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

  1. Overview

  2. Performance Plots

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

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

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


Overview

Metric Pknots MCFold
MCC 0.588 > 0.522
Average MCC ± 95% Confidence Intervals 0.649 ± 0.054 > 0.599 ± 0.061
Sensitivity 0.535 = 0.535
Positive Predictive Value 0.652 > 0.516
Total TP 1611 = 1611
Total TN 350165 > 349515
Total FP 979 < 1767
Total FP CONTRA 111 < 194
Total FP INCONS 749 < 1316
Total FP COMP 119 < 257
Total FN 1398 = 1398
P-value 3.56938820447e-08

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Performance plots


  1. Comparison of performance of Pknots and MCFold. 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 Pknots and MCFold).

  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 Pknots and MCFold).

  3. Comparison of average Matthews Correlation Coefficients (MCCs) for Pknots and MCFold. 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 Pknots and MCFold).

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Performance of Pknots - scored higher in this pairwise comparison

1. Total counts & total scores for Pknots

Total Base Pair Counts
Total TP 1611
Total TN 350165
Total FP 979
Total FP CONTRA 111
Total FP INCONS 749
Total FP COMP 119
Total FN 1398
Total Scores
MCC 0.588
Average MCC ± 95% Confidence Intervals 0.649 ± 0.054
Sensitivity 0.535
Positive Predictive Value 0.652
Nr of predictions 131

