Representing part-whole relations and effectively using them in domain ontologies and conceptual data models poses multiple challenges. In this paper we face the issue of imposing temporal constraints on part-whole relationships, introducing a way to account for "essential" and "immutable" parts (and wholes) in addition to the usual "mandatory" parts (and wholes). Our approach is based on i) an explicit temporalization of the part-whole relation, which allows us to introduce a novel notion of "status" for part-whole relationships; ii) an explicit account of the ontological nature of the classes involved in a part-whole relationships, which distinguishes between "rigid" and "anti-rigid" classes. The main novelty in this paper is to resort to a temporal logic approach to capture the above mentioned notions. The formalization proposed here is grounded on the temporal description logic DLRUS and is based on previous successful efforts to formalize temporal conceptual models.