To be eligible to participate in this study candidates must meet the following eligibility criteria at Screening: Inclusion: For the treatment group * Male and females between the ages of 4 months to 72 months at the time of screening. * Molecularly-confirmed diagnosis of SPG50 (confirmed by a CLIA certified, CE-marked, or equivalent lab): Genomic DNA mutation analysis demonstrating bi-allelic pathogenic variants in the AP4M1 gene. * Subjects must have features of neurologic dysfunction by clinical history and physical examination. * Stable doses of concomitant medications such as anti-spasticity medications, anti-seizure medications, behavioral management medications, sleep medications, and special diets, supplements, or nutritional support for at least 3 months prior to Screening. If recent changes (\< 3 months) in medications, the subject may be allowed per Investigator judgement. * Parent/legal guardian willing to provide written informed consent for their child prior to participation in the study, * Subjects and caregivers must demonstrate ability to travel to the study center. For the 30 days post treatment subjects must reside within 100 miles (approximately 160 km) of the clinical site. For the control group * Male and females between the ages of 4 to 72 months at the time of screening. * A molecularly confirmed diagnosis of SPG47, SPG50 or SPG52 (confirmed by a CLIA certified, CE-marked, or equivalent lab). Genomic DNA mutation analysis demonstrating bi-allelic pathogenic variants in the AP4B1, AP4M1, or AP4S1 gene, * Subjects must have features of neurologic dysfunction by clinical history and physical examination. * Parent/legal guardian willing to provide written informed consent for their child prior to participation in the study. * Subject able to comply with all protocol requirements and procedures. * Subjects and caregivers must demonstrate the ability to travel to the study center. Exclusion: For the treatment group * Inability to participate in the clinical evaluation as determined by the principal investigators. * Clinically significant abnormal laboratory values (hemoglobin \< 6 or \> 20 g/dL; white blood cell \> 20,000 per cmm, platelets count \< 100,000 per cmm; INR \> ULN; GGT, ALT, and AST or total bilirubin \> 1.5 × ULN, creatinine ≥ 1.5 mg/dL) prior to gene replacement therapy. * Presence of a concomitant medical condition (eg, scoliosis or bleeding disorder) that precludes a lumbar puncture or use of anesthetics for sedated procedures. * Documented cardiomyopathy or significant congenital heart abnormalities. * History of severe/life-threatening allergic reaction to sirolimus, tacrolimus, corticosteroids, or gadolinium. * Concomitant illness or requirement for chronic drug treatment that in the opinion of the PI creates unnecessary risks for gene transfer, or interactions with the immunosuppressive agents. * Any item which would exclude the subject from being able to undergo MRI according to local institutional policy, or any other procedure. * The presence of significant AP-4 related CNS impairment or behavioral disturbances that would confound the scientific rigor or interpretation of results of the study. * Recent or planned elective surgical procedures (within 6 months) that would confound the scientific rigor or interpretation of results of the study. * Failure to obtain appropriate informed consent. * Reason to believe that the subject or parents of the subject will not comply with the study procedures outlined in the study protocol. * Have received an investigational drug within 30 days prior to screening or plan to receive an investigational drug (other than gene therapy) during the study. * Enrollment and participation in another interventional clinical trial 90 days before first visit (screening). For the control group * Inability to participate in the clinical evaluation as determined by the principal investigators. * Any other situation that would exclude the subject from undergoing any other procedure required in this study. * The presence of significant AP-4 related CNS impairment or behavioral disturbances that would confound the scientific rigor or interpretation of results of the study. * Recent or planned elective surgical procedures that would confound the scientific rigor or interpretation of results of the study. * Failure to obtain appropriate informed consent. * Reason to believe that the subject or parents of the subject will not comply with the study procedures outlined in the study protocol. * Have received an investigational drug within 30 days prior to screening or plans to receive an investigational drug (other than gene therapy) during the study. * Enrollment and participation in another interventional clinical trial 90 days before first visit (screening).