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

RNA Chain Rfam family MCC SENS PPV TP TN FP FP CONTRA FP INCONS FP COMP FN
2K5Z_A - 0.90 0.82 1.00 9 397 0 0 0 0 2
2K63_A - 0.95 0.90 1.00 9 397 0 0 0 0 1
2K66_A - 0.95 0.90 1.00 9 222 0 0 0 0 1
2KD8_A - 1.00 1.00 1.00 9 222 0 0 0 0 0
2KDQ_B 0.95 0.91 1.00 10 396 0 0 0 0 1
2KE6_A 0.97 0.95 1.00 18 1110 1 0 0 1 1
2KEZ_A - 0.89 0.80 1.00 8 268 0 0 0 0 2
2KFC_A - 0.11 0.10 0.17 2 618 10 0 10 0 18
2KGP_A - 0.81 0.67 1.00 6 294 1 0 0 1 3
2KMJ_A - 1.00 1.00 1.00 11 367 0 0 0 0 0
2KPV_A - 0.96 0.92 1.00 12 549 0 0 0 0 1
2KU0_A - -0.02 0.00 0.00 0 693 10 0 10 0 16
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 - -0.02 0.00 0.00 0 346 5 0 5 0 10
2KZL_A - 0.59 0.53 0.67 10 1470 5 1 4 0 9
2L1F_A 0.98 0.96 1.00 23 2057 0 0 0 0 1
2L1F_B 0.98 0.96 1.00 24 2121 0 0 0 0 1
2L2K_A - 0.94 0.89 1.00 17 844 0 0 0 0 2
2L3C_B - 0.87 0.76 1.00 13 548 0 0 0 0 4
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 - 0.46 0.26 0.83 5 624 1 0 1 0 14
2LBS_A - 1.00 1.00 1.00 14 482 0 0 0 0 0
2LC8_A 0.79 0.75 0.83 15 1522 3 1 2 0 5
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 - 0.44 0.44 0.45 17 6067 22 4 17 1 22
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 - 0.33 0.23 0.50 3 490 3 0 3 0 10
2M58_A - 0.51 0.41 0.64 7 1642 4 1 3 0 10
2RP0_A - 0.66 0.58 0.78 7 342 2 1 1 0 5
2RPT_A - 1.00 1.00 1.00 7 183 0 0 0 0 0
2RRC_A - 0.77 0.60 1.00 6 225 0 0 0 0 4
2WRQ_Y 0.57 0.59 0.56 10 2832 12 5 3 4 7
2WW9_F - 0.59 0.60 0.60 6 290 4 0 4 0 4
2WW9_D - 0.35 0.31 0.42 8 1934 11 2 9 0 18
2WW9_E - 0.39 0.29 0.57 4 554 3 0 3 0 10
2WWQ_V 0.24 0.21 0.29 6 2905 16 1 14 1 22
2XKV_B 0.23 0.25 0.22 5 4537 31 1 17 13 15
2XQD_Y 0.88 0.78 1.00 21 2829 0 0 0 0 6
2XXA_G 0.32 0.31 0.34 13 5113 25 1 24 0 29
2Y8W_B - 1.00 1.00 1.00 6 184 1 0 0 1 0
2Y9C_V - 0.59 0.50 0.71 12 2261 6 1 4 1 12
2YIE_Z - -0.01 0.00 0.00 0 1525 15 3 12 0 12
2YIE_X - 0.36 0.33 0.40 4 1368 6 2 4 0 8
2ZY6_A - 0.58 0.58 0.58 7 691 5 1 4 0 5
2ZZN_D 0.86 0.78 0.95 21 2463 1 0 1 0 6
3A2K_C 0.45 0.43 0.48 12 2901 13 2 11 0 16
3A3A_A 0.87 0.76 1.00 28 3627 0 0 0 0 9
3AKZ_H 0.84 0.75 0.95 21 2679 1 1 0 0 7
3AM1_B - 0.91 0.83 1.00 29 3211 0 0 0 0 6
3AMU_B 0.86 0.74 1.00 20 2983 2 0 0 2 7
3DW4_A - 0.70 0.50 1.00 5 346 0 0 0 0 5
3GCA_A - 0.35 0.24 0.57 4 521 3 0 3 0 13
3GX2_A 0.47 0.40 0.55 16 4342 14 1 12 1 24
3HAY_E - 0.86 0.83 0.88 15 2261 5 0 2 3 3
3HJW_D - 0.94 0.89 1.00 16 1637 3 0 0 3 2
