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

  1. Overview

  2. Performance Plots

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

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

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


Overview

Metric Sfold MCFold
MCC 0.663 > 0.522
Average MCC ± 95% Confidence Intervals 0.694 ± 0.049 > 0.599 ± 0.061
Sensitivity 0.574 > 0.535
Positive Predictive Value 0.771 > 0.516
Total TP 1728 > 1611
Total TN 350395 > 349515
Total FP 653 < 1767
Total FP CONTRA 57 < 194
Total FP INCONS 456 < 1316
Total FP COMP 140 < 257
Total FN 1281 < 1398
P-value 3.56938820447e-08

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Performance plots


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

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

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Performance of Sfold - scored higher in this pairwise comparison

1. Total counts & total scores for Sfold

Total Base Pair Counts
Total TP 1728
Total TN 350395
Total FP 653
Total FP CONTRA 57
Total FP INCONS 456
Total FP COMP 140
Total FN 1281
Total Scores
MCC 0.663
Average MCC ± 95% Confidence Intervals 0.694 ± 0.049
Sensitivity 0.574
Positive Predictive Value 0.771
Nr of predictions 131

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2. Individual counts for Sfold [ 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 0 0 0 0 0
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 0.92 0.89 0.94 17 1110 2 0 1 1 2
2KEZ_A - 0.89 0.80 1.00 8 268 0 0 0 0 2
2KFC_A - -0.02 0.00 0.00 0 623 7 0 7 0 20
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 0 0 0 0 0
2KPV_A - 0.96 0.92 1.00 12 549 0 0 0 0 1
2KU0_A - 0.83 0.69 1.00 11 692 0 0 0 0 5
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 - 0.89 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_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 - 0.91 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 0.45 0.35 0.58 7 1528 5 0 5 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 - 0.89 0.79 1.00 31 6074 1 0 0 1 8
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 - 0.29 0.18 0.50 3 1647 3 1 2 0 14
2RP0_A - 0.64 0.42 1.00 5 346 0 0 0 0 7
2RPT_A - 1.00 1.00 1.00 7 183 0 0 0 0 0
2RRC_A - 0.83 0.70 1.00 7 224 0 0 0 0 3
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 - 0.29 0.12 0.75 3 1949 1 0 1 0 23
2WW9_E - 0.53 0.29 1.00 4 557 0 0 0 0 10
2WWQ_V 0.80 0.68 0.95 19 2906 2 0 1 1 9
2XKV_B 0.51 0.50 0.53 10 4541 23 0 9 14 10
2XQD_Y 0.77 0.67 0.90 18 2830 2 0 2 0 9
2XXA_G 0.36 0.31 0.43 13 5121 17 1 16 0 29
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_Z - -0.01 0.00 0.00 0 1525 15 3 12 0 12
2YIE_X - 0.58 0.33 1.00 4 1374 0 0 0 0 8
2ZY6_A - 0.58 0.58 0.58 7 691 5 1 4 0 5
2ZZN_D 0.82 0.78 0.88 21 2461 3 0 3 0 6
3A2K_C 0.42 0.39 0.46 11 2902 13 2 11 0 17
