Heparan Sulfate Proteoglycan Modulation of Wnt5A Signal Transduction in Metastatic Melanoma Cells

Heparan sulfate proteoglycans (HSPGs) are important modulators for optimizing signal transduction of many pathways, including the Wnt pathways. We demonstrate that HSPG glycosaminoglycan levels increased with increasing metastatic potential of melanoma cells. Previous studies have demonstrated that Wnt5A increases the invasiveness of melanoma cells. We further demonstrate that HSPGs potentiate Wnt5A signaling, since enzymatic removal of the HSPG backbone resulted in a decrease in cellular Wnt5A levels, an increase in secreted Wnt5A in cell media, a decrease in downstream signaling, and ultimately, a decrease in invasiveness. Specifically, syndecan 1 and syndecan 4 expression correlated to Wnt5A expression and melanoma malignancy. Knockdown of syndecan 1 or 4 caused decreases in cell invasion, which could be restored by treating the cells with recombinant Wnt5A. These data indicate that syndecan 1 and 4 correlate to increased metastatic potential in melanoma patients and are an important component of the Wnt5A autocrine signaling loop, the activation of which leads to increased metastasis of melanoma.The American Cancer Society estimates that in 2009 there will be 68,720 new cases of melanoma in this country with ∼8,650 deaths. Recent studies have demonstrated that the non-canonical Wnt pathway, also known as the Wnt/Ca²⁺ pathway, plays an important role in increasing the metastatic potential of melanoma cells (1–5). Studies from our laboratory demonstrated that increasing Wnt5A, which mediates the non-canonical Wnt/Ca²⁺ signaling pathway, increased melanoma metastasis (1–3), and silencing Wnt5A levels via siRNA³ decreased invasion (2, 3). In addition, we have shown that Wnt5A acts via protein kinase C (PKC) to mediate the motility of melanoma cells via the inhibition of metastasis suppressors and an initiation of the epithelial to mesenchymal transition, characterized by the loss of E-cadherin and the up-regulation of Snail (2).Wnt signaling can be mediated by heparan sulfate proteoglycans (HSPGs) which are important signal transduction modulators. They mediate fibroblast growth factor, Hedgehog, epidermal growth factor, transforming growth factor-β, and WNT signaling pathways (6–11). HSPGs consist of two types, cell surface and basement membrane-associated HSPGs (12). Cell surface HSPGs are glycoproteins with covalently attached unbranched and modified sugar chains known as glycosaminoglycans (GAGs). There are two types of cell surface HSPGs, known as glypicans and syndecans (11, 13). Glypicans are attached to the cell surface via a glycosylphosphatidylinositol anchor, whereas syndecans are type 1 transmembrane proteins. HSPG GAG side chains are unbranched chains of modified repeating disaccharide units of N-acetylglucosamine and glucuronic acid. They are joined to the core protein via a tetrasaccharide linker attached to a serine residue. Following synthesis, these chains undergo modification with the addition of sulfates by N- and O-sulfation (14). The sulfation status determines to which specific portion of the GAG chains ligands, such as Wnt, will attach. The heparan sulfate endosulfatases Sulf1 and Sulf2 are cell surface enzymes that control growth factor signaling. The regulation of the 6-O-sulfation states by these endosulfatases changes the affinity of the GAG chains for ligand binding (15–17). Following sulfation modification, HSPGs can regulate signaling by dimerization (with other HSPGs or canonical signaling receptors), stabilization, or transport of the ligand to or away from the high affinity receptors (18–20). In addition, studies have suggested that the core proteins themselves may also play an important role in cell phenotype and function (21).HSPGs have been implicated in a number of pathological conditions, such as Simpson-Golabi-Behmel overgrowth syndrome (22), fibrodysplasia ossificans progressiva (23), and Alzheimer disease (24). In addition, HSPGs are overexpressed in many forms of cancer, including prostate cancer and melanoma (25, 26). Importantly, in cancer, proteoglycans can have both tumor-promoting and tumor-suppressing activities. This depends on the type of protein core, the GAGs attached, and the localization of the proteoglycan and the molecules they associate with. In addition, the tumor subtype, stages, and degree of tumor differentiation also affect the function of HSPGs (27). HSPGs are cleaved by heparanases or heparin lyases (heparinases), which have been shown to have differing effects on tumor cell activity. For example, treating cancer cells with heparanase-1, which cleaves heparin-like regions (specifically HLGAG sites with O-sulfated l-iduronic acid residues), results in an increase in both tumor growth and metastatic dissemination (28). However, treating tumor cells with heparinase III, which more specifically cleaves HSPGs (i.e. unsulfated d-glucorinic acid, heparan-sulfate-like regions) results in an inhibition of their metastatic capacity (29). Importantly, heparanase I cleaves only certain side chains, where heparinase III treatment cleaves the entire backbone of the HSPG. It is likely that cleavage of specific side chains facilitates cell motility by releasing cells from adhesion to neighboring cells, whereas cleavage of the entire molecule decreases the availability of secreted ligands to their receptors, especially those involved in autocrine signaling, such as Wnt5A.Historically, Wnt5A has been quite difficult to purify from cell culture media, despite the fact that it is a secreted protein. Further, in melanoma cells, Wnt5A appears to be signaling in an autocrine fashion (1, 2). These two observations, together with the fact that Wnt5A undergoes glycosylation (30), led us to hypothesize that HSPGs might be involved in increasing the availability of Wnt5A to its receptor, resulting in an increase in autocrine signaling and ultimately an increase in cellular invasion. In this study, we explore this hypothesis and investigate the role of HSPGs in the Wnt5A signaling cascade in metastatic melanoma cells.     

