### Table 5: Experiment result on real biological data The data are collected from the SCPD. For each set of data, we look for the motifs with length equals to the published motif and d equals to 1. Transcription Factor Published Motif pattern Motif Pattern Found

"... In PAGE 9: ... The lengths of the motifs were same as those of the published motifs and d was 1. Experimental results are showed in Table5 . The Voting Algorithm could find the motifs for these data sets.... ..."

### Table 1: Network motifs and their ID

"... In PAGE 2: ... It must also be noted that the total number of motifs of a given type is counted and possible isomorphisms are considered the same motif type. Table1 in the Appendix lists all 3-node connection patterns in directed graphs, including auto-connections, up to isomorphism. We shall later refer to particular... ..."

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### Table 1: The Find-1-Motif-Brute-Force algorithm

2002

"... In PAGE 6: ... 4. EFFICIENT MOTIF DISCOVERY Recall that the brute force motif discovery algorithm introduced in Table1 requires O(m2) calculations of the distance function. As previously mentioned, the symmetric property of the Euclidean distance measure could be used to half the number of calculations by storing D(Q,C) and re-using the value when it is necessary to find D(C,Q).... ..."

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### Table 3. Results of PatternBranching on biological samples

2003

"... In PAGE 6: ... We set l = 20 and k = 5, and modified the algorithm to save 20 motifs with lowest total distance score. Table3 shows that PatternBranching finds the known reference motif(s) in each sample. We note that, for the preproinsulin sample, some of the motifs returned by PatternBranching have a better total distance score than any of the reference motifs.... In PAGE 6: ... We also tested the ProfileBranching algorithm on these biological samples, again using l = 20 and k = 5. For each sample, the consensus pattern of the motif profile returned by ProfileBranching similarly matches one of the reference motifs from Table3 . Running times ranged from less than 1 second for DHFR to 18 seconds for preproinsulin on a 1.... ..."

### Table 3. Structural Motif Number and Networks Optimized for Functional Motif Number

2004

"... In PAGE 4: ... Convergence was robust and consistent structural features of optimized connection matrices were observed. Figure 3B, Figure 5, Table3 , and Table 4 summarize results obtained from the optimizations. When maximizing func- tional motif number (Figure 5A), we obtained networks that closely resembled real brain networks with respect to their structural and functional motif number, motif diversity (unpublished data), structural motif frequency spectrum, and the specific structural motifs that occurred with significantly increased frequency (Tables 3 and 4).... In PAGE 7: ...75.36 (36.34) z = 8.94 Compare motif ID with those shown in Figure 3 and Table 2. As in Table3 , all networks were optimized for high functional motif number (M =3,N = 30, K = 311, mean and standard deviation for n = 10 exemplars). Optimizations and comparisons of macaque and cat matrices produce similar results (unpublished data).... ..."

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### Table 3. Results of PatternBranching on biological samples. We list the motif(s) from PatternBranching output which match the reference motif(s), underlining the areas which match. References: (A) (Stormo and Hartzell III, 1989), (B) (Blanchette, 2001), (C) (Buhler and Tompa, 2001). Running times ranged from less than 1 second for DHFR to 6 seconds for preproinsulin on a 1.0GHz processor.

"... In PAGE 6: ... We set l = 20 and k = 5, and modified the algorithm to save 20 motifs with lowest total distance score. Table3 shows that PatternBranching finds the known reference motif(s) in each sample. We note that, for the preproinsulin sample, some of the motifs returned by PatternBranching have a better total distance score than any of the reference motifs.... In PAGE 6: ... We also tested the ProfileBranching algorithm on these biological samples, again using l = 20 and k = 5. For each sample, the consensus pattern of the motif profile returned by ProfileBranching similarly matches one of the reference motifs from Table3 . Running times ranged from less than 1 second for DHFR to 18 seconds for preproinsulin on a 1.... ..."

