{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"michaelis-menten"},"result":{"entity_type":"calculator","id":"michaelis-menten","calculator_id":"michaelis-menten","canonical_url":"https://tttkmbb.com/science/michaelis-menten","name":"Michaelis-Menten Enzyme Kinetics Calculator","title":"Michaelis-Menten Calculator – Reaction Rate from Vmax, Km and Substrate Concentration","category":"science","category_name":"Biology, Earth & Space","tool_name":"calculate_michaelis_menten_rate","featured_mcp_tool":false,"description":"Computes the initial reaction velocity of an enzyme with Michaelis–Menten kinetics, v = Vmax × [S] / (Km + [S]), the fraction of Vmax reached, the substrate concentrations for 50 % and 90 % of Vmax, and optionally a rate table for a list of substrate concentