Final answer:
Troponin is the regulatory protein that binds calcium ions (Ca++) on the thin myofilaments in muscle cells, causing a shift in tropomyosin and allowing muscle contraction to begin.
Step-by-step explanation:
The regulatory protein molecule of the thin myofilament that binds Ca++ is Troponin. When calcium ions are released from the sarcoplasmic reticulum, they bind to the troponin complex on the thin filaments. This binding causes a conformational change in the troponin complex, which shifts the tropomyosin molecule that covers the binding sites for the myosin heads on the actin filament. As a result, the myosin-binding sites on actin become exposed, allowing the myosin heads to attach to the actin filament and initiate muscle contraction.