### Table 1: Parameterized vertices

"... In PAGE 9: ... How- ever, each vertex is given as the intersection of hyperplanes defined by the constraints with faces of A1BG. Figure 2 shows the general position of these intersection points, and Table1 presents them as a list. The third column of Table 1 states the conditions on the parameters for when the intersection point is an actual vertex of the polytope.... In PAGE 9: ... Figure 2 shows the general position of these intersection points, and Table 1 presents them as a list. The third column of Table1 states the conditions on the parameters for when the intersection point is an actual vertex of the polytope. Such conditions we will subsequently encounter in great numbers; they are formally introduced by the following definition.... In PAGE 10: ...(r)=0.4, I(s)=0.2 (0,0,1,0) (0,0,1,0) I(r)=0, I(s)=0 (0,0,1,0) Figure 1: Polytopes for different parameter values 3 5 2 4 8 6 7 1 Figure 2: General vertex positions problem statement for generating a complete parameterized vertex list can now be refined as follows: given input constraints BV, we have to find a list DACY BM APCY B4BD AK CY AK C5B5 (13) where each DACY is a parameterized vertex as in (7), and the APCY are lists of p-constraints, such that for every parameter instantiation C1 the set of vertices of A1B4C1B4BV B5B5 is just CUC1B4DACYB5 CY C1 satisfies APCYCV (where, naturally, C1 satisfies AP iff for every APCX AH D4CX AO BC BE AP: C1B4D4CXB5 AO BC). Table1 provides this list for the input constraints (9) and (10). To obtain a systematic method for generating such a list it is convenient to consider one by one the different faces of A1BEC3, in... In PAGE 12: ...icularly suitable method is fraction free Gaussian elimination (see e.g. [7]). This is a variant of Gaussian elimination that avoids divisions, which is useful for us, as otherwise we would have to divide by symbolic expressions that might be zero for some parameter values and nonzero for others, thereby requiring us to make a number of case distinctions. As an illustration for the working of the algorithm we retrace how vertex 8 in Table1 was generated. This vertex is the solution of the system (14)-(16) defined by CS BP BE, C0 BP CUBDBN BEBN BFCV and the (then mandatory) selection of both constraints CRBDBN CRBE for (16).... In PAGE 14: ... To illustrate the general method, we continue with our example, taking C8 B4BMBT CY BUB5 to be the target probability of the inference rule to be derived. The probability of BMBT given BU at the vertices listed in Table1 is evaluated by computing DABFBPB4DABDB7DABFB5, which leads to the values listed in Table 4. Note that the possible values of C8 B4BMBT CY BUB5 are still annotated with the parameter constraints on the vertices at which they are attained, and that for vertices 5 and 8 the new p-constraint D7 BO BD has been added.... In PAGE 21: ... Minimal irredundant sets of values for minimization and maximization of C8 B4BT CY BU CM BWB5 are indicated by the +-marks in the columns 8 and 9, respectively. The final bound functions we obtain now are C4B4D6BN D8BN D9BN DAB5 BP minCJD6BPDA BM AQ BN DA BQ BCCL (40) CDB4D6BN D8BN D9BN DAB5 BP maxCJBC BM D6 BP BCBN DA BP BDBN BD BM AQ BN D9 AK D6BN D6 BQ BCBN BD BM AQ BN D9 AK D8BN D8 AK DABN D8 BQ BCBN BD BM AQ BN D6 AK D9BN D9 AK D8BN D9 BQ BCBN BD BM AQ BN DA AK D8BN DA BQ BCBN D8BPD9 BM AQ BN D8 AK D9BN D9 BQ BCCLBM (41) where the p-constraints AQ suppressed in Table1 have been reinstated. Remembering the con- ventions min BN BP BDBN max BN BP BC, and taking into account that the conditions D6 AK D8BN D9 AK DA are taken for granted in (29), these functions can be seen to be the same as (30) and (31).... ..."

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### Table 2 Experimental results on real biological data

"... In PAGE 12: ... For Algorithm exVote, we let l = 6 and d = 1 for all data set. Table2 shows the results of these algorithms. Even though we did not input the lengths of published motifs, Algorithm exVote could discover all the published motifs.... ..."

### TABLE I COMPARISON OF THE PERFORMANCE AND RUNNING TIME BY DIFFERENT ALGORITHMS ON THE EXACT DATASET FOR FINDING (15;4) MOTIF.

### Table 1. Table 2. Comparison of Performance in finding a (9,1)-motif for a (15,2)-motif problem.

"... In PAGE 7: ... Task 1 - (l; d) known In this task, the HMD algorithm returns (15,2) motifs. The results are provided in Table1 and 2. We notice that HMD correctly found the correct motif in both cases.... ..."