What is the difference between folate, folic acid, and folinic acid? And how do each of them contribute or are involved in the methotrexate pathway?

Folate is the naturally occurring salt form of vitamin B9. Folic acid is a synthetic water-soluble acid form of vitamin B9. Folate and folic acid are metabolised by dihydrofolate reductase, the enzyme inhibited by methotrexate, to FH2 and then FH4, which in turn is converted to N5, N10-methylene-FH4.

Folinic acid (also known as leucovorin) is N5-formyl-FH4 and can be converted rapidly to N5, N10-methylene-FH4 without the need for dihydrofolate reductase and so bypasses the inhibition of the pathway by methotrexate.

In the treatment of rheumatoid arthritis with methotrexate, adverse effects include nausea and vomiting, mouth and gastrointestinal ulcers, and hair loss are caused by the reduction in N5, N10-methylene-FH4, which is required for the synthesis of amino acids and nucleic acids necessary for cell proliferation.  Folic acid at high enough doses to overcome the methotrexate inhibition of dihydrofolate reductase or folinic acid is therefore used to reduce the adverse effects of methotrexate.