.157 0.073 0.088 0.116 0.137 0.111 0.054 0.077 0.063 0.059 0.054 0.051 0.051 0.036
0.455 0.522 0.385 0.248 0.141 0.158 0.081 0.080 0.114 0.107 0.069 0.061 0.059 0.070 0.069 0.037 0.099 0.122 0.069 0.040
0.502 0.425 0.304 0.275 0.184 0.151 0.093 0.091 0.108 0.102 0.079 0.054 0.052 0.080 0.079 0.044 0.127 0.109 0.079 0.038
0.498 0.371 0.422 0.272 0.104 0.149 0.139 0.110 0.127 0.100 0.060 0.052 0.067 0.063 0.093 0.058 0.168 0.092 0.078 0.036
0.578 0.181 0.313 0.353 0.138 0.131 0.097 0.073 0.070 0.106 0.065 0.056 0.038 0.067 0.099 0.062 0.208 0.143 0.098 0.040
0.394 0.198 0.295 0.424 0.150 0.114 0.050 0.128 0.076 0.070 0.010 0.005 0.022 0.055 0.126 0.050 0.295 0.205 0.106 0.054
In Table 3, we listed the 20 most intense gel spots. Observe already from the first few rows of Table 3 that intensities of spots regularly decrease only in the control group. However, once spots are ordered we maintain the same ordering of spots in the remaining six columns corresponding to different concentrations of LY1711883 (percent in diet). The D/D matrix is constructed by calculating the Euclidean distance through the space for any pair of spots (i, j) and then calculating the distance between the same two spots i and j along the zigzag line. The matrix element is given as the quotient DE(i,j)/ DL(i,j), where DE and DL are the distances measured through the space and along the zigzag line, respectively. Leading Eigenvalue of D/D Matrix. Once we have associated a matrix with a map we can consider various matrix invariants as potential descriptors for the map. As is known from chemical graph theory,13-16,24 where mathematical invariants are used for characterization of molecular structures, there are no general rules of how to construct and how to select one set of invariants over the other set. Invariants, which are known in QSAR as topological indices, are judged by their utility: If they offer useful correlation between structure and property they became important, if not, they fade away and tend to be forgotten. However, it is important to realize (1) that judgment on novel invariants comes after they have been proposed and (2) that often it not easy to predict in advance whether an invariant will be useful or not for a particular task. This situation extends to characterization of proteomics maps. In our earlier work we selected the leading eigenvalue of D/D matrix as an invariant of choice even though the question of utility of numerical characterizations of proteomics maps has yet to be
Table 4. Leading Eigenvalues for Different Concentrations and Different Powers of nD/nD Matrixes
1 2 3 4 5 6 7 8 9 10 15 20 25 30 35 40 50 60 70 80
0
0.003
0.01
0.03
0.1
0.3
0.6
9.7811 4.2507 3.4662 3.1789 3.0265 2.9237 2.8447 2.7797 2.7386 2.7310 2.7074 2.6924 2.6800 2.6687 2.6582 2.6483 2.6297 2.6124 2.5960 2.5804
9.6913 4.0793 3.2282 2.8988 2.7658 2.7267 2.7001 2.6799 2.6634 2.6492 2.5944 2.5514 2.5144 2.4815 2.4520 2.4254 2.3793 2.3406 2.3076 2.2792
9.7450 4.2667 3.4098 3.0611 2.8654 2.7545 2.7312 2.7142 2.7006 2.6893 2.6479 2.6161 2.5879 2.5617 2.5371 2.5138 2.4707 2.4317 2.3961 2.3636
9.7638 4.2661 3.4293 3.1042 2.9270 2.8062 2.7225 2.7046 2.6903 2.6782 2.6329 2.5974 2.5657 2.5364 2.5088 2.4829 2.4352 2.3924 2.3537 2.3188
9.7180 4.1557 3.2978 2.9428 2.7732 2.7353 2.7097 2.6904 2.6747 2.6613 2.6110 2.5728 2.5401 2.5111 2.4850 2.4613 2.4198 2.3844 2.3538 2.3269
9.6679 4.2476 3.5104 3.2337 3.0866 2.9884 2.9136 2.8523 2.8002 2.7548 2.5945 2.5010 2.4452 2.4236 2.4069 2.3936 2.3738 2.3596 2.3485 2.3393
9.6418 4.2040 3.3889 3.0461 2.8556 2.7308 2.6403 2.5701 2.5133 2.4658 2.3713 2.3502 2.3354 2.3230 2.3119 2.3016 2.2825 2.2649 2.2486 2.2333
better explored. In this paper, we will present an application of the leading eigenvalue of D/D matrixes for characterization of proteomics maps, which demonstrates use of the leading eigenvalue of D/D matrixes as useful map descriptor. Let us comment on why we selected the leading eigenvalue of a matrix instead of several other possibilities, such as the average matrix element that is related to the Wiener index25 that has some prominence in QSAR, or an index analogous to Balaban’s J index,26 also having use in QSAR. The mentioned alternatives, as well as a dozen other descriptors, are legitimate choices, and there is no doubt that they also will be examined as descriptors for characterizations of maps. While many topological indices may have somewhat unclear structural interpretation, the leading eigenvalues of D/D matrixes17-21,27-29 have been found to have rather interesting structural interpretation as descriptors. They offer, at least in the case of chain structures (as is the case with zigzag line considered here), a measure of the degree of bending of a structure. By extension, the leading eigenvalue of cyclic structures is some measure of the compactness of a structure. Because the zigzag 3-D curve of the control group is descending regularly, one expects that the leading eigenvalue of this curve will be accompanied with the largest leading eigenvalue in comparison with the leading eigenvalues of curves corresponding to different concentrations of LY171883. The other zigzag curves will show some oscillatory variations in spot abundances along the zigzag line and, hence, will induce a greater “bending” of the line, thus reducing the corresponding leading eigenvalue. To what extent this is the case can be seen from the first numerical row of Table 4 in Journal of Proteome Research • Vol. 1, No. 3, 2002 219
research articles which we have listed the leading eigenvalue of D/D matrixes for the control group and the six concentrations of LY171883 considered. Higher Order nD/nD Matrixes. A single invariant, even if it offers some interpretation, is by far too limited descriptor for useful characterization of 2-D maps. We need one dozen, two dozen, and even more such descriptors if we hope to capture more information on similarities and dissimilarities of different maps. A possible route to additional matrix invariants is to perform some matrix operation on the D/D matrix, such as raising matrixes to higher power, and thus generating additional map matrixes. This can be accomplished in two ways, either (1) to consider the Kronecker (or Hadamard) multiplication of matrixes in which individual matrix elements are multiplied (or raised to higher powers) or (2) to consider the standard matrix multiplication. The Kronecker (or Hadamard) product of matrixes is referred to as “element-by-element array multiplication” in the MATLAB tutorial.30 MATLAB is software tool suitable for manipulation of matrixes and arrays and visualization of the results of such operations.31 Since the matrix elements of D/D matrixes are already less than, or at most equal to 1, we decided to construct additional invariants applying the Kronecker product on the D/D matrix. In this way, we generate nD/nD matrixes of ever increasing power. In the remaining rows of Table 4 we have listed the leading eigenvalues of so-constructed higher order nD/nD matrixes starting with the exponent n ) 1 (which is the already considered D/D matrix) and stopping at n ) 80. Each column in Table 4 corresponds to the leading eigenvalue of the seven maps (the control map and the six maps corresponding to the six different concentrations of LY171883). We included in Table 4 the results only for the powers 1-10, and then we show the eigenvalues after increasing the powers by 5 (from 10 to 40), and finally we show the eigenvalues after increasing the powers by 10 (from 40 to 80). We carried our calculations using all 80 powers, that is, an 80-component vector (column) represents each map. There are two question to consider: (1) how well the 80-component vectors characterize the seven individual maps and (2) what is the smallest number of components of a vector that offers useful characterization of selected properties. The plot of the leading eigenvalues against the exponent n that defines the nth order nD/nD matrix is shown in Figure 4. The curve correspond to the leading eigenvalues of the control group while the leading eigenvalues of the six different doses of LY171883 show very similar dependence on n. Hence, all seven curves show the same general shape with some but not pronounced differences in details of the curves. To magnify these minor differences in Figure 5a-f we show plots in which instead of the leading eigenvalues we chosen the quotient of the leading eigenvalues for various concentrations and the corresponding leading eigenvalue of the control group. As has been outlined in ref 5, this approach may for different compounds and concentration produced considerably different curves, but as we see from Figure 5-f no simple pattern of quotient curves emerges that would suggest a regular increase of dose concentrations in mouse diets. Hence, we have to resort to novel avenues of processing the numerical data of Table 4. Principal Component Analysis (PCA) on nD/nD Eigenvalues. Principal component analysis is a multivariate technique for examining relationships among several quantitative variables. The method originated 100 years ago by Pearson32 and was fully developed later by Hotelling.33 We will use the principal 220
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Randic´ et al.
Figure 4. Plot of the leading eigenvalue of nD/nD matrix against the exponent n for the control group.
component analysis to evaluate derived numerical characterizations of the 2-D proteomics maps. In the Appendix, we have outlined a brief mathematical description of PCA. To analyze the eigenvalues of Table 4 and their quotients illustrated in Figure 5a-f we will simplify the input data and consider vectors having 24 rather than 80 components. Besides the first 10 eigenvalues associated with powers 1-10, we included for the remaining components only powers that increase by five: 15, 20, 25, ..., 75, 80. However, to test use of novel map descriptors, we would need to have a large number of experimental points for fitting. In view that we have in all seven input dose concentrations (including the control group which has zero concentration of LY171883 in food diet), we have to reduce further the number of vector components. We decided to use vectors with six components. To select descriptors we have examined more closely the leading eigenvalues of Table 4 and noticed that the relative magnitudes vary as we change n, the exponent, which runs from n ) 1 to n ) 80. In Table 5, we have listed the variation of the relative magnitudes by using labels 1-7, where 1 belongs to the largest leading eigenvalue and 7 to the smallest leading eigenvalue. As we see from Table 5, in all there are eight different relative orderings associated with the displayed powers. These are: n ) 1, 2, 3-6, 7-10, 15, 20-30, 35-50, and 60-80. For the six components of vectors characterizing maps, we have selected n ) 1, 5, 10, 20, 40, and 80; thus, after n ) 5 each successive choice for the component involves the exponent n, which has been doubled. Hence, we have dropped the eigenvalue for n ) 2 and n ) 15 and kept only one of the egenvalues of each of the remaining six groups. As will be seen from the following paragraph, the approach appears robust and alternative selections produce similar results. The next question to decide is how many descriptors will be used in PCA. We performed the PCA using as input the seven vectors corresponding to seven dose concentrations having from three to five components, selected from the following six powers (1, 5, 10, 20, 40, 80). The components correspond to the leading eigenvalues of concentrations of LY171883 in mouse diet from 0 to 0.6. The selection of the six powers (1, 5, 10, 20, 40, 80) is to some degree arbitrary, because equally other choices are possible, such as, for example, (3, 6, 15, 30, 50, 70).
2-D Proteomics Maps by Matrix Invariants
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Figure 5. Plot of the quotient of the leading eigenvalue of nD/nD matrixes corresponding to various concentrations of LY171883 in diet and the leading eigenvalue of the control group against the exponent n.
