We report on observations of "zippering" that occurs when two sickle hemoglobin fibers come together side by side. A transient Y-shaped object is formed which "zips " closed. We have been able to show how the strength of the interactions that drive this may be estimated by studying the frustrated structures sometimes formed between several fibers. Our measurements, when combined with mechanical constants determined by an analysis of bending fluctuations, allow us to make the first estimate of the magnitude of these interactions, of the order of 7kBT microm(-1). Hemoglobin volume fractions of tens of %, lead to significant depletion forces. We estimate the magnitude of both the depletion and Van der Waals forces between pairs of single fibers. We study how these are effected by the helical nature of the fibers and renormalised by bending fluctuations, calculations that could have wider applications beyond sickle hemoglobin fibers. Our theoretical analysis of single fibers is in encouraging, although not fully quantitative, agreement with our measurements. We conclude that the physics and rheology of the hemoglobin gel, as well as the pathology of sickle cell anemia itself, may be influenced by depletion interactions.