Splitting the BLOSUM Score into Numbers of Biological Significance

Mathematical tools developed in the context of Shannon information theory were used to analyze the meaning of the BLOSUM score, which was split into three components termed as the BLOSUM spectrum (or BLOSpectrum). These relate respectively to the sequence convergence (the stochastic similarity of the two protein sequences), to the background frequency divergence (typicality of the amino acid probability distribution in each sequence), and to the target frequency divergence (compliance of the amino acid variations between the two sequences to the protein model implicit in the BLOCKS database). This treatment sharpens the protein sequence comparison, providing a rationale for the biological significance of the obtained score, and helps to identify weakly related sequences. Moreover, the BLOSpectrum can guide the choice of the most appropriate scoring matrix, tailoring it to the evolutionary divergence associated with the two sequences, or indicate if a compositionally adjusted matrix could perform better.[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]     

��-Defensins in Enteric Innate Immunity: FUNCTIONAL PANETH CELL ��-DEFENSINS IN MOUSE COLONIC LUMEN*

Paneth cells are a secretory epithelial lineage that release dense core granules rich in host defense peptides and proteins from the base of small intestinal crypts. Enteric α-defensins, termed cryptdins (Crps) in mice, are highly abundant in Paneth cell secretions and inherently resistant to proteolysis. Accordingly, we tested the hypothesis that enteric α-defensins of Paneth cell origin persist in a functional state in the mouse large bowel lumen. To test this idea, putative Crps purified from mouse distal colonic lumen were characterized biochemically and assayed in vitro for bactericidal peptide activities. The peptides comigrated with cryptdin control peptides in acid-urea-PAGE and SDS-PAGE, providing identification as putative Crps. Matrix-assisted laser desorption ionization time-of-flight mass spectrometry experiments showed that the molecular masses of the putative α-defensins matched those of the six most abundant known Crps, as well as N-terminally truncated forms of each, and that the peptides contain six Cys residues, consistent with identities as α-defensins. N-terminal sequencing definitively revealed peptides with N termini corresponding to full-length, (des-Leu)-truncated, and (des-Leu-Arg)-truncated N termini of Crps 1–4 and 6. Crps from mouse large bowel lumen were bactericidal in the low micromolar range. Thus, Paneth cell α-defensins secreted into the small intestinal lumen persist as intact and functional forms throughout the intestinal tract, suggesting that the peptides may mediate enteric innate immunity in the colonic lumen, far from their upstream point of secretion in small intestinal crypts.Antimicrobial peptides (AMPs)² are released by epithelial cells onto mucosal surfaces as effectors of innate immunity (1–5). In mammals, most AMPs derive from two major families, the cathelicidins and defensins (6). The defensins comprise the α-, β-, and θ-defensin subfamilies, which are defined by the presence of six cysteine residues paired in characteristic tridisulfide arrays (7). α-Defensins are highly abundant in two primary cell lineages: phagocytic leukocytes, primarily neutrophils, of myeloid origin and Paneth cells, which are secretory epithelial cells located at the base of the crypts of Lieberkühn in the small intestine (8–10). Neutrophil α-defensins are stored in azurophilic granules and contribute to non-oxidative microbial cell killing in phagolysosomes (11, 12), except in mice whose neutrophils lack defensins (13). In the small bowel, α-defensins and other host defense proteins (14–18) are released apically as components of Paneth cell secretory granules in response to cholinergic stimulation and after exposure to bacterial antigens (19). Therefore, the release of Paneth cell products into the crypt lumen is inferred to protect mitotically active crypt cells from colonization by potential pathogens and confer protection against enteric infection (7, 20, 21).Under normal, homeostatic conditions, Paneth cells are not found outside the small bowel, although they may appear ectopically in response to local inflammation throughout the gastrointestinal tract (22, 23). Paneth cell numbers increase progressively throughout the small intestine, occurring at highest numbers in the distal ileum (24). Mouse Paneth cells express numerous α-defensin isoforms, termed cryptdins (Crps) (25), that have broad spectrum antimicrobial activities (6, 26). Collectively, α-defensins constitute approximately seventy percent of the bactericidal peptide activity in mouse Paneth cell secretions (19), selectively killing bacteria by membrane-disruptive mechanisms (27–30). The role of Paneth cell α-defensins in gastrointestinal mucosal immunity is evident from studies of mice transgenic for human enteric α-defensin-5, HD-5, which are immune to infection by orally administered Salmonella enterica sv. typhimurium (S. typhimurium) (31).The biosynthesis of mature, bactericidal α-defensins from their inactive precursors requires activation by lineage-specific proteolytic convertases. In mouse Paneth cells, inactive ∼8.4-kDa Crp precursors are processed intracellularly into microbicidal ∼4-kDa Crps by specific cleavage events mediated by matrix metalloproteinase-7 (MMP-7) (32, 33). MMP-7 null mice exhibit increased susceptibility to systemic S. typhimurium infection and decreased clearance of orally administered non-invasive Escherichia coli (19, 32). Although the α-defensin proregions are sensitive to proteolysis, the mature, disulfide-stabilized peptides resist digestion by their converting enzymes in vitro, whether the convertase is MMP-7 (32), trypsin (34), or neutrophil serine proteinases (35). Because α-defensins resist proteolysis in vitro, we hypothesized that Paneth cell α-defensins resist degradation and remain in a functional state in the large bowel, a complex, hostile environment containing varied proteases of both host and microbial origin.Here, we report on the isolation and characterization of a population of enteric α-defensins from the mouse colonic lumen. Full-length and N-terminally truncated Paneth cell α-defensins were identified and are abundant in the distal large bowel lumen.     

