Catalyzes the phosphorylation of 1D-myo-inositol 1,4,5-trisphosphate (InsP3) into 1D-myo-inositol 1,3,4,5-tetrakisphosphate and participates to the regulation of calcium homeostasis (PubMed:12649294). Can phosphorylate inositol 2,4,5-triphosphate to inositol 2,4,5,6-tetraphosphate. 1D-myo-inositol 1,4,5-trisphosphate + ATP = 1D-myo-inositol 1,3,4,5-tetrakisphosphate + ADP + H(+) Activated by calcium/calmodulin. Inhibited by high concentrations of the substrate Ins(1,2,4)P3, and allosterically activated by the product Ins(1,3,4,5)P4. Shuttles actively between nucleus and cytoplasm with both nuclear import and nuclear export activity. Highly expressed in tongue epithelium, taste papilla, heart, brain and testis. Weakly expressed in lung, liver and kidney. Belongs to the inositol phosphokinase (IPK) family.