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.     

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]     

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]     

Characterization of the Human Sulfatase Sulf1 and Its High Affinity Heparin/Heparan Sulfate Interaction Domain

The extracellular sulfatases Sulf1 and Sulf2 remodel the 6O-sulfation state of heparan sulfate proteoglycans on the cell surface, thereby modulating growth factor signaling. Different from all other sulfatases, the Sulfs contain a unique, positively charged hydrophilic domain (HD) of about 320 amino acid residues. Using various HD deletion mutants and glutathione S-transferase (GST)-HD fusion proteins, this study demonstrates that the HD is required for enzymatic activity and acts as a high affinity heparin/heparan sulfate interaction domain. Association of the HD with the cell surface is sensitive to heparinase treatment, underlining specificity toward heparan sulfate chains. Correspondingly, isolated GST-HD binds strongly to both heparin and heparan sulfate in vitro and also to living cells. Surface plasmon resonance studies indicate nanomolar affinity of GST-HD toward immobilized heparin. The comparison of different mutants reveals that especially the outer regions of the HD mediate heparan sulfate binding, probably involving “tandem” interactions. Interestingly, binding to heparan sulfate depends on the presence of 6O-sulfate substrate groups, suggesting that substrate turnover facilitates release of the enzyme from its substrate. Deletion of the inner, less conserved region of the HD drastically increases Sulf1 secretion without affecting enzymatic activity or substrate specificity, thus providing a tool for the in vitro modulation of HS-dependent signaling as demonstrated here for the signal transduction of fibroblast growth factor 2. Taken together, the present study shows that specific regions of the HD influence different aspects of HS binding, cellular localization, and enzyme function.The human sulfatases represent a family of 17 enzymes responsible for the turnover and remodeling of sulfate esters and sulfamates. Their reaction mechanism relies on a special amino acid residue, Cα-formylglycine, which is generated post-translationally via oxidation of a conserved cysteine residue in the active site (1–3). Besides the lysosomal sulfatases involved in the cellular degradation of various sulfated substrates (4), two extracellular sulfatases, Sulf1 and Sulf2 (the Sulfs), have been described (5, 6). The Sulfs are endosulfatases with restricted substrate specificity toward 6O-sulfate groups of heparan sulfate (HS),² an information-rich glycosaminoglycan (GAG) polymer attached to proteoglycans at the cell surface and in the extracellular matrix (6–8). HS proteoglycans (HSPGs) act as co-receptors in cell signaling pathways and provide binding sites for growth factors and morphogens via specific sulfation patterns on their HS chains. By enzymatically removing 6O-sulfate groups from HSPGs on the cell surface, Sulf1 and Sulf2 differentially regulate the activity of FGF, vascular endothelial growth factor, Wnt, and other HS ligands, thereby modulating important processes such as development, cell growth, and differentiation (9–12). Misregulation of the Sulfs has been linked with both tumor progression and suppression, depending on either activating or inhibitory effects upon cell signaling (13–16).To investigate the physiological role of Sulf1 and Sulf2, single and double knock-out mice were generated (17–21). Both Sulf1 and Sulf2 knock-out mice are characterized by increased embryonic lethality, impaired neurite outgrowth, and other neurological abnormalities in the developing and adult nervous system (22). The corresponding double knock-out mice display an obvious reduction in body weight and developmental malformations, including skeletal and renal defects (18, 19, 23). Together