Analysis of Free Energy Signals Arising from Nucleotide Hybridization Between rRNA and mRNA Sequences during Translation in Eubacteria

A decoding algorithm is tested that mechanistically models the progressive alignments that arise as the mRNA moves past the rRNA tail during translation elongation. Each of these alignments provides an opportunity for hybridization between the single-stranded, -terminal nucleotides of the 16S rRNA and the spatially accessible window of mRNA sequence, from which a free energy value can be calculated. Using this algorithm we show that a periodic, energetic pattern of frequency 1/3 is revealed. This periodic signal exists in the majority of coding regions of eubacterial genes, but not in the non-coding regions encoding the 16S and 23S rRNAs. Signal analysis reveals that the population of coding regions of each bacterial species has a mean phase that is correlated in a statistically significant way with species () content. These results suggest that the periodic signal could function as a synchronization signal for the maintenance of reading frame and that codon usage provides a mechanism for manipulation of signal phase.[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]     

Chimeric Plant Virus Particles Administered Nasally or Orally Induce Systemic and Mucosal Immune Responses in Mice

The humoral immune responses to the D2 peptide of fibronectin-binding protein B (FnBP) of Staphylococcus aureus, expressed on the plant virus cowpea mosaic virus (CPMV), were evaluated after mucosal delivery to mice. Intranasal immunization of these chimeric virus particles (CVPs), either alone or in the presence of ISCOM matrix, primed CPMV-specific T cells and generated high titers of CPMV- and FnBP-specific immunoglobulin G (IgG) in sera. Furthermore, CPMV- and FnBP-specific IgA and IgG could also be detected in the bronchial, intestinal, and vaginal lavage fluids, highlighting the ability of CVPs to generate antibody at distant mucosal sites. IgG2a and IgG2b were the dominant IgG subclasses in sera to both CPMV and FnBP, demonstrating a bias in the response toward the T helper 1 type. The sera completely inhibited the binding of human fibronectin to the S. aureus FnBP. Oral immunization of the CVPs also generated CPMV- and FnBP-specific serum IgG; however, these titers were significantly lower and more variable than those generated by the intranasal route, and FnBP-specific intestinal IgA was undetectable. Neither the ISCOM matrix nor cholera toxin enhanced these responses. These studies demonstrate for the first time that recombinant plant viruses have potential as mucosal vaccines without the requirement for adjuvant and that the nasal route is most effective for the delivery of these nonreplicating particles.Replicating vaccines such as live-attenuated bacterial (13) and virus (36, 40, 45) vaccines, as well as naked DNA vaccines (31), induce stronger and longer-lasting immune responses than conventional killed-subunit vaccines and also elicit protective cell-mediated immunity, often without the need for adjuvant. There are however, safety concerns over the use of these vaccines (24, 49), where persistence or reversion to virulence of the live vaccine strains and integration of the naked DNA vaccine into the host chromosome are of major concern. Recent technological advances, such as the use of more-effective adjuvants for both mucosal and systemic delivery (12, 16), liposome and ISCOM encapsulation of proteins and peptides (3, 19, 27), multiple antigenic peptides (35), and virus-like particles (VLPs) (1), have led to the development of more-effective subunit vaccines. To circumvent the safety concerns of replicating vaccines and to avoid the need for peptide synthesis and chemical coupling to a carrier such as keyhole limpet hemocyanin, we have been examining the utility of the plant virus cowpea mosaic virus (CPMV) as a carrier of peptides for immune recognition. CPMV is composed of 2 subunits, the small (S) and large (L) coat proteins, of which there are 60 copies of each per virus particle (46). Foreign peptides up to 37 amino acids in length can be expressed on either the L or S proteins; hence, 60 to 120 copies of a peptide can be displayed on a single virus particle (4b, 34). A peptide from the human immunodeficiency virus (HIV) gp41 glycoprotein is highly immunogenic when displayed on CPMV, eliciting high titers of HIV neutralizing antibodies (28, 29). Furthermore, a peptide derived from the VP2 protein of canine parvovirus (CPV) expressed on CPMV is immunogenic when administered to mink and subsequently protected the mink from a lethal challenge with the CPV-related mink enteritis virus (10).Most infectious viral and bacterial diseases involve colonization or invasion through mucosal surfaces by the pathogen, and hence it is important to develop vaccines that induce strong protective mucosal immune responses as a first line of defense. Where the organism, such as Vibrio cholerae and enterotoxigenic Escherichia coli, is restricted to the mucosa, strong mucosal immunity is often sufficient. However, when the organism disseminates from the mucosa into the bloodstream, a strong systemic response is also required to engender sterile immunity. Hence, the ideal mucosal vaccine should generate local immune responses at mucosal surfaces but also elicit generalized vaccine-specific immunity in the systemic lymphoid organs. The potential of CPMV-based vaccines for mucosal vaccination has not previously been determined.Oral immunization with particulate antigens, especially when presented as viable organisms, which can colonize the mucosa better than killed organisms, is effective at inducing local and generalized secretory and systemic immune responses (5, 43). However, the acidic pH and the presence of degradative enzymes in the gastrointestinal tract mean that when nonreplicating antigens are used, high concentrations are often required to elicit high levels of immunity (6). Another way to elicit mucosal immunity but circumvent the problems of oral immunization is to vaccinate via the intranasal route (2). Intranasal immunization requires up to 10-fold less immunogen for effective immunization and avoids the problems of low pH. Live vaccines (15, 37), virus-like particles (4, 25, 32), and synthetic peptides (17, 33, 44) in the absence of adjuvant have been shown to stimulate strong immunity when administered by this route. Furthermore, stimulation of the nasal mucosa, like stimulation of the intestinal mucosa, has been shown to be effective at generating protective immunity at distant mucosal sites (reviewed in reference 2).To assess the potential of CVPs as mucosal vaccines, mice were immunized intranasally or orally with CPMV expressing a peptide derived from the fibronectin-binding protein B (FnBP) D2 motif of Staphylococcus aureus (14, 42). The three fibronectin-binding domains, termed D1, D2, and D3, of FnBP have been shown to be immunogenic in mice and rats (7, 41). The CVPs were shown to be more immunogenic when administered (without adjuvant) via the intranasal route than when administered by the oral route, generating high titers of D2-specific antibody in serum and mucosa, and the serum antibody inhibited fibronectin binding to FnBP.     

Role of Partitioning-defective 1/Microtubule Affinity-regulating Kinases in the Morphogenetic Activity of Helicobacter pylori CagA

Helicobacter pylori CagA plays a key role in gastric carcinogenesis. Upon delivery into gastric epithelial cells, CagA binds and deregulates SHP-2 phosphatase, a bona fide oncoprotein, thereby causing sustained ERK activation and impaired focal adhesions. CagA also binds and inhibits PAR1b/MARK2, one of the four members of the PAR1 family of kinases, to elicit epithelial polarity defect. In nonpolarized gastric epithelial cells, CagA induces the hummingbird phenotype, an extremely elongated cell shape characterized by a rear retraction defect. This morphological change is dependent on CagA-deregulated SHP-2 and is thus thought to reflect the oncogenic potential of CagA. In this study, we investigated the role of the PAR1 family of kinases in the hummingbird phenotype. We found that CagA binds not only PAR1b but also other PAR1 isoforms, with order of strength as follows: PAR1b > PAR1d ≥ PAR1a > PAR1c. Binding of CagA with PAR1 isoforms inhibits the kinase activity. This abolishes the ability of PAR1 to destabilize microtubules and thereby promotes disassembly of focal adhesions, which contributes to the hummingbird phenotype. Consistently, PAR1 knockdown potentiates induction of the hummingbird phenotype by CagA. The morphogenetic activity of CagA was also found to be augmented through inhibition of non-muscle myosin II. Because myosin II is functionally associated with PAR1, perturbation of PAR1-regulated myosin II by CagA may underlie the defect of rear retraction in the hummingbird phenotype. Our findings reveal that CagA systemically inhibits PAR1 family kinases and indicate that malfunctioning of microtubules and myosin II by CagA-mediated PAR1 inhibition cooperates with deregulated SHP-2 in the morphogenetic activity of CagA.Infection with Helicobacter pylori strains bearing cagA (cytotoxin-associated gene A)-positive strains is the strongest risk factor for the development of gastric carcinoma, the second leading cause of cancer-related death worldwide (1–3). The cagA gene is located within a 40-kb DNA fragment, termed the cag pathogenicity island, which is specifically present in the genome of cagA-positive H. pylori strains (4–6). In addition to cagA, there are ∼30 genes in the cag pathogenicity island, many of which encode a bacterial type IV secretion system that delivers the cagA-encoded CagA protein into gastric epithelial cells (7–10). Upon delivery into gastric epithelial cells, CagA is localized to the plasma membrane, where it undergoes tyrosine phosphorylation at the C-terminal Glu-Pro-Ile-Tyr-Ala motifs by Src family kinases or c-Abl kinase (11–14). The C-terminal Glu-Pro-Ile-Tyr-Ala-containing region of CagA is noted for the structural diversity among distinct H. pylori isolates. Oncogenic potential of CagA has recently been confirmed by a study showing that systemic expression of CagA in mice induces gastrointestinal and hematological malignancies (15).When expressed in gastric epithelial cells, CagA induces morphological transformation termed the hummingbird phenotype, which is characterized by the development of one or two long and thin protrusions resembling the beak of the hummingbird. It has been thought that the hummingbird phenotype is related to the oncogenic action of CagA (7, 16–19). Pathophysiological relevance for the hummingbird phenotype in gastric carcinogenesis has recently been provided by the observation that infection with H. pylori carrying CagA with greater ability to induce the hummingbird phenotype is more closely associated with gastric carcinoma (20–23). Elevated motility of hummingbird cells (cells showing the hummingbird phenotype) may also contribute to invasion and metastasis of gastric carcinoma.In host cells, CagA interacts with the SHP-2 phosphatase, C-terminal Src kinase, and Crk adaptor in a tyrosine phosphorylation-dependent manner (16, 24, 25) and also associates with Grb2 adaptor and c-Met in a phosphorylation-independent manner (26, 27). Among these CagA targets, much attention has been focused on SHP-2 because the phosphatase has been recognized as a bona fide oncoprotein, gain-of-function mutations of which are found in various human malignancies (17, 18, 28). Stable interaction of CagA with SHP-2 requires CagA dimerization, which is mediated by a 16-amino acid CagA-multimerization (CM)² sequence present in the C-terminal region of CagA (29). Upon complex formation, CagA aberrantly activates SHP-2 and thereby elicits sustained ERK MAP kinase activation that promotes mitogenesis (30). Also, CagA-activated SHP-2 dephosphorylates and inhibits focal adhesion kinase (FAK), causing impaired focal adhesions. It has been shown previously that both aberrant ERK activation and FAK inhibition by CagA-deregulated SHP-2 are involved in induction of the hummingbird phenotype (31).Partitioning-defective 1 (PAR1)/microtubule affinity-regulating kinase (MARK) is an evolutionally conserved serine/threonine kinase originally isolated in C. elegans (32–34). Mammalian cells possess four structurally related PAR1 isoforms, PAR1a/MARK3, PAR1b/MARK2, PAR1c/MARK1, and PAR1d/MARK4 (35–37). Among these, PAR1a, PAR1b, and PAR1c are expressed in a variety of cells, whereas PAR1d is predominantly expressed in neural cells (35, 37). These PAR1 isoforms phosphorylate microtubule-associated proteins (MAPs) and thereby destabilize microtubules (35, 38), allowing asymmetric distribution of molecules that are involved in the establishment and maintenance of cell polarity.In polarized epithelial cells, CagA disrupts the tight junctions and causes loss of apical-basolateral polarity (39, 40). This CagA activity involves the interaction of CagA with PAR1b/MARK2 (19, 41). CagA directly binds to the kinase domain of PAR1b in a tyrosine phosphorylation-independent manner and inhibits the kinase activity. Notably, CagA binds to PAR1b via the CM sequence (19). Because PAR1b is present as a dimer in cells (42), CagA may passively homodimerize upon complex formation with the PAR1 dimer via the CM sequence, and this PAR1-directed CagA dimer would form a stable complex with SHP-2 through its two SH2 domains.Because of the critical role of CagA in gastric carcinogenesis (7, 16–19), it is important to elucidate the molecular basis underlying the morphogenetic activity of CagA. In this study, we investigated the role of PAR1 isoforms in induction of the hummingbird phenotype by CagA, and we obtained evidence that CagA-mediated inhibition of PAR1 kinases contributes to the development of the morphological change by perturbing microtubules and non-muscle myosin II.     

