CompaRNA - on-line benchmarks of RNA structure prediction methods
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Table of contents:

  1. Overview

  2. Performance Plots

  3. Performance of CentroidFold - scored higher in this pairwise comparison

  4. Performance of MaxExpect - scored lower in this pairwise comparison

  5. Compile and download dataset for CentroidFold & MaxExpect [.zip] - may take several seconds...


Overview

Metric CentroidFold MaxExpect
MCC 0.554 > 0.549
Average MCC ± 95% Confidence Intervals 0.654 ± 0.050 > 0.644 ± 0.053
Sensitivity 0.455 < 0.477
Positive Predictive Value 0.674 > 0.631
Total TP 2785 < 2920
Total TN 6889574 > 6889079
Total FP 1489 < 1897
Total FP CONTRA 145 < 166
Total FP INCONS 1201 < 1540
Total FP COMP 143 < 191
Total FN 3331 > 3196
P-value 3.09925167801e-06

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Performance plots


  1. Comparison of performance of CentroidFold and MaxExpect. Positive Predictive Value (PPV) is plotted against sensitivity. Each dot represents a single test of each method. See tables below for raw data (individual counts for CentroidFold and MaxExpect).

  2. Average Matthews Correlation Coefficients (MCC) with 95% confidence intervals (CIs) were plotted for different RNA families, for which at least 3 members were present in the benchmarking dataset. 'n' denotes the number of MCCs used to calculate the average and CI. See tables below for raw data (individual counts for CentroidFold and MaxExpect).

  3. Comparison of average Matthews Correlation Coefficients (MCCs) for CentroidFold and MaxExpect. The whiskers correspond to 95% confidence intervals (CIs). 'n' denotes the number of MCCs used to calculate average MCCs and CIs. See tables below for raw data (individual counts for CentroidFold and MaxExpect).

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Performance of CentroidFold - scored higher in this pairwise comparison

1. Total counts & total scores for CentroidFold

Total Base Pair Counts
Total TP 2785
Total TN 6889574
Total FP 1489
Total FP CONTRA 145
Total FP INCONS 1201
Total FP COMP 143
Total FN 3331
Total Scores
MCC 0.554
Average MCC ± 95% Confidence Intervals 0.654 ± 0.050
Sensitivity 0.455
Positive Predictive Value 0.674
Nr of predictions 130

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2. Individual counts for CentroidFold [ download as .csv ]