with biochemical analyses on the impact of Sulf loss on HS sulfation, the phenotypic observations suggest a functional cooperativity between Sulf1 and Sulf2 in modulating the 6O-sulfation of UA(2S)-GlcNS(6S) disaccharide units within the S-domains of HS chains (17, 24). Moreover, analyses of heparan sulfate disaccharide compositions from Sulf1 and Sulf2 knock-out mice cell lines have indicated dynamic influences of Sulf loss also on non-substrate N-, 2O-, and 6O-sulfate groups via modulation of sulfotransferase expression, which may contribute to the developmental defects associated with the Sulf knock-out mice (24).From the biochemical perspective, it is an important question how the Sulfs are able to recognize their HSPG substrates and how cell surface localization is achieved, despite a lack of transmembrane domains or lipid anchors. Classical GAG-binding proteins, such as antithrombin III (25) or FGF1 (26), interact with their negatively charged GAG partners via small clusters of positively charged amino acid residues. Although some consensus sequences for heparin binding have been identified (XBBXBX, XBBBXXBX, and XBBXXBBBXXBBX, where B is a basic residue and X a hydropathic) (27–29), they are neither required nor sufficient. Unlike these classical GAG-binding proteins, Sulf1 and Sulf2 contain a large hydrophilic domain (HD), located between the N-terminal catalytic domain and the C-terminal domain. The HD is a unique feature of the extracellular sulfatases that is neither found in other sulfatases nor shows any homology with other known protein domains. According to sequence alignments, the HD of human Sulf1 has a size of ∼320 amino acid residues, 27% of which are basic and 14% acidic, resulting in a strong positive charge at neutral pH and a high theoretical pI of 9.8. Remarkably, the C-terminal end of the HD is composed of a cluster of 12 basic amino acid residues. Whereas the outer regions of the HD are highly conserved between Sulf1 and Sulf2 as well as between human, murine, and avian orthologs, the inner region, encoded by exons 13 and 14 in the case of human Sulf1 (6), is significantly less conserved.The role of the HD has previously been investigated for the avian ortholog QSulf2 (30). Results from this study indicated that the HD binds to negatively charged ligands and might serve to anchor the enzyme on the cell surface. Sulfate release assays indicated the necessity of the avian HD for enzymatic activity. Moreover, a very recent analysis of the HD of human Sulf1/Sulf2 revealed the presence of two furin-type proteinase cleavage sites within the inner region, explaining their partial processing into disulfide-linked subunits of 75 and 50 kDa (31). Sulf1/2 mutants, in which these sites were deleted, retained enzymatic activity but failed to potentiate Wnt signaling when overexpressed in human embryonic kidney 293 cells.Due to the observed differences in enzyme secretion and detergent solubility between the human and avian orthologs (24, 30) and the likely importance of this domain for mammalian Sulf localization and activity, we analyzed the function of the HD of human Sulf1 in mediating enzyme activity, cell surface targeting, secretion, and substrate recognition. Using different Sulf1 deletion mutants and glutathione S-transferase (GST)-HD fusion proteins, this study demonstrates that specific regions of the HD, especially at the conserved N and C termini, are responsible for heparin/HS binding, cell surface localization, and enzymatic activity of human Sulf1. Interaction analyses show that binding of the HD to heparin is significantly stronger compared with other typical heparin-binding proteins, suggesting a new mode of GAG binding. The deletion of the inner region of the HD leads to significantly increased secretion of the enzyme, allowing the purification of an active variant that is able to modulate FGF signaling in cell culture experiments.     

