Systems biology focuses on the study of complex interactions in biological systems, rather than the study of individual molecules such as DNA, RNA, proteins and metabolites [1]. One of the goals of systems biology is understanding the structures of all molecules and their interactions in a system level. Therefore major challenges are understanding the dynamic structures of small molecules and determining their functions in a living cell. Various types of biological interactions have been expressed in networks, which include transcriptional regulatory networks, signaling pathways, metabolic networks and protein-protein interaction (PPI) networks. Biological networks share some of structural properties of other complex networks, or have specific features of scale-free and small-world effect [2]. However, the properties have been questioned by Lacroix et al. [3] with a number of reasons including the incompleteness of networks and inconsistent link generation for the graphs. Therefore, the analysis extends to other network properties such as network clusters and network motifs.

As biological networks are massive and the size is still increasing, dividing the network into a number of clusters helps reveal specific local properties. Network motif, as another concept describing local properties of a network, is defined as a small connected subgraph appearing frequently and uniquely in a network. Similar to a protein sequence motif, network motif is defined as a over-repeated pattern, but it requires much more computation as the process involves isomorphic testing and repeated processes for uniqueness determination. Network alignment [4] and network querying [5] are analogous to network motifs, but while network motifs are defined with only structural information, network alignment and network querying require both of the topological and biological information. Previous network motif discovery algorithms include exact counting and approximation algorithms: Exhaustive recursive search (ERS) [6], enumerate subgraphs (ESU) [7] and compact topological motifs [8] are exact counting algorithms. For efficient detection, several approximation algorithms have been provided including edge sampling (MFINDER) [6], randomized version of ESU from a search tree (RAND-ESU) [9], and tree-filtering search which is NEMOFINDER[10]. Furthermore, parallel search algorithms have been developed to realize feasible exact counting algorithms [11, 12].

Network motifs are used for many applications in biological networks. Feed-forward-loop (FFL) and bifan network motifs are identified as the typical patterns in different types of biological networks [13, 14]. Przulj et al. [15] used network motifs as a relative graphlet frequency distance to distinguish different protein-protein interaction networks. Also motif frequencies are exploited as classifiers for network model selection [16]. Milo et al. [17] studied that networks of different biological and technological domains have been classified into different superfamilies on the basis of motif significance profiles. To predict protein-protein interactions, Albert I. and Albert R. [18] used network motifs successfully. In the study by Conant and Wagner [19], network motifs in transcriptional regulatory networks are not evolutionary conserved while network motifs in PPI networks are evolutionary related. On the other hand, network motifs are extended to 'motif modes' each of which has a certain topology and a specific functional property [20].

Through a number of network motif applications, however, we notice several problems regarding the biological meanings of network motifs, on top of the computational challenge for the detection. First, the biological quality of network motifs are not validated thoroughly. A network motif is selected only by its structural uniqueness and just small number of instances of the type are biologically exemplified. Second, only small portion of network motif instances are used for applications and others are ignored. Third, non-motifs, that is, structurally insignificant subgraphs, have not been analyzed in any studies, which are filtered out before applying to any applications. Fourth, it is still questionable what the network motifs really represent in biological networks.

As we believe that the biological quality of network motifs are also significant, we define a biological network motif in this paper. Throughout this paper, we refer a network motif as a **structural network motif** to distinguish it from a biological network motif. Unlike structural network motifs, biological network motifs are biologically significant small connected subgraphs regardless of the structure. The biological significance is unspecified in the definition, as it will be assigned flexibly by a goal of the application. We introduce EDGEGO-BNM, EDGEBETWEENNESS-BNM, NMF-BNM, NMFGO-BNM and VOLTAGE-BNM algorithms for efficient discovery of biological network motifs, and design new evaluation measures named, 'motifs included in complex', 'motifs included in functional module' and 'GO term clustering score'. Our algorithms compete with existing algorithms including ESU, RAND-ESU and MFINDER, and the performance are compared based on the new measures introduced in this paper. The main idea for our algorithms is to reduce the number of subgraphs to search by removing a number of edges from the original network and, at the same time, increase the discovery rate for biological network motifs. Experimental results with a couple of S. cerevisiae PPI networks demonstrate that EDGEGO-BNM and EDGEBETWEENNESS-BNM algorithms perform better than other algorithms in most of the measures. In addition, we show that all of our algorithms are applicable to the discovery of structural network motifs as well.

The work has three contributions to the study of network motifs: 1)We question biological meanings of network motifs which have not been focused by existing detection algorithms. New motif search algorithms and evaluation measures are developed based on these questions. 2)We design several algorithms combining the topological and biological information in a network. The algorithms further enrich existing algorithms in a biological context. 3)We develop a number of evaluation measures which qualify biological importance of network motifs. As we know of, this is the first time to suggest systematical evaluation measures for network motifs. With these contributions, we hope that our work gives some guidelines for the researches of network motifs in biological networks.