RNA Chain Rfam family MCC SENS PPV TP TN FP FP CONTRA FP INCONS FP COMP FN
2KD8_A - 1.00 1.00 1.00 9 222 0 0 0 0 0
2KE6_A 0.97 0.95 1.00 18 1110 1 0 0 1 1
2KPV_A - 0.96 0.92 1.00 12 549 0 0 0 0 1
2KRL_A - 0.76 0.60 0.96 24 5126 5 0 1 4 16
2KU0_A - 0.47 0.31 0.71 5 696 2 0 2 0 11
2KUR_A 0.95 0.90 1.00 19 1109 0 0 0 0 2
2KUU_A 0.92 0.86 1.00 18 1110 1 0 0 1 3
2KUV_A 0.93 0.86 1.00 19 1109 0 0 0 0 3
2KUW_A 0.95 0.90 1.00 19 1109 0 0 0 0 2
2KX8_A 0.94 0.89 1.00 16 845 0 0 0 0 2
2KXM_A - 0.89 0.80 1.00 8 343 0 0 0 0 2
2KZL_A - 0.86 0.74 1.00 14 1471 0 0 0 0 5
2L1F_B 0.98 0.96 1.00 24 2121 0 0 0 0 1
2L1F_A 0.98 0.96 1.00 23 2057 0 0 0 0 1
2L2K_A - 0.94 0.89 1.00 17 844 0 0 0 0 2
2L3C_B - 0.91 0.82 1.00 14 547 0 0 0 0 3
2L3E_A - 0.96 0.92 1.00 12 583 1 0 0 1 1
2L3J_B 0.94 0.88 1.00 30 2455 0 0 0 0 4
2L5Z_A - 0.86 0.75 1.00 9 316 0 0 0 0 3
2L94_A 0.92 0.90 0.95 18 971 1 0 1 0 2
2LA5_A - 0.42 0.26 0.71 5 623 2 0 2 0 14
2LBS_A - 1.00 1.00 1.00 14 482 0 0 0 0 0
2LC8_A 0.49 0.35 0.70 7 1530 3 1 2 0 13
2LDL_A - 0.90 0.82 1.00 9 342 1 0 0 1 2
2LDT_A - 0.85 0.73 1.00 11 454 0 0 0 0 4
2LHP_A - 0.97 0.94 1.00 15 651 0 0 0 0 1
2LI4_A - 0.93 0.88 1.00 14 482 0 0 0 0 2
2LJJ_A - 1.00 1.00 1.00 8 343 2 0 0 2 0
2LK3_A - 0.95 0.90 1.00 9 267 0 0 0 0 1
2LKR_A - 0.82 0.74 0.91 29 6073 4 0 3 1 10
2LQZ_A - 0.85 0.82 0.90 9 341 1 1 0 0 2
2LU0_A - 0.93 0.88 1.00 14 1162 0 0 0 0 2
2LWK_A - 0.92 0.92 0.92 12 483 1 0 1 0 1
2M58_A - 0.09 0.06 0.14 1 1646 6 0 6 0 16
2RRC_A - 0.70 0.50 1.00 5 226 0 0 0 0 5
2WRQ_Y 0.57 0.59 0.56 10 2832 12 5 3 4 7
2WW9_D - 0.29 0.12 0.75 3 1949 1 0 1 0 23
2WW9_E - 0.00 0.00 0.00 0 561 0 0 0 0 14
2WW9_F - 0.59 0.60 0.60 6 290 4 0 4 0 4
2WWQ_V 0.80 0.68 0.95 19 2906 3 0 1 2 9
2XKV_B 0.51 0.50 0.53 10 4541 25 0 9 16 10
2XQD_Y 0.75 0.67 0.86 18 2829 3 0 3 0 9
2XXA_G 0.15 0.12 0.20 5 5126 20 1 19 0 37
2Y8W_B - 1.00 1.00 1.00 6 184 1 0 0 1 0
2Y9C_V - 0.66 0.63 0.71 15 2257 8 1 5 2 9
2YIE_Z - 0.43 0.42 0.45 5 1529 6 3 3 0 7
2YIE_X - 0.67 0.58 0.78 7 1369 3 1 1 1 5
3A2K_C 0.44 0.43 0.46 12 2900 14 2 12 0 16
3ADB_C - 0.92 0.84 1.00 32 4154 0 0 0 0 6
3AKZ_H 0.43 0.39 0.48 11 2678 12 2 10 0 17
3AM1_B - 0.68 0.63 0.73 22 3210 8 1 7 0 13
3AMU_B 0.70 0.59 0.84 16 2984 5 0 3 2 11
3GX2_A 0.79 0.63 1.00 25 4346 1 0 0 1 15
3IAB_R - 0.86 0.86 0.86 12 1021 4 1 1 2 2
3IVN_B 0.78 0.61 1.00 19 2327 0 0 0 0 12
3IWN_A 0.80 0.67 0.96 22 4255 1 0 1 0 11
3IYQ_A 0.36 0.37 0.34 35 60624 71 17 50 4 59
3IZ4_A 0.55 0.45 0.68 59 70789 33 4 24 5 73
3IZF_C 0.68 0.61 0.77 33 6860 10 1 9 0 21
3J0L_2 - 0.25 0.24 0.28 8 6187 24 0 21 3 25
3J0L_7 - -0.01 0.00 0.00 0 1220 5 0 5 0 17
3J0L_h - 0.77 0.65 0.90 28 6074 5 0 3 2 15
3J0L_g - 0.00 0.00 0.00 0 465 0 0 0 0 4
3J0L_1 - 0.65 0.47 0.90 9 1215 2 0 1 1 10
3J0L_a - 0.41 0.31 0.56 5 1119 4 2 2 0 11