3IAB_R - 0.86 0.86 0.86 12 1021 4 1 1 2 2
3ID5_D - -0.01 0.00 0.00 0 491 5 1 4 0 9
3IVN_B 0.78 0.65 0.95 20 2325 1 0 1 0 11
3IWN_A 0.21 0.18 0.25 6 4254 18 1 17 0 27
3IZF_C 0.70 0.61 0.80 33 6862 8 1 7 0 21
3J0L_8 - 0.93 0.88 1.00 7 183 0 0 0 0 1
3J0L_2 - 0.36 0.36 0.36 12 6183 23 3 18 2 21
3J0L_g - -0.01 0.00 0.00 0 460 5 2 3 0 4
3J0L_7 - 0.28 0.29 0.29 5 1208 12 0 12 0 12
3J0L_h - 0.58 0.49 0.70 21 6075 11 0 9 2 22
3J0L_1 - 0.65 0.47 0.90 9 1215 2 0 1 1 10
3J0L_a - 0.35 0.31 0.42 5 1116 7 1 6 0 11
3J16_L 0.34 0.30 0.41 9 2753 13 0 13 0 21
3J20_0 0.84 0.70 1.00 21 2829 1 0 0 1 9
3J2L_3 0.62 0.51 0.75 27 7839 12 0 9 3 26
3J3D_C 0.46 0.43 0.50 12 2751 12 1 11 0 16
3J3E_8 0.07 0.06 0.08 2 7479 33 2 20 11 31
3J3E_7 0.43 0.35 0.54 19 7105 16 1 15 0 35
3J3F_7 0.27 0.24 0.32 12 7222 27 1 25 1 38
3J3F_8 0.36 0.36 0.37 13 12211 34 2 20 12 23
3JYV_7 0.81 0.66 1.00 21 2829 0 0 0 0 11
3JYX_3 0.34 0.37 0.31 10 6296 24 9 13 2 17
3JYX_4 0.18 0.21 0.16 7 12203 41 13 23 5 26
3KTW_C - 0.49 0.44 0.56 19 4526 16 2 13 1 24
3LA5_A 0.80 0.65 1.00 22 2463 0 0 0 0 12
3NDB_M - 0.77 0.69 0.88 42 9132 7 1 5 1 19
3NKB_B - 0.55 0.50 0.62 13 1995 8 0 8 0 13
3NMU_E - 0.54 0.50 0.60 3 556 5 0 2 3 3
3O58_3 0.27 0.31 0.24 11 12357 38 11 24 3 24
3O58_2 0.83 0.74 0.93 28 7230 3 0 2 1 10
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 0.54 0.44 0.65 32 12831 19 0 17 2 40
3R4F_A - 0.92 0.88 0.96 22 2122 1 0 1 0 3
3R9X_C - 0.73 0.70 0.78 7 586 5 0 2 3 3
3RKF_A 0.77 0.62 0.95 21 2189 1 0 1 0 13
3SD1_A 0.65 0.52 0.81 22 3889 5 0 5 0 20
3SIU_F - 0.73 0.55 1.00 6 372 0 0 0 0 5
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 - 0.26 0.22 0.32 8 3135 17 0 17 0 29
3VJR_D - 0.96 0.92 1.00 12 618 0 0 0 0 1
3ZEX_G - 0.36 0.32 0.41 24 16413 37 3 31 3 50
3ZEX_F - 0.00 0.00 0.00 0 2619 12 2 7 3 12
3ZEX_C 0.07 0.08 0.07 4 14141 54 4 47 3 48
3ZEX_H - 0.20 0.21 0.20 8 9005 32 5 27 0 30
3ZEX_D 0.27 0.24 0.30 12 6981 28 0 28 0 37
3ZND_W 0.20 0.22 0.19 5 2977 23 1 20 2 18
4A1C_2 0.24 0.24 0.25 8 11749 36 3 21 12 25
4A1C_3 0.25 0.22 0.29 12 7099 29 1 28 0 42
4A4U_A - 1.00 1.00 1.00 9 222 0 0 0 0 0
4ATO_G - 0.30 0.30 0.33 3 519 6 2 4 0 7
4ENB_A 0.83 0.79 0.88 15 1258 2 1 1 0 4
4ENC_A 0.86 0.79 0.94 15 1310 1 1 0 0 4
4FNJ_A - -0.02 0.00 0.00 0 585 10 0 10 0 16
4FRG_B 0.48 0.38 0.63 12 3467 7 0 7 0 20
4FRN_A 0.51 0.42 0.63 15 5127 9 1 8 0 21
4HXH_A - 1.00 1.00 1.00 6 319 0 0 0 0 0
4JF2_A 0.81 0.77 0.86 24 2822 4 3 1 0 7
4JRC_A - 0.81 0.70 0.94 16 1523 1 1 0 0 7