However, as long as a choice covers the whole range of the powers, as the both above choices do, one may expect similar results. This is because the leading eigenvalues of alternative choices themselves are highly interrelated. Let consider more closely the above two choices for the case of the control group (given in the first column of eigenvalues of Table 4), which
translate into the following two vectors with components of the leading eigenvalues: (9.7811, 3.0265, 2.7310, 2.65924, 2.6483, 2.5804) and (3.4662, 2.9237, 2.7074, 2.6687, 2.6297, 2.5960), respectively. A correlation between the corresponding components of the two vectors is shown in Figure 6. When the plot is fitted by cubic polynomial we obtain for the correlation Journal of Proteome Research • Vol. 1, No. 3, 2002 221
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Randic´ et al. Table 7. Principal Component Regression Based on Use of Three to Five Components as Descriptorsa
3 4 5 PC1
69.38% 0.162 533 0.0030 3 0.1355 4 0.0057 5 0.0025
r
s
F
0.3738 0.9925 0.9997
0.296 0.048 0.014
0.2 32.9 326.3
PC2
PC3
PC4
22.02% 7.06% 0.398 297 -0.937 406 0.0100 0.030 -0.0223 0.117 0.0089 0.028 0.0091 0.033
PC5
Cons.
1.50% 0.04% 6.820 919 -5.685 923 0.100 0.300 0.214 0.203 0.065 0.273 0.105 0.289
0.149 000 0.600 0.187 0.612 0.604
a R, S, and F designate the correlation coefficient, the standard error, and the Fisher ratio, respectively.
Table 8. Multivariate Linear Regression Using from Two to Five Descriptorsa Figure 6. Fitting of cubic polynomial to plot of components of vector (3, 6, 15, 30, 50, 70) against the corresponding components of vector (1, 5, 10, 20, 40, 80).
2 3 4 5
Table 5. Relative Magnitudes of the Leading Eigenvalues for Different Concentrations and Different Powers of nD/nD Matrices power
0
0.003
0.01
0.03
0.1
0.3
0.6
1 2 3 4 5 6 7 8 9 10 15 20 25 30 35 40 50 60 70 80
1 3 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1
5 7 7 7 7 7 6 6 6 6 6 5 5 5 5 5 5 6 6 6
3 1 4 4 4 4 3 3 3 3 2 2 2 2 2 2 2 2 2 2
2 2 3 3 3 3 4 4 4 4 3 3 3 3 3 3 3 3 4 4
4 6 6 6 6 6 5 5 5 5 4 4 4 4 4 4 4 4 3 3
6 4 1 1 1 1 1 1 1 1 5 6 6 6 6 6 6 5 5 5
7 5 5 5 5 5 7 7 7 7 7 7 7 7 7 7 7 7 7 7
Table 6. Correlation Coefficient (r), the Standard Error (s), and the Fisher Ratio (F) for Correlation among Vectors Using D/D Matrxes with Powers (1, 5, 10, 20, 40, 80) and Powers (3, 6, 15, 30, 50, 70) for Different Concentrations of LY17188 Based on Cubic Polynomial concn (M)
r
s
F
0 0.003 0.01 0.03 0.1 0.3 0.6
1.000 00 0.999 96 1.000 00 1.000 00 0.999 99 0.999 96 0.999 99
0.0055 0.0419 0.0128 0.0042 0.0206 0.0399 0.0244
465 072 8123 86 579 665 644 33 656 8772 24 258
coefficient r ) 1.0000, the standard error s ) 0.005, and the Fisher ratio F ) 465072. In Table 6, we show the r, s, and F values for the similar correlations of the remaining six vectors corresponding to different concentrations of LY17188 in mice food diet constructed by alternative choice of the powers of D/D matrixes. As we see all cases show very great degree of interrelation that well exceed in quality the correlations be222
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parameter
2
A B A B A B A B
3 4 5 a
r
s
F
0.9234 0.9390 0.9702 0.9994
0.106 0.110 0.095 0.018
12 7 8 185
value
error
t value
probability
0.02195 0.85266 0.01762 0.88176 0.00875 0.94127 0.00016 0.99946
0.04068 0.15851 0.03706 0.14440 0.02698 0.10514 0.00377 0.01471
0.47543 6.10617 0.32428 8.95223 0.04274 67.92877
0.65453 0.00171 0.75885 0.00029 0.96752