3J0L_8 - 0.93 0.88 1.00 7 183 0 0 0 0 1
3J16_L 0.50 0.40 0.63 12 2756 7 0 7 0 18
3J20_1 0.91 0.87 0.95 20 2905 2 0 1 1 3
3J20_0 0.44 0.40 0.50 12 2826 13 0 12 1 18
3J20_2 0.56 0.48 0.67 303 1116310 158 8 144 6 330
3J2C_M - 0.40 0.29 0.57 59 106387 47 5 40 2 148
3J2C_O - 0.76 0.65 0.89 41 10250 6 0 5 1 22
3J2L_3 0.53 0.49 0.58 26 7830 21 0 19 2 27
3J3D_C 0.49 0.43 0.57 12 2754 9 0 9 0 16
3J3E_8 0.00 0.00 0.00 0 7485 21 2 16 3 33
3J3E_7 0.46 0.37 0.59 20 7106 14 1 13 0 34
3J3F_8 0.34 0.33 0.34 12 12211 34 4 19 11 24
3J3F_7 0.20 0.18 0.23 9 7221 30 1 29 0 41
3J3V_B 0.47 0.37 0.60 21 6986 14 1 13 0 36
3JYV_7 0.77 0.59 1.00 19 2831 0 0 0 0 13
3JYX_3 0.29 0.30 0.30 8 6301 20 8 11 1 19
3JYX_4 0.23 0.21 0.25 7 12218 25 5 16 4 26
3KIY_A - 0.53 0.41 0.69 505 4053393 246 24 206 16 736
3KTW_C - 0.45 0.40 0.53 17 4528 16 2 13 1 26
3LA5_A 0.78 0.62 1.00 21 2464 0 0 0 0 13
3NDB_M - 0.80 0.70 0.91 43 9133 5 0 4 1 18
3NKB_B - 0.57 0.50 0.65 13 1996 7 0 7 0 13
3NMU_E - 0.71 0.50 1.00 3 558 3 0 0 3 3
3NPB_A 0.77 0.70 0.86 32 6984 7 1 4 2 14
3O58_2 0.82 0.76 0.88 29 7227 7 1 3 3 9
3O58_3 0.38 0.29 0.50 10 12383 10 0 10 0 25
3OVB_D - 1.00 1.00 1.00 11 584 1 0 0 1 0
3OVS_D - 0.96 0.92 1.00 12 549 0 0 0 0 1
3P22_G - 0.96 0.92 1.00 11 730 1 0 0 1 1
3PDR_A 0.73 0.60 0.90 43 12832 7 0 5 2 29
3R4F_A - 0.81 0.76 0.86 19 2123 3 1 2 0 6
3R9X_C - 0.84 0.80 0.89 8 586 2 0 1 1 2
3RKF_A 0.76 0.59 1.00 20 2191 0 0 0 0 14
3SD1_A 0.58 0.48 0.71 20 3888 8 1 7 0 22
3SIU_F - 0.79 0.64 1.00 7 371 0 0 0 0 4
3SN2_B 0.96 0.92 1.00 11 395 0 0 0 0 1
3TRZ_Z - 0.91 0.83 1.00 5 205 1 0 0 1 1
3TS0_U - 1.00 1.00 1.00 6 247 1 0 0 1 0
3TS2_V - 1.00 1.00 1.00 5 271 0 0 0 0 0
3U4M_B - 0.57 0.46 0.71 17 3136 7 0 7 0 20
3UZL_B 0.72 0.54 0.95 20 3549 1 0 1 0 17
3VJR_D - 0.96 0.92 1.00 12 618 0 0 0 0 1
3W1K_J 0.56 0.50 0.63 19 4156 11 2 9 0 19
3W3S_B 0.87 0.78 0.97 31 4721 2 0 1 1 9
3ZEX_C 0.47 0.27 0.82 14 14179 3 1 2 0 38
3ZEX_G - 0.36 0.28 0.46 21 16425 29 2 23 4 53
3ZEX_E - 0.00 0.00 0.00 0 21895 52 3 47 2 77
3ZEX_D 0.73 0.59 0.91 29 6989 3 0 3 0 20
3ZEX_B - 0.23 0.16 0.34 87 1072121 178 9 163 6 471
3ZEX_H - 0.19 0.18 0.21 7 9011 27 4 23 0 31
3ZEX_F - 0.00 0.00 0.00 0 2619 9 2 7 0 12
3ZND_W 0.20 0.22 0.19 5 2977 23 1 20 2 18
4A1C_3 0.67 0.56 0.81 30 7103 7 1 6 0 24
4A1C_2 0.16 0.15 0.17 5 11752 29 5 19 5 28
4A4U_A - 1.00 1.00 1.00 9 222 0 0 0 0 0
4AOB_A 0.45 0.33 0.61 14 4348 10 1 8 1 28
4ATO_G - 0.31 0.20 0.50 2 524 2 0 2 0 8
4ENB_A 0.76 0.58 1.00 11 1264 0 0 0 0 8
4ENC_A 0.73 0.58 0.92 11 1314 1 1 0 0 8
4FNJ_A - 0.15 0.13 0.22 2 586 7 0 7 0 14
4FRG_B 0.24 0.22 0.28 7 3461 18 3 15 0 25
4FRN_A 0.40 0.28 0.59 10 5134 7 2 5 0 26
4HXH_A - 1.00 1.00 1.00 6 319 0 0 0 0 0
4JF2_A 0.78 0.61 1.00 19 2831 0 0 0 0 12
4JRC_A - 0.81 0.70 0.94 16 1523 1 0 1 0 7

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Performance of MaxExpect - scored lower in this pairwise comparison

1. Total counts & total scores for MaxExpect

Total Base Pair Counts
Total TP 2920
Total TN 6889079
Total FP 1897
Total FP CONTRA 166
Total FP INCONS 1540
Total FP COMP 191
Total FN 3196
Total Scores
MCC 0.549
Average MCC ± 95% Confidence Intervals 0.644 ± 0.053
Sensitivity 0.477
Positive Predictive Value 0.631
Nr of predictions 130

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2. Individual counts for MaxExpect [ download as .csv ]

RNA Chain Rfam family MCC SENS PPV TP TN FP FP CONTRA FP INCONS FP COMP FN
2KD8_A - 1.00 1.00 1.00 9 222 0 0 0 0 0
2KE6_A 0.92 0.89 0.94 17 1110 2 0 1 1 2
2KPV_A - 0.96 0.92 1.00 12 549 0 0 0 0 1
2KRL_A - 0.76 0.60 0.96 24 5126 5 0 1 4 16
2KU0_A - -0.02 0.00 0.00 0 694 9 0 9 0 16
2KUR_A 0.90 0.86 0.95 18 1109 1 0 1 0 3
2KUU_A 0.87 0.81 0.94 17 1110 2 0 1 1 4
2KUV_A 0.88 0.82 0.95 18 1109 1 0 1 0 4
2KUW_A 0.90 0.86 0.95 18 1109 1 0 1 0 3
2KX8_A 0.94 0.89 1.00 16 845 0 0 0 0 2
2KXM_A - 0.83 0.70 1.00 7 344 0 0 0 0 3
2KZL_A - 0.86 0.74 1.00 14 1471 0 0 0 0 5
2L1F_B 0.98 0.96 1.00 24 2121 0 0 0 0 1
2L1F_A 0.98 0.96 1.00 23 2057 0 0 0 0 1
2L2K_A - 0.94 0.89 1.00 17 844 0 0 0 0 2
2L3C_B - 0.91 0.82 1.00 14 547 0 0 0 0 3
2L3E_A - 0.96 0.92 1.00 12 583 1 0 0 1 1
2L3J_B 0.94 0.88 1.00 30 2455 0 0 0 0 4
2L5Z_A - 0.86 0.75 1.00 9 316 0 0 0 0 3
2L94_A 0.97 0.95 1.00 19 971 0 0 0 0 1
2LA5_A - 0.39 0.26 0.63 5 622 3 1 2 0 14
2LBS_A - 1.00 1.00 1.00 14 482 0 0 0 0 0
2LC8_A -0.01 0.00 0.00 0 1525 15 2 13 0 20
2LDL_A - 0.90 0.82 1.00 9 342 1 0 0 1 2
2LDT_A - 0.85 0.73 1.00 11 454 0 0 0 0 4
2LHP_A - 0.97 0.94 1.00 15 651 0 0 0 0 1
2LI4_A - 0.93 0.88 1.00 14 482 0 0 0 0 2
2LJJ_A - 1.00 1.00 1.00 8 343 2 0 0 2 0
2LK3_A - 0.95 0.90 1.00 9 267 0 0 0 0 1
2LKR_A - 0.81 0.74 0.88 29 6072 6 0 4 2 10
2LQZ_A - 0.85 0.82 0.90 9 341 1 1 0 0 2
2LU0_A - 1.00 1.00 1.00 16 1160 0 0 0 0 0
2LWK_A - 0.88 0.85 0.92 11 484 1 0 1 0 2
2M58_A - -0.01 0.00 0.00 0 1640 13 0 13 0 17
2RRC_A - 0.70 0.50 1.00 5 226 0 0 0 0 5
2WRQ_Y 0.53 0.53 0.53 9 2833 13 5 3 5 8
