Instruments and methods facilitate reconstruction, repair, and the closure of apertures in soft tissues, including the anulus fibrosus (AF), heart, lung, abdomen, thorax, vascular structures and other organs. Flexible longitudinal fixation components (i.e., sutures) are positioned across apertures in soft tissues, preferably to closes the inlets and/or outlets of such apertures. Tension on the flexible longitudinal fixation components may be used to narrow or close the apertures. Knotless fixation technologies such as suture welding are preferably used to fasten the ends of the flexible longitudinal fixation components. Certain embodiments include an intra-aperture component dimensioned for positioning within a defect in the AF, with one or more components being used to maintain the intra-aperture component in position. For example, the flexible longitudinal fixation component(s) may be anchored to one of the upper and lower vertebral bodies.