The invention provides alpha -mimetic structure of formula (I) wherein A is -(C=O)-CHR-, or-(C=O), B is N-R- or -CHR-, D is -(C=O)-(CHR)- or- (C=O) E is -(ZR)- or (C=O), G is -(XR)n-, -(CHR)-(NR)-, -(C=O)-(XR)-,-(C=N-W-R)-, -(C=O)-, X-(C=O)-R, X-(C=O)-NRR,X-(SO)-R, or X-(C=O)-OR, W is -Y(C=O)-,-(C=O)NH-, -(SO)-, -CHR, (C=O)-(NR)-,substituted or unsubstituted oxadiazole, substituted or unsubstituted triazole,substituted or unsubstituted thiadiazole, substituted or unsubstituted 4,5dihydrooxazole, substituted or unsubstituted 4,5 dihydrothiazole, substitutedor unsubstituted 4,5 dihydroimidazole, or nothing, Y is oxygen or sulfur, X andZ is independently nitrogen or CH, n=0 or 1; and R R, R,R, R, R, R, R, RR, R, R, R, R, andR are the same or different and independently selected from an aminoacid side chain moiety or derivative thereof, the remainder of the molecule, alinker and a solid support, and stereoisomers, salts, and prodrugs thereof, providedthat where B is CHR and W is -Y(C=O)-, -(C=O)NH-, -(SO), -CHR,or (C=O)-(NR)-, G cannot be CHR, NR, (C=O)-CHR,(C=O)-NR, or no atom at all. Additionally, the invention providesmethods wherein α-mimetic compounds are used to treat cancer stem cells.