This study will look into insulin sensitivity (how the body responds to insulin) and effects of meals on type 2 diabetics comparing insulin peglispro to insulin glargine. The study has two treatment periods, each of which will last about four weeks. One drug (insulin peglispro or insulin glargine) will be administered in each period. Participants will receive both drugs during the study. Participants may remain on stable dose metformin, as prescribed by their personal physician.