2WW9_F - 0.69 0.70 0.70 7 290 3 0 3 0 3
2WW9_D - 0.14 0.12 0.19 3 1937 13 0 13 0 23
2WW9_E - -0.01 0.00 0.00 0 556 5 0 5 0 14
2WWQ_V 0.76 0.64 0.90 18 2906 3 0 2 1 10
2XKV_B 0.51 0.50 0.53 10 4541 24 0 9 15 10
2XQD_Y 0.88 0.78 1.00 21 2829 0 0 0 0 6
2XXA_G 0.25 0.24 0.27 10 5114 27 1 26 0 32
2Y8W_B - 1.00 1.00 1.00 6 184 1 0 0 1 0
2Y9C_V - 0.65 0.63 0.68 15 2256 8 1 6 1 9
2YIE_X - 0.36 0.33 0.40 4 1368 6 2 4 0 8
2YIE_Z - -0.01 0.00 0.00 0 1525 15 3 12 0 12
3A2K_C 0.42 0.39 0.46 11 2902 13 2 11 0 17
3ADB_C - 0.80 0.74 0.88 28 4154 4 0 4 0 10
3AKZ_H 0.84 0.75 0.95 21 2679 1 1 0 0 7
3AM1_B - 0.78 0.71 0.86 25 3211 4 0 4 0 10
3AMU_B 0.64 0.59 0.70 16 2980 9 0 7 2 11
3GX2_A 0.77 0.63 0.96 25 4345 2 0 1 1 15
3IAB_R - 0.92 0.86 1.00 12 1023 0 0 0 0 2
3IVN_B 0.78 0.61 1.00 19 2327 0 0 0 0 12
3IWN_A 0.74 0.67 0.81 22 4251 6 0 5 1 11
3IYQ_A 0.29 0.31 0.27 29 60617 84 19 61 4 65
3IZ4_A 0.54 0.46 0.64 61 70781 39 3 31 5 71
3IZF_C 0.71 0.61 0.83 33 6863 7 1 6 0 21
3J0L_7 - -0.01 0.00 0.00 0 1213 12 0 12 0 17
3J0L_g - -0.01 0.00 0.00 0 461 4 1 3 0 4
3J0L_2 - 0.25 0.24 0.26 8 6185 25 2 21 2 25
3J0L_8 - 0.93 0.88 1.00 7 183 0 0 0 0 1
3J0L_a - 0.39 0.31 0.50 5 1118 5 2 3 0 11
3J0L_1 - 0.76 0.63 0.92 12 1212 2 0 1 1 7
3J0L_h - 0.81 0.65 1.00 28 6077 2 0 0 2 15
3J16_L 0.29 0.23 0.37 7 2756 12 0 12 0 23
3J20_2 0.49 0.44 0.56 276 1116272 221 10 207 4 357
3J20_0 0.44 0.40 0.50 12 2826 13 0 12 1 18
3J20_1 0.73 0.70 0.76 16 2905 5 0 5 0 7
3J2C_M - 0.38 0.31 0.47 64 106354 76 10 63 3 143
3J2C_O - 0.63 0.51 0.78 32 10255 10 0 9 1 31
3J2L_3 0.62 0.53 0.74 28 7837 12 0 10 2 25
3J3D_C 0.47 0.43 0.52 12 2752 11 1 10 0 16
3J3E_7 0.46 0.37 0.59 20 7106 14 1 13 0 34
3J3E_8 0.07 0.06 0.08 2 7478 32 2 21 9 31
3J3F_8 0.32 0.31 0.33 11 12213 36 3 19 14 25
3J3F_7 0.79 0.68 0.92 34 7223 4 0 3 1 16
3J3V_B 0.50 0.40 0.62 23 6984 14 1 13 0 34
3JYV_7 0.41 0.31 0.56 10 2832 8 0 8 0 22
3JYX_3 0.64 0.63 0.65 17 6302 19 1 8 10 10
3JYX_4 0.31 0.30 0.31 10 12214 34 4 18 12 23
3KIY_A - 0.54 0.45 0.65 553 4053271 314 23 281 10 688
3KTW_C - 0.54 0.47 0.65 20 4529 12 2 9 1 23
3LA5_A 0.78 0.62 1.00 21 2464 0 0 0 0 13
3NDB_M - 0.78 0.69 0.89 42 9133 6 0 5 1 19
3NKB_B - 0.59 0.54 0.67 14 1995 7 0 7 0 12
3NMU_E - -0.01 0.00 0.00 0 553 10 1 7 2 6
3NPB_A 0.71 0.61 0.82 28 6987 9 0 6 3 18
3O58_3 0.35 0.34 0.35 12 12369 35 2 20 13 23
3O58_2 0.75 0.76 0.74 29 7221 11 3 7 1 9
3OVB_D - 1.00 1.00 1.00 11 584 1 0 0 1 0
3OVS_D - 0.96 0.92 1.00 12 549 0 0 0 0 1
3P22_G - 0.96 0.92 1.00 11 730 0 0 0 0 1
3PDR_A 0.76 0.63 0.94 45 12832 5 0 3 2 27
3R4F_A - 0.96 0.92 1.00 23 2122 0 0 0 0 2
3R9X_C - 0.84 0.80 0.89 8 586 2 0 1 1 2
3RKF_A 0.76 0.59 1.00 20 2191 0 0 0 0 14
3SD1_A 0.58 0.48 0.71 20 3888 8 1 7 0 22
3SIU_F - 0.85 0.73 1.00 8 370 0 0 0 0 3
3SN2_B 1.00 1.00 1.00 12 394 0 0 0 0 0
3TRZ_Z - 0.91 0.83 1.00 5 205 1 0 0 1 1
3TS0_U - 1.00 1.00 1.00 6 247 1 0 0 1 0
3TS2_V - 1.00 1.00 1.00 5 271 0 0 0 0 0
3U4M_B - 0.45 0.35 0.59 13 3138 9 1 8 0 24
3UZL_B 0.77 0.62 0.96 23 3546 1 0 1 0 14
3VJR_D - 0.96 0.92 1.00 12 618 0 0 0 0 1
3W1K_J 0.87 0.79 0.97 30 4155 1 1 0 0 8
3W3S_B 0.88 0.78 1.00 31 4722 1 0 0 1 9
3ZEX_H - 0.19 0.18 0.19 7 9009 29 5 24 0 31
3ZEX_B - 0.30 0.26 0.35 147 1071957 289 29 247 13 411
3ZEX_D 0.69 0.61 0.79 30 6983 8 1 7 0 19
3ZEX_E - 0.00 0.00 0.00 0 21893 54 2 50 2 77
3ZEX_G - 0.77 0.66 0.91 49 16417 11 0 5 6 25
3ZEX_C 0.41 0.27 0.64 14 14174 16 2 6 8 38
3ZEX_F - 0.00 0.00 0.00 0 2617 11 2 9 0 12
3ZND_W 0.20 0.22 0.19 5 2977 23 1 20 2 18
4A1C_3 0.69 0.57 0.84 31 7103 6 0 6 0 23
4A1C_2 0.14 0.15 0.14 5 11746 41 4 26 11 28
4A4U_A - 1.00 1.00 1.00 9 222 0 0 0 0 0
4AOB_A 0.50 0.40 0.63 17 4344 11 2 8 1 25
4ATO_G - 0.38 0.30 0.50 3 522 3 0 3 0 7
4ENB_A 0.76 0.58 1.00 11 1264 0 0 0 0 8
4ENC_A 0.31 0.26 0.38 5 1313 8 1 7 0 14
4FNJ_A - 0.79 0.63 1.00 10 585 0 0 0 0 6
4FRG_B 0.30 0.22 0.41 7 3469 10 1 9 0 25
4FRN_A 0.45 0.36 0.57 13 5128 10 2 8 0 23
4HXH_A - 1.00 1.00 1.00 6 319 0 0 0 0 0
4JF2_A 0.78 0.61 1.00 19 2831 0 0 0 0 12
4JRC_A - 0.29 0.26 0.35 6 1523 11 0 11 0 17

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Matthews Correlation Coeffient, Sensitivity and Positive Predictive Value have been calculated based on the paper by Gardener & Giegerich, 2004.