An enhanced triclustering $ \delta $-Trimax method with fuzzy cuckoo search based on Lévy flight and Gaussian distribution for gene expression data

Titin Siswantining; Muhamad Ido Raskapati; Nisa Nurul Hidayah; Gianinna Ardaneswari; Saskya Mary Soemartojo; Siti Nurrohmah; Devvi Sarwinda; Setia Pramana; Titin Siswantining; Muhamad Ido Raskapati; Nisa Nurul Hidayah; Gianinna Ardaneswari; Saskya Mary Soemartojo; Siti Nurrohmah; Devvi Sarwinda; Setia Pramana

doi:10.3934/mbe.2026015

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 2: 366-387. doi: 10.3934/mbe.2026015

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Research article

An enhanced triclustering $ \delta $-Trimax method with fuzzy cuckoo search based on Lévy flight and Gaussian distribution for gene expression data

1.
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok 16424, Indonesia
2.
Politeknik Statistika STIS, Jakarta 13330, Indonesia

Received: 05 March 2025 Revised: 30 October 2025 Accepted: 11 November 2025 Published: 06 January 2026

The triclustering method employed in this study integrates the $ \delta $-Trimax approach with the fuzzy cuckoo search (FCS), thereby leveraging the Lévy flight and Gaussian distribution to analyze gene expression data in three dimensions. In this framework, the initial triclusters produced by $ \delta $-Trimax are further optimized using FCS, where the Lévy flight enhances global exploration and the Gaussian distribution intensifies local exploitation, thus achieving a balanced search for optimal solutions. Each tricluster set is evaluated using the tricluster quality index (TQI) to ensure coherence across genes, conditions, and time points. The method was applied to gene expression datasets from primary fibroblast cells and heart disease samples. In the fibroblast dataset, the best tricluster set was obtained with $ \delta = 0.015 $ and yielded the lowest average TQI value. For the heart disease dataset, the most optimal solution was achieved with $ \delta = 0.026 $, which yielded the lowest average TQI, and the best tricluster showed large gene coverage across multiple time points. A functional analysis of the selected triclusters using gene ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathways uncovered significant enrichment in pathways such as the NF-$ \kappa $B signaling pathway (hsa04064), TGF-$ \beta $ signaling pathway (hsa04350), and calcium signaling pathway (hsa04020), all of which are mechanistically relevant to immune modulation, extracellular matrix organization, and cardiac muscle function. These findings highlight the utility of the proposed hybrid framework in uncovering biologically meaningful gene modules and provide valuable insights into the molecular mechanisms underlying fibrotic and cardiovascular diseases.
- fuzzy cuckoo search,
- gene expression data,
- gene ontology,
- triclustering quality index,
- triclustering analysis
Citation: Titin Siswantining, Muhamad Ido Raskapati, Nisa Nurul Hidayah, Gianinna Ardaneswari, Saskya Mary Soemartojo, Siti Nurrohmah, Devvi Sarwinda, Setia Pramana. An enhanced triclustering $ \delta $-Trimax method with fuzzy cuckoo search based on Lévy flight and Gaussian distribution for gene expression data[J]. Mathematical Biosciences and Engineering, 2026, 23(2): 366-387. doi: 10.3934/mbe.2026015

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Abstract

The triclustering method employed in this study integrates the $ \delta $-Trimax approach with the fuzzy cuckoo search (FCS), thereby leveraging the Lévy flight and Gaussian distribution to analyze gene expression data in three dimensions. In this framework, the initial triclusters produced by $ \delta $-Trimax are further optimized using FCS, where the Lévy flight enhances global exploration and the Gaussian distribution intensifies local exploitation, thus achieving a balanced search for optimal solutions. Each tricluster set is evaluated using the tricluster quality index (TQI) to ensure coherence across genes, conditions, and time points. The method was applied to gene expression datasets from primary fibroblast cells and heart disease samples. In the fibroblast dataset, the best tricluster set was obtained with $ \delta = 0.015 $ and yielded the lowest average TQI value. For the heart disease dataset, the most optimal solution was achieved with $ \delta = 0.026 $, which yielded the lowest average TQI, and the best tricluster showed large gene coverage across multiple time points. A functional analysis of the selected triclusters using gene ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathways uncovered significant enrichment in pathways such as the NF-$ \kappa $B signaling pathway (hsa04064), TGF-$ \beta $ signaling pathway (hsa04350), and calcium signaling pathway (hsa04020), all of which are mechanistically relevant to immune modulation, extracellular matrix organization, and cardiac muscle function. These findings highlight the utility of the proposed hybrid framework in uncovering biologically meaningful gene modules and provide valuable insights into the molecular mechanisms underlying fibrotic and cardiovascular diseases.

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