Compounds which inhibit beta-secretase activity and methods of use thereof

A compound having the following structural formula:

• or a pharmaceutically acceptable salt thereof,
• wherein:
• X1 is —O— or a covalent bond;
• R7 is an unsubstituted C1-12alkylene;
• m is 0, 1, 2, or 3;
• R8 is a substituted or unsubstituted C1-4aliphatic group, —OR9, —R23—O—R9, a halogen, a cyano, a nitro, NR9R10, guanidino, —OPO3−2, —PO3−2, —OSO3−, —S(O)pR9, —OC(O)R9, —C(O)R9, —C(O)2R9, —NR9C(O)R10, —C(O)NR9R10, —OC(O)NR9R10, —NR9C(O)2R10, a substituted or unsubstituted aryl, a substituted or unsubstituted aralkyl, a substituted or unsubstituted heteroaryl, a substituted or unsubstituted heteroaralkyl, a substituted or unsubstituted heterocycle, or a substituted or unsubstituted heterocycloalkyl;
• p is 0, 1 or 2;
• and R9 and R10 are each, independently, H, a C1-4aliphatic group, an aryl, an aralkyl, a heterocycle, a heterocycloalkyl, a heteroaryl or a heteroaralkyl, wherein the aliphatic group, aryl, aralkyl, heterocycle, heterocyclalkyl, heteroaryl or heteroaralkyl are optionally substituted with one or more aliphatic groups;
• R23 is a substituted or unsubstituted C1-4alkylene;
• P1 is isobutyl, hydroxymethyl, cyclopropylmethyl, cyclobutylmethyl, phenylmethyl, cyclopentylmethyl, or heterocycloalkyl;
• P1′ is a substituted or unsubstituted C1-4aliphatic group, a substituted or unsubstituted heteroaralkyl, a substituted or unsubstituted aryl, a substituted or unsubstituted aralkyl, a substituted or unsubstituted heteroaryl, a substituted or unsubstituted heteroaralkyl, a substituted or unsubstituted heterocycle, or a substituted or unsubstituted heterocycloalkyl;
• P2 is R11SR12, —R11S(O)R12, —R11S(O)2R12, —R11C(O)NR12R13, —R11OR12, —R11OR14OR12, or a heterocycloalkyl, wherein
• R11 and R14 are each, independently, a C1-4alkylene;
• R12 and R13 are each, independently, H, an C1-4aliphatic group, an aryl, a heterocycle, a heterocycloalkyl, a heteroaryl, or a heteroaralkyl;
• P2′ is isopropyl or isobutyl;
• R is H;
• R4 is H;
• R2 is H;
• and
• R3 is selected from the group consisting of 2-furanylmethyl, phenylmethyl, indan-2yl, n-butyl, isopropyl, isobutyl, 1-fluoromethyl-2-fluoroethyl, indol-3yl, and 3-pyridylmethyl.