Dbf4-Cdc7 Phosphorylation of Mcm2 Is Required for Cell Growth

The Dbf4-Cdc7 kinase (DDK) is required for the activation of the origins of replication, and DDK phosphorylates Mcm2 in vitro. We find that budding yeast Cdc7 alone exists in solution as a weakly active multimer. Dbf4 forms a likely heterodimer with Cdc7, and this species phosphorylates Mcm2 with substantially higher specific activity. Dbf4 alone binds tightly to Mcm2, whereas Cdc7 alone binds weakly to Mcm2, suggesting that Dbf4 recruits Cdc7 to phosphorylate Mcm2. DDK phosphorylates two serine residues of Mcm2 near the N terminus of the protein, Ser-164 and Ser-170. Expression of mcm2-S170A is lethal to yeast cells that lack endogenous MCM2 (mcm2Δ); however, this lethality is rescued in cells harboring the DDK bypass mutant mcm5-bob1. We conclude that DDK phosphorylation of Mcm2 is required for cell growth.The Cdc7 protein kinase is required throughout the yeast S phase to activate origins (1, 2). The S phase cyclin-dependent kinase also activates yeast origins of replication (3–5). It has been proposed that Dbf4 activates Cdc7 kinase in S phase, and that Dbf4 interaction with Cdc7 is essential for Cdc7 kinase activity (6). However, it is not known how Dbf4-Cdc7 (DDK)² acts during S phase to trigger the initiation of DNA replication. DDK has homologs in other eukaryotic species, and the role of Cdc7 in activation of replication origins during S phase may be conserved (7–10).The Mcm2-7 complex functions with Cdc45 and GINS to unwind DNA at a replication fork (11–15). A mutation of MCM5 (mcm5-bob1) bypasses the cellular requirements for DBF4 and CDC7 (16), suggesting a critical physiologic interaction between Dbf4-Cdc7 and Mcm proteins. DDK phosphorylates Mcm2 in vitro with proteins purified from budding yeast (17, 18) or human cells (19). Furthermore, there are mutants of MCM2 that show synthetic lethality with DBF4 mutants (6, 17), suggesting a biologically relevant interaction between DBF4 and MCM2. Nevertheless, the physiologic role of DDK phosphorylation of Mcm2 is a matter of dispute. In human cells, replacement of MCM2 DDK-phosphoacceptor residues with alanines inhibits DNA replication, suggesting that Dbf4-Cdc7 phosphorylation of Mcm2 in humans is important for DNA replication (20). In contrast, mutation of putative DDK phosphorylation sites at the N terminus of Schizosaccharomyces pombe Mcm2 results in viable cells, suggesting that phosphorylation of S. pombe Mcm2 by DDK is not critical for cell growth (10).In budding yeast, Cdc7 is present at high levels in G₁ and S phase, whereas Dbf4 levels peak in S phase (18, 21, 22). Furthermore, budding yeast DDK binds to chromatin during S phase (6), and it has been shown that Dbf4 is required for Cdc7 binding to chromatin in budding yeast (23, 24), fission yeast (25), and Xenopus (9). Human and fission yeast Cdc7 are inert on their own (7, 8), but Dbf4-Cdc7 is active in phosphorylating Mcm proteins in budding yeast (6, 26), fission yeast (7), and human (8, 10). Based on these data, it has been proposed that Dbf4 activates Cdc7 kinase in S phase and that Dbf4 interaction with Cdc7 is essential for Cdc7 kinase activity (6, 9, 18, 21–24). However, a mechanistic analysis of how Dbf4 activates Cdc7 has not yet been accomplished. For example, the multimeric state of the active Dbf4-Cdc7 complex is currently disputed. A heterodimer of fission yeast Cdc7 (Hsk1) in complex with fission yeast Dbf4 (Dfp1) can phosphorylate Mcm2 (7). However, in budding yeast, oligomers of Cdc7 exist in the cell (27), and Dbf4-Cdc7 exists as oligomers of 180 and 300 kDa (27).DDK phosphorylates the N termini of human Mcm2 (19, 20, 28), human Mcm4 (10), budding yeast Mcm4 (26), and fission yeast Mcm6 (10). Although the sequences of the Mcm N termini are poorly conserved, the DDK sites identified in each study have neighboring acidic residues. The residues of budding yeast Mcm2 that are phosphorylated by DDK have not yet been identified.In this study, we find that budding yeast Cdc7 is weakly active as a multimer in phosphorylating Mcm2. However, a low molecular weight form of Dbf4-Cdc7, likely a heterodimer, has a higher specific activity for phosphorylation of Mcm2. Dbf4 or DDK, but not Cdc7, binds tightly to Mcm2, suggesting that Dbf4 recruits Cdc7 to Mcm2. DDK phosphorylates two serine residues of Mcm2, Ser-164 and Ser-170, in an acidic region of the protein. Mutation of Ser-170 is lethal to yeast cells, but this phenotype is rescued by the DDK bypass mutant mcm5-bob1. We conclude that DDK phosphorylation of Ser-170 of Mcm2 is required for budding yeast growth.     

Proinsulin and the Genetics of Diabetes Mellitus

Insulin plays a central role in the regulation of vertebrate metabolism. The hormone, the post-translational product of a single-chain precursor, is a globular protein containing two chains, A (21 residues) and B (30 residues). Recent advances in human genetics have identified dominant mutations in the insulin gene causing permanent neonatal-onset DM² (1–4). The mutations are predicted to block folding of the precursor in the ER of pancreatic β-cells. Although expression of the wild-type allele would in other circumstances be sufficient to maintain homeostasis, studies of a corresponding mouse model (5–7) suggest that the misfolded variant perturbs wild-type biosynthesis (8, 9). Impaired β-cell secretion is associated with ER stress, distorted organelle architecture, and cell death (10). These findings have renewed interest in insulin biosynthesis (11–13) and the structural basis of disulfide pairing (14–19). Protein evolution is constrained not only by structure and function but also by susceptibility to toxic misfolding.Insulin plays a central role in the regulation of vertebrate metabolism. The hormone, the post-translational product of a single-chain precursor, is a globular protein containing two chains, A (21 residues) and B (30 residues). Recent advances in human genetics have identified dominant mutations in the insulin gene causing permanent neonatal-onset DM² (1–4). The mutations are predicted to block folding of the precursor in the ER of pancreatic β-cells. Although expression of the wild-type allele would in other circumstances be sufficient to maintain homeostasis, studies of a corresponding mouse model (5–7) suggest that the misfolded variant perturbs wild-type biosynthesis (8, 9). Impaired β-cell secretion is associated with ER stress, distorted organelle architecture, and cell death (10). These findings have renewed interest in insulin biosynthesis (11–13) and the structural basis of disulfide pairing (14–19). Protein evolution is constrained not only by structure and function but also by susceptibility to toxic misfolding.     

Algorithms for Finding Small Attractors in Boolean Networks

A Boolean network is a model used to study the interactions between different genes in genetic regulatory networks. In this paper, we present several algorithms using gene ordering and feedback vertex sets to identify singleton attractors and small attractors in Boolean networks. We analyze the average case time complexities of some of the proposed algorithms. For instance, it is shown that the outdegree-based ordering algorithm for finding singleton attractors works in time for , which is much faster than the naive time algorithm, where is the number of genes and is the maximum indegree. We performed extensive computational experiments on these algorithms, which resulted in good agreement with theoretical results. In contrast, we give a simple and complete proof for showing that finding an attractor with the shortest period is NP-hard.[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]