Pegloticase and Methotrexate Co-administered in Participants With Uncontrolled Gout Who Previously Failed Pegloticase Monotherapy

Inclusion Criteria: 1. Willing and able to give informed consent. 2. Willing and able to comply with the prescribed treatment protocol and evaluations for the duration of the trial. 3. Adult men or women ≥18 years of age. 4. Uncontrolled gout, defined by the following criteria: * Hyperuricemia during the Screening Period, defined as sUA ≥6 mg/dL, and; * Failure to maintain normalization of sUA with xanthine oxidase inhibitors at the maximum medically appropriate dose or with a contraindication to xanthine oxidase inhibitor therapy based on medical record review or subject interview, and; * Symptoms of gout, including at least 1 of the following: * Presence of at least 1 tophus * Recurrent flares, defined as 2 or more flares in the 12 months prior to Screening * Presence of chronic gouty arthritis 5. Subject was previously treated with pegloticase without concomitant immunomodulation and stopped pegloticase due to failure to maintain sUA reduction response (had ≥1 sUA \>6 mg/dL within 2 weeks post pegloticase infusion) and did not experience an IR (Cohort 1) and/or stopped pegloticase treatment due to pegloticase-related clinically mild IR (Cohort 2). 6. Subject for whom the last pegloticase infusion occurred \>6 months prior to Screening. 7. Willing to discontinue any oral urate-lowering therapy for at least 7 days prior to Day 1 and remain off other urate-lowering therapy during the Pegloticase + MTX Treatment Period. 8. Women of childbearing potential (including those with an onset of menopause \<2 years prior to Screening, non-therapy-induced amenorrhea for \<12 months prior to Screening or not surgically sterile \[absence of ovaries and/or uterus\]) must have negative serum pregnancy tests during Screening: Subjects must agree to use 2 reliable forms of contraception during the trial, 1 of which is recommended to be hormonal, such as an oral contraceptive. Hormonal contraception must be started ≥1 full cycle prior to Week -6 (start of MTX) and continue for 30 days after the last dose of pegloticase, or at least 1 ovulatory cycle after the last dose of MTX (whichever is the longer duration after the last dose of pegloticase). Highly effective contraceptive methods (with a failure rate \<1% per year), when used consistently and correctly, include implants, injectables, combined oral contraceptives, some intrauterine devices, sexual abstinence or vasectomized partner. 9. Men who are not vasectomized must agree to use appropriate contraception so as to not impregnate a female partner of reproductive potential during the trial, beginning with the initiation of MTX at Week -6 and continuing for at least 3 months after the last dose of MTX. 10. Able to tolerate MTX at SC doses of at least 15 mg during the MTX Run-in Period, regardless of estimated glomerular filtration rate (eGFR) status. Exclusion Criteria: 1. Known history of medically confirmed prior anaphylactic reaction. 2. Known history of moderate or severe IR (including but not limited to difficulty in breathing, hypotension, generalized urticaria, generalized erythema, angioedema and/or required treatment with IV steroids or epinephrine; or other serious adverse events (SAEs) related to pegloticase or any other pegylated product treatment. 3. Weight \>160 kg (352 pounds) at Screening. 4. Any serious acute bacterial infection, unless treated and completely resolved with antibiotics at least 2 weeks prior to the Week 6 Visit. 5. Severe chronic or recurrent bacterial infections, such as recurrent pneumonia or chronic bronchiectasis. 6. Current or chronic treatment with systemic immunosuppressive agents, such as MTX, azathioprine, cyclosporine, leflunomide, cyclophosphamide or mycophenolate mofetil. 7. Current treatment with prednisone \>10 mg/day or equivalent dose of another corticosteroid on a chronic basis (defined as 3 months or longer). 8. Known history of any solid organ transplant surgery requiring maintenance immunosuppressive therapy. 9. Known history of hepatitis B virus surface antigen positivity or hepatitis B DNA positivity, unless treated, viral load is negative and no chronic or active infection confirmed by hepatitis B virus serology. 10. Known history of hepatitis C virus RNA positivity, unless treated and viral load is negative. 11. Known history of human immunodeficiency virus positivity. 12. glucose-6-phosphate dehydrogenase (G6PD) deficiency (tested at the Screening Visit). 13. Severe chronic renal impairment (eGFR \<30 mL/min/1.73 m\^2) at the Screening Visit based on 4 variable Modification of Diet in Renal Disease \[MDRD\] formula or currently on dialysis. 14. Non-compensated congestive heart failure, hospitalization for congestive heart failure or treatment for acute coronary syndrome (myocardial infarction or unstable angina) within 3 months of the Screening Visit, current uncontrolled arrhythmia or current uncontrolled blood pressure (\>160/100 mm Hg) prior to Week -6. 15. Pregnant, planning to become pregnant, breastfeeding, planning to impregnate female partner, or not on an effective form of birth control, as determined by the Investigator. 16. Prior treatment with another recombinant uricase (rasburicase) or concomitant therapy with a PEG-conjugated drug. 17. Known allergy to pegylated products or history of anaphylactic reaction to a recombinant protein or porcine product. 18. Contraindication to MTX treatment or MTX treatment considered inappropriate. 19. Known intolerance to MTX. 20. Receipt of an investigational drug within 4 weeks or 5 half-lives, whichever is longer, prior to MTX administration at Week -6 or plan to take an investigational drug during the trial. 21. Current liver disease, as determined by alanine transaminase (ALT) or aspartate transaminase (AST) \>1.25 × upper limit of normal (ULN) or albumin \<lower limit of normal at the Screening Visit. 22. Currently receiving systemic or radiologic treatment for ongoing cancer, excluding nonmelanoma skin cancer. 23. History of malignancy within 5 years other than non-melanoma skin cancer or in situ carcinoma of cervix. 24. White blood cell count \<4.0 × 10\^9/L, hematocrit \<32% or platelet count \<75 × 10\^9/L. 25. Diagnosis of osteomyelitis. 26. Known history of hypoxanthine-guanine phosphoribosyl-transferase deficiency, such as Lesch-Nyhan and Kelley-Seegmiller syndrome. 27. Unsuitable candidate for the trial (e.g., cognitive impairment), based on the opinion of the Investigator, such that participation might create undue risk to the subject or interfere with the subject's ability to comply with the protocol requirements or complete the trial. 28. Alcohol use in excess of 3 alcoholic beverages per week. 29. A known intolerance to all protocol standard gout flare prophylaxis regimen (i.e., unable to tolerate any of the following 3 agents: colchicine,nonsteroidal anti-inflammatory drugs (NSAIDs) or low- dose prednisone (≤10 mg/day). 30. Current pulmonary fibrosis, bronchiectasis or interstitial pneumonitis. If deemed necessary by the Investigator, a chest x-ray may be performed during Screening.