Biochemical Characterization and Function of Complexes Formed by Hyaluronan and the Heavy Chains of Inter-��-inhibitor (HC��HA) Purified from Extracts of Human Amniotic Membrane

Clinically, amniotic membrane (AM) suppresses inflammation, scarring, and angiogenesis. AM contains abundant hyaluronan (HA) but its function in exerting these therapeutic actions remains unclear. Herein, AM was extracted sequentially with buffers A, B, and C, or separately by phosphate-buffered saline (PBS) alone. Agarose gel electrophoresis showed that high molecular weight (HMW) HA (an average of ∼3000 kDa) was predominantly extracted in isotonic Extract A (70.1 ± 6.0%) and PBS (37.7 ± 3.2%). Western blot analysis of these extracts with hyaluronidase digestion or NaOH treatment revealed that HMW HA was covalently linked with the heavy chains (HCs) of inter-α-inhibitor (IαI) via a NaOH-sensitive bond, likely transferred by the tumor necrosis factor-α stimulated gene-6 protein (TSG-6). This HC·HA complex (nHC·HA) could be purified from Extract PBS by two rounds of CsCl/guanidine HCl ultracentrifugation as well as in vitro reconstituted (rcHC·HA) by mixing HMW HA, serum IαI, and recombinant TSG-6. Consistent with previous reports, Extract PBS suppressed transforming growth factor-β1 promoter activation in corneal fibroblasts and induced mac ro phage apo pto sis. However, these effects were abolished by hyaluronidase digestion or heat treatment. More importantly, the effects were retained in the nHC·HA or rcHC·HA. These data collectively suggest that the HC·HA complex is the active component in AM responsible in part for clinically observed anti-inflammatory and anti-scarring actions.Hyaluronan (HA)⁴ is widely distributed in extracellular matrices, tissues, body fluids, and even in intracellular compartments (reviewed in Refs. 1 and 2). The molecular weight of HA ranges from 200 to 10,000 kDa depending on the source (3), but can also exist as smaller fragments and oligosaccharides under certain physiological or pathological conditions (1). Investigations over the last 15 years have suggested that low M_r HA can induce the gene expression of proinflammatory mediators and proangiogenesis, whereas high molecular weight (HMW) HA inhibits these processes (4–7).Several proteins have been shown to bind to HA (8) such as aggrecan (9), cartilage link protein (10), versican (11), CD44 (12, 13), inter-α-inhibitor (IαI) (14, 15), and tumor necrosis factor-α stimulated gene-6 protein (TSG-6) (16, 17). IαI consists of two heavy chains (HCs) (HC1 and HC2), both of which are linked through ester bonds to a chondroitin sulfate chain that is attached to the light chain, i.e. bikunin. Among all HA-binding proteins, only the HCs of IαI have been clearly demonstrated to be covalently coupled to HA (14, 18). However, TSG-6 has also been reported to form stable, possibly covalent, complexes with HA, either alone (19, 20) or when associated with HC (21).The formation of covalent bonds between HCs and HA is mediated by TSG-6 (22–24) where its expression is often induced by inflammatory mediators such as tumor necrosis factor-α and interleukin-1 (25, 26). TSG-6 is also expressed in inflammatory-like processes, such as ovulation (21, 27, 28) and cervical ripening (29). TSG-6 interacts with both HA (17) and IαI (21, 24, 30–33), and is essential for covalently transferring HCs on to HA (22–24). The TSG-6-mediated formation of the HC·HA complex has been demonstrated to play a crucial role in female fertility in mice. The HC·HA complex is an integral part of an expanded extracellular “cumulus” matrix around the oocyte, which plays a critical role in successful ovulation and fertilization in vivo (22, 34). HC·HA complexes have also been found at sites of inflammation (35–38) where its pro- or anti-inflammatory role remain arguable (39, 40).Immunostaining reveals abundant HA in the avascular stromal matrix of the AM (41, 42).⁵ In ophthalmology, cryopreserved AM has been widely used as a surgical graft for ocular surface reconstruction and exerts clinically observable actions to promote epithelial wound healing and to suppress inflammation, scarring, and angiogenesis (for reviews see Refs. 43–45). However, it is not clear whether HA in AM forms HC·HA complex, and if so whether such an HC·HA complex exerts any of the above therapeutic actions. To address these questions, we extracted AM with buffers of increasing salt concentration. Because HMW HA was found to form the HC·HA complex and was mainly extractable by isotonic solutions, we further purified it from the isotonic AM extract and reconstituted it in vitro from three defined components, i.e. HMW HA, serum IαI, and recombinant TSG-6. Our results showed that the HC·HA complex is an active component in AM responsible for the suppression of TGF-β1 promoter activity, linkable to the scarring process noted before by AM (46–48) and by the AM soluble extract (49), as well as for the promotion of macrophage death, linkable to the inflammatory process noted by AM (50) and the AM soluble extract (51).     