3A3A_A 0.87 0.76 1.00 28 3627 0 0 0 0 9
3AKZ_H 0.44 0.39 0.50 11 2679 11 2 9 0 17
3AM1_B - 0.69 0.63 0.76 22 3211 7 1 6 0 13
3AMU_B 0.70 0.59 0.84 16 2984 5 0 3 2 11
3DW4_A - -0.02 0.00 0.00 0 345 6 0 6 0 10
3GCA_A - 0.54 0.29 1.00 5 523 0 0 0 0 12
3GX2_A 0.72 0.55 0.96 22 4348 2 0 1 1 18
3HAY_E - 0.91 0.83 1.00 15 2263 6 0 0 6 3
3HJW_D - 0.94 0.89 1.00 16 1637 2 0 0 2 2
3IAB_R - 0.92 0.86 1.00 12 1023 0 0 0 0 2
3ID5_D - 0.49 0.33 0.75 3 492 1 0 1 0 6
3IVN_B 0.78 0.61 1.00 19 2327 0 0 0 0 12
3IWN_A 0.63 0.58 0.70 19 4251 9 0 8 1 14
3IZF_C 0.71 0.61 0.83 33 6863 7 0 7 0 21
3J0L_8 - 0.93 0.88 1.00 7 183 0 0 0 0 1
3J0L_2 - 0.36 0.33 0.39 11 6188 19 2 15 2 22
3J0L_g - -0.01 0.00 0.00 0 460 5 1 4 0 4
3J0L_7 - -0.01 0.00 0.00 0 1222 3 0 3 0 17
3J0L_h - 0.75 0.60 0.93 26 6077 4 0 2 2 17
3J0L_1 - 0.76 0.63 0.92 12 1212 3 0 1 2 7
3J0L_a - 0.51 0.50 0.53 8 1113 7 3 4 0 8
3J16_L 0.29 0.23 0.37 7 2756 12 0 12 0 23
3J20_0 0.43 0.40 0.48 12 2825 14 0 13 1 18
3J2L_3 0.62 0.53 0.74 28 7837 12 0 10 2 25
3J3D_C 0.36 0.25 0.54 7 2762 6 0 6 0 21
3J3E_7 0.60 0.50 0.73 27 7103 10 1 9 0 27
3J3E_8 0.06 0.06 0.08 2 7477 32 2 22 8 31
3J3F_7 0.69 0.60 0.79 30 7222 8 1 7 0 20
3J3F_8 0.35 0.33 0.38 12 12214 34 3 17 14 24
3JYV_7 -0.01 0.00 0.00 0 2834 16 0 16 0 32
3JYX_4 0.35 0.30 0.42 10 12222 23 3 11 9 23
3JYX_3 0.62 0.63 0.61 17 6300 19 1 10 8 10
3KTW_C - 0.74 0.63 0.87 27 4529 5 2 2 1 16
3LA5_A 0.78 0.62 1.00 21 2464 0 0 0 0 13
3NDB_M - 0.81 0.69 0.95 42 9136 3 0 2 1 19
3NKB_B - 0.64 0.54 0.78 14 1998 4 0 4 0 12
3NMU_E - 0.61 0.50 0.75 3 557 4 0 1 3 3
3O58_3 0.43 0.34 0.55 12 12381 17 0 10 7 23
3O58_2 0.74 0.74 0.74 28 7222 11 3 7 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 0.69 0.56 0.85 40 12833 9 0 7 2 32
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 0.65 0.50 0.84 21 3891 4 1 3 0 21
3SIU_F - 0.85 0.73 1.00 8 370 0 0 0 0 3
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 2 0 0 2 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 - 0.38 0.32 0.46 12 3134 14 0 14 0 25
3VJR_D - 0.96 0.92 1.00 12 618 0 0 0 0 1
3ZEX_F - 0.00 0.00 0.00 0 2623 7 2 3 2 12
3ZEX_G - 0.78 0.68 0.91 50 16416 9 0 5 4 24
3ZEX_D 0.74 0.63 0.86 31 6985 5 0 5 0 18
3ZEX_H - 0.19 0.18 0.19 7 9009 29 5 24 0 31
3ZEX_C 0.32 0.21 0.48 11 14173 26 1 11 14 41
3ZND_W 0.20 0.22 0.19 5 2977 22 1 20 1 18
4A1C_2 0.16 0.15 0.17 5 11751 34 2 23 9 28
4A1C_3 0.69 0.57 0.84 31 7103 6 1 5 0 23
4A4U_A - 1.00 1.00 1.00 9 222 0 0 0 0 0
4ATO_G - 0.31 0.10 1.00 1 527 0 0 0 0 9
4ENB_A 0.76 0.58 1.00 11 1264 0 0 0 0 8
4ENC_A 0.51 0.26 1.00 5 1321 0 0 0 0 14
4FNJ_A - 0.79 0.63 1.00 10 585 0 0 0 0 6
4FRG_B 0.62 0.47 0.83 15 3468 3 1 2 0 17
4FRN_A 0.63 0.44 0.89 16 5133 2 2 0 0 20
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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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.