Table 2.

Summary of transition invariants in the PN model.

Transition invariants	Remarks
p0 → p1 → k1 → β2 → γ2	Synthesis of OPG-RANKL from OPG and RANKL, and its interaction with active osteoblasts
pO → dO	Synthesis and degradation of OPG
p1 → ak3 → β3 → γ3	Closed-loop interactions among RANKL, RANK-RANKL, and osteoclast precursors
p1 → ak3 → β3 → α4 → γ4	Interactions among RANKL, RANK-RANKL, osteoclast precursors, and active osteoclasts
ak3 → k4	Reversible process between RANKL and RANK-RANKL
k2 → k1	Reversible process among OPG, RANKL, and OPG-RANKL
p1 → dL	Synthesis and degradation of RANKL
α1 → α2 → γ2	Interaction between osteoblast precursors and active osteoblasts
α1 → γ1	Synthesis and degradation of osteoblast precursors
α3 → γ3	Synthesis and degradation of osteoclast precursors
α3 → α4 → γ4	Interaction between osteoclast precursors and active osteoclasts