A 3-year-old boy is brought to the physician by his mother because of a recurring rash and belly pain. The current rash has been present for the past 3 weeks; but she states that he has been suffering from rashes for 2 years. Physical examination shows a raised crusty reddish-brown rash involving the chest, abdomen, back, and groin area. There is also diffuse lymphadenopathy. He has pain upon palpation of the abdomen, scalp and upper and lower extremities. X-rays of the skull and long bones show osteolytic lesions. A diagnostic lymph node biopsy is performed which shows a sinusoidal infiltrate. Electron microscopy reveals club shaped structures in the lesional cells. Immunohistochemical studies of the abnormal cells show CD1a, Langerin and S-100 antigen expression. The abnormal cells in this patient are most likely derived from which of the following?

A 20-year-old woman presents to the urgent care due to a recurrence of painful joints that have resulted in the inability to bend her fingers and arms. It has also become increasingly difficult to sit and stand. Gram stain examination of synovial fluid taken from the left knee joint shows a marked leukocytosis with numerous extracellular Gram-positive cocci. The CLS noted that she had S. aureus isolated from the right knee 6 months prior. A deficiency in which of the following proteins of the complement system is likely responsible for her recurrent infections?