Adoptive TReg Cell for Suppression of aGVHD After UCB HSCT for Heme Malignancies

Inclusion Criteria: * Must be ≥18, but \< 70 years of age with no matched 7/8 or 8/8 sibling donor - patients ≥ 70 and ≤ 75 years of age may be eligible if they have a Co-Morbidity score ≤ 2 (http://www.qxmd.com/calculate-online/hematology/hct-ci) * UCB unit(s) composing the graft will be selected according to the current University of Minnesota umbilical cord blood graft selection algorithm and an additional cord blood unit to be used as the source to manufacture the Treg product. This UCB unit must be matched at 4-6/6 to the patient, considering HLA-A, B at the antigen level and DRB1 at the allele level * Acute Leukemias: Must be in remission by morphology. Also a small percentage of blasts that is equivocal between marrow regeneration versus early relapse are acceptable provided there are no associated cytogenetic markers consistent with relapse. Refer to Section 5.2 for complete definitions. * Burkitt's Lymphoma in CR2 or subsequent CR * Natural Killer Cell Malignancies * Chronic Myelogenous Leukemia: all types except refractory blast crisis. Chronic phase patients must have failed at least two different tyrosine-kinase inhibitors (TKIs), or been intolerant to all available TKIs or have T315I mutation. * Myelodysplastic Syndrome: IPSS INT-2 or High Risk; R-IPSS High or Very High; WHO classification: RAEB-1, RAEB-2; Severe Cytopenias: ANC \< 0.8, Anemia or thrombocytopenia requiring transfusion; Poor or very poor risk cytogenetics based on IPSS or R-IPSS definitions; ther…