A Novel Function of Heparan Sulfate in the Regulation of Cell-Cell Fusion

Despite the important contribution of cell-cell fusion in the development and physiology of eukaryotes, little is known about the mechanisms that regulate this process. Our study shows that glycosaminoglycans and more specifically heparan sulfate (HS) expressed on the cell surface and extracellular matrix may act as negative regulator of cell-cell fusion. Using herpes simplex virus type-1 as a tool to enhance cell-cell fusion, we demonstrate that the absence of HS expression on the cell surface results in a significant increase in cell-cell fusion. An identical phenomenon was observed when other viruses or polyethylene glycol was used as fusion enhancer. Cells deficient in HS biosynthesis showed increased activity of two Rho GTPases, RhoA and Cdc42, both of which showed a correlation between increased activity and increased cell-cell fusion. This could serve as a possible explanation as to why HS-deficient cells showed significantly enhanced cell-cell fusion and suggests that HS could regulate fusion via fine tuning of RhoA and Cdc42 activities.Cell-cell fusion is an important physiological process widespread in organisms ranging from yeast to humans (1). It is critical for several biological phenomena including fertilization, placenta formation, skeletal muscle and bone development, tumorigenesis, immune response, and stem cell differentiation (1–9). Defects in cell-cell fusion can lead to serious diseases, such as myotonic dystrophy, centronuclear myopathy, preeclampsia, and osteopetrosis (10–13). Defects in sperm-egg fusion are a major cause of infertility (5). Cell-cell fusion has also been utilized for therapeutic applications, including the generation of monoclonal antibody-producing hybridomas (14) as well as new agents for cancer immunotherapy (15–17).Because of its critical nature, many studies have looked at the mechanism by which cell-cell fusion occurs. Although it can occur in a variety of different biological processes, many of the fusion events share common characteristics (8). For example, tetraspanin proteins function in gamete-, myoblast-, macrophage-, and virus-mediated fusion events (18–21). Although many mediators of cell-cell fusion are known, little is known about the fine-tuning mechanisms that may regulate the membrane fusion process.Viruses have been a useful tool for studying cell-cell fusion since the discovery that they could induce the fusion of somatic cells in vitro (22). Enveloped viruses, like herpes simplex virus type-1 (HSV-1),² use transmembrane viral proteins to mediate fusion with the host cell during entry and spread (23–25). For HSV-1, fusion occurs after the virus has attached to host cells by binding to heparan sulfate (HS) using glycoproteins gB and gC (26). Fusion of the virus envelope with the plasma membrane requires that an additional glycoprotein, gD, binds to one of its receptors, a process that also requires HSV-1 gB, gH, and gL (27–29). During HSV-1-mediated cell-cell fusion, gB, gD, gH, and gL are expressed on the surface of infected cells, allowing them to bind and fuse with surrounding uninfected cells, forming syncytia.Heparan sulfate proteoglycans are ubiquitously expressed cell surface molecules composed of a protein core, commonly syndecan, covalently attached to one or more HS glycosaminoglycan (GAG) side chains via a linker region (30). HS polysaccharide chains are composed of alternating hexuronic acid and d-glucosamine units (30, 31). HS chains undergo extensive modifications during their biosynthesis, including sulfation and epimerization, resulting in a variety of structurally diverse HS chains (30, 32–33). This diversity allows HS to interact with an array of functionally unrelated proteins and participate in various processes, such as the regulation of embryonic development, angiogenesis, blood coagulation, growth factor/cytokine interactions, cell adhesion, and lipid metabolism (30).Much remains to be learned about the cell-cell fusion mechanism and regulation of this phenomenon. The purpose of our study was to examine the effect of HS on cell-cell fusion and how it may function in the fusion mechanism. Using HSV-1 as a tool, we discovered that the absence of HS from the cell surface significantly enhanced the ability of cells to fuse with each other. This effect was also seen independently of HSV-1 in cells that neither expressed HSV-1 glycoproteins nor their receptors. This suggests a novel role for HS as a negative regulator and a fine-tuner of cell-cell fusion events.     

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]