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Performance of MCFold - scored lower in this pairwise comparison

1. Total counts & total scores for MCFold

Total Base Pair Counts
Total TP 1611
Total TN 349515
Total FP 1767
Total FP CONTRA 194
Total FP INCONS 1316
Total FP COMP 257
Total FN 1398
Total Scores
MCC 0.522
Average MCC ± 95% Confidence Intervals 0.599 ± 0.061
Sensitivity 0.535
Positive Predictive Value 0.516
Nr of predictions 131

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

RNA Chain Rfam family MCC SENS PPV TP TN FP FP CONTRA FP INCONS FP COMP FN
2K5Z_A - 1.00 1.00 1.00 11 395 1 0 0 1 0
2K63_A - 0.90 0.90 0.90 9 396 4 0 1 3 1
2K66_A - 1.00 1.00 1.00 10 221 0 0 0 0 0
2KD8_A - 1.00 1.00 1.00 9 222 1 0 0 1 0
2KDQ_B 0.95 0.91 1.00 10 396 1 0 0 1 1
2KE6_A 1.00 1.00 1.00 19 1109 2 0 0 2 0
2KEZ_A - 0.90 0.90 0.90 9 266 1 0 1 0 1
2KFC_A - 0.28 0.25 0.36 5 616 9 2 7 0 15
2KGP_A - 1.00 1.00 1.00 9 291 1 0 0 1 0
2KMJ_A - 1.00 1.00 1.00 11 367 1 0 0 1 0
2KPV_A - 1.00 1.00 1.00 13 548 2 0 0 2 0
2KU0_A - 0.97 0.94 1.00 15 688 0 0 0 0 1
2KUR_A 1.00 1.00 1.00 21 1107 0 0 0 0 0
2KUU_A 0.98 0.95 1.00 20 1108 1 0 0 1 1
2KUV_A 0.98 0.95 1.00 21 1107 0 0 0 0 1
2KUW_A 0.95 0.95 0.95 20 1107 1 0 1 0 1
2KX8_A 0.91 0.89 0.94 16 844 2 0 1 1 2
2KXM_A - 1.00 1.00 1.00 10 341 1 0 0 1 0
2KZL_A - 0.80 0.84 0.76 16 1464 9 1 4 4 3
2L1F_A 0.72 0.75 0.69 18 2054 9 0 8 1 6
2L1F_B 0.73 0.76 0.70 19 2118 9 0 8 1 6
2L2K_A - 1.00 1.00 1.00 19 842 1 0 0 1 0
2L3C_B - 0.97 0.94 1.00 16 545 0 0 0 0 1
2L3E_A - 1.00 1.00 1.00 13 582 1 0 0 1 0
2L3J_B 1.00 1.00 1.00 34 2451 0 0 0 0 0
2L5Z_A - 0.96 0.92 1.00 11 314 0 0 0 0 1
2L94_A 0.97 0.95 1.00 19 971 0 0 0 0 1
2LA5_A - -0.03 0.00 0.00 0 615 15 0 15 0 19
2LBS_A - 1.00 1.00 1.00 14 482 1 0 0 1 0
2LC8_A 0.41 0.45 0.39 9 1517 16 0 14 2 11
2LDL_A - 0.95 0.91 1.00 10 341 2 0 0 2 1
2LDT_A - 0.75 0.73 0.79 11 451 3 0 3 0 4
2LHP_A - 1.00 1.00 1.00 16 650 1 0 0 1 0
2LI4_A - 0.97 0.94 1.00 15 481 0 0 0 0 1
2LJJ_A - 1.00 1.00 1.00 8 343 2 0 0 2 0
2LK3_A - 1.00 1.00 1.00 10 266 0 0 0 0 0
2LKR_A - 0.93 0.92 0.95 36 6067 14 0 2 12 3
2LQZ_A - 0.91 0.91 0.91 10 340 1 1 0 0 1
2LU0_A - 1.00 1.00 1.00 16 1160 5 0 0 5 0
2LWK_A - 0.92 0.92 0.92 12 483 2 0 1 1 1
2M58_A - 0.20 0.24 0.18 4 1631 19 2 16 1 13
2RP0_A - 0.44 0.42 0.50 5 341 5 1 4 0 7
2RPT_A - 1.00 1.00 1.00 7 183 2 0 0 2 0
2RRC_A - 0.95 0.90 1.00 9 222 0 0 0 0 1
2WRQ_Y 0.27 0.35 0.22 6 2823 25 8 13 4 11
2WW9_F - 0.59 0.60 0.60 6 290 4 0 4 0 4
2WW9_D - -0.01 0.00 0.00 0 1925 28 5 23 0 26
2WW9_E - -0.02 0.00 0.00 0 547 14 0 14 0 14
2WWQ_V 0.15 0.18 0.15 5 2892 30 4 25 1 23
2XKV_B 0.20 0.25 0.17 5 4530 37 2 23 12 15
2XQD_Y 0.34 0.37 0.32 10 2819 22 1 20 1 17
2XXA_G 0.23 0.24 0.24 10 5110 32 1 30 1 32
2Y8W_B - 1.00 1.00 1.00 6 184 2 0 0 2 0
2Y9C_V - 0.62 0.67 0.59 16 2251 13 2 9 2 8
2YIE_Z - 0.36 0.50 0.27 6 1518 18 4 12 2 6
2YIE_X - 0.06 0.08 0.06 1 1361 21 1 15 5 11
2ZY6_A - 0.51 0.58 0.47 7 688 8 3 5 0 5
2ZZN_D 0.61 0.63 0.61 17 2457 13 1 10 2 10
3A2K_C 0.44 0.46 0.42 13 2895 18 2 16 0 15
3A3A_A 0.89 0.86 0.91 32 3620 3 1 2 0 5
3AKZ_H 0.40 0.43 0.39 12 2670 19 3 16 0 16
3AM1_B - 0.93 0.89 0.97 31 3208 2 0 1 1 4
3AMU_B 0.44 0.48 0.42 13 2972 18 2 16 0 14
3DW4_A - 0.90 0.90 0.90 9 341 3 0 1 2 1
3GCA_A - 0.04 0.06 0.07 1 514 13 1 12 0 16
3GX2_A 0.47 0.48 0.48 19 4331 22 0 21 1 21
3HAY_E - 1.00 1.00 1.00 18 2260 11 0 0 11 0
3HJW_D - 0.94 0.94 0.94 17 1635 8 0 1 7 1
3IAB_R - 0.84 0.93 0.76 13 1018 8 2 2 4 1
3ID5_D - 0.34 0.33 0.38 3 488 9 0 5 4 6
3IVN_B 0.39 0.39 0.40 12 2316 18 0 18 0 19
3IWN_A 0.19 0.21 0.18 7 4238 33 8 25 0 26
3IZF_C 0.71 0.69 0.74 37 6853 14 0 13 1 17
3J0L_8 - 1.00 1.00 1.00 8 182 0 0 0 0 0
3J0L_2 - 0.21 0.24 0.18 8 6172 39 7 29 3 25
3J0L_g - 0.13 0.25 0.08 1 452 12 8 4 0 3
3J0L_7 - -0.01 0.00 0.00 0 1206 19 4 15 0 17
3J0L_h - 0.91 0.91 0.91 39 6062 13 0 4 9 4
3J0L_1 - 0.79 0.79 0.79 15 1206 6 0 4 2 4
3J0L_a - 0.16 0.19 0.17 3 1110 15 1 14 0 13
3J16_L 0.45 0.47 0.44 14 2743 19 1 17 1 16
3J20_0 0.59 0.57 0.63 17 2823 12 1 9 2 13
3J2L_3 0.58 0.57 0.60 30 7825 23 2 18 3 23
3J3D_C 0.42 0.43 0.43 12 2747 18 2 14 2 16
3J3E_8 0.12 0.12 0.12 4 7470 46 3 26 17 29
3J3E_7 0.40 0.39 0.42 21 7090 29 3 26 0 33
3J3F_7 0.74 0.74 0.74 37 7210 17 2 11 4 13
3J3F_8 0.13 0.17 0.11 6 12191 61 9 40 12 30
3JYV_7 0.21 0.22 0.21 7 2817 26 0 26 0 25
3JYX_3 0.41 0.52 0.33 14 6285 34 12 17 5 13
3JYX_4 0.20 0.24 0.17 8 12199 43 16 23 4 25
3KTW_C - 0.40 0.40 0.43 17 4520 24 4 19 1 26
3LA5_A 0.32 0.32 0.34 11 2453 21 1 20 0 23
3NDB_M - 0.23 0.23 0.24 14 9121 46 1 44 1 47
3NKB_B - 0.56 0.58 0.56 15 1989 13 0 12 1 11
3NMU_E - 0.28 0.33 0.25 2 553 12 0 6 6 4
3O58_3 0.22 0.26 0.19 9 12355 45 9 30 6 26
3O58_2 0.20 0.24 0.17 9 7208 44 5 38 1 29
3OVB_D - 1.00 1.00 1.00 11 584 2 0 0 2 0
3OVS_D - 1.00 1.00 1.00 13 548 0 0 0 0 0
3P22_G - 1.00 1.00 1.00 12 729 5 0 0 5 0
3PDR_A 0.66 0.61 0.72 44 12819 19 0 17 2 28
3R4F_A - 0.92 0.92 0.92 23 2120 5 0 2 3 2
3R9X_C - 0.69 0.70 0.70 7 585 7 0 3 4 3
3RKF_A 0.70 0.65 0.76 22 2182 7 1 6 0 12
3SD1_A 0.33 0.33 0.35 14 3876 26 0 26 0 28
3SIU_F - 0.42 0.45 0.42 5 366 7 1 6 0 6
3SN2_B 0.55 0.58 0.54 7 393 6 0 6 0 5
3TRZ_Z - -0.04 0.00 0.00 0 201 9 0 9 0 6
3TS0_U - 1.00 1.00 1.00 6 247 2 0 0 2 0
3TS2_V - -0.02 0.00 0.00 0 268 9 0 8 1 5
3U4M_B - 0.59 0.59 0.59 22 3123 15 0 15 0 15
3VJR_D - 1.00 1.00 1.00 13 617 1 0 0 1 0
3ZEX_G - 0.00 0.00 0.00 0 16456 15 1 14 0 74
3ZEX_F - -0.01 0.00 0.00 0 2610 29 4 14 11 12
3ZEX_C 0.24 0.21 0.28 11 14156 29 3 26 0 41
3ZEX_H - 0.14 0.18 0.12 7 8987 52 14 37 1 31
3ZEX_D 0.17 0.18 0.18 9 6970 42 4 38 0 40
3ZND_W 0.19 0.22 0.18 5 2975 26 1 22 3 18
4A1C_2 0.13 0.15 0.11 5 11735 56 10 31 15 28
4A1C_3 0.68 0.67 0.71 36 7089 17 1 14 2 18
4A4U_A - 1.00 1.00 1.00 9 222 0 0 0 0 0
4ATO_G - 0.30 0.30 0.33 3 519 8 0 6 2 7
4ENB_A 0.61 0.63 0.60 12 1255 8 2 6 0 7
4ENC_A 0.28 0.32 0.27 6 1304 17 2 14 1 13
4FNJ_A - 0.83 0.81 0.87 13 580 3 0 2 1 3
4FRG_B 0.32 0.34 0.31 11 3450 25 0 25 0 21
4FRN_A 0.12 0.14 0.12 5 5110 38 1 35 2 31
4HXH_A - 1.00 1.00 1.00 6 319 4 0 0 4 0
4JF2_A 0.66 0.68 0.66 21 2818 11 0 11 0 10
4JRC_A - 0.20 0.22 0.21 5 1516 19 0 19 0 18

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