How does vitamin K deficiency reduce the blood's ability to clot?

Vitamin K is essential for the synthesis of six proteins involved in blood coagulation, including factors II (prothrombin), VII, IX, and X, as well as proteins C and S.123

In the liver, these proteins are initially synthesized in a precursor form. Vitamin K acts as a necessary cofactor for an enzyme-mediated reaction called $\gamma$-carboxylation, which converts specific glutamic acid residues into $\gamma$-carboxyglutamic acid. This modified amino acid is critical because it creates calcium-binding sites on the proteins.1

When vitamin K is deficient, $\gamma$-carboxylation is inhibited, resulting in the production of proteins that lack $\gamma$-carboxyglutamic acid. Because these defective proteins cannot bind to calcium, they are unable to interact with cell membranes and possess no biological activity. Consequently, the blood's ability to form clots is significantly impaired, leading to an increase in clotting time and a tendency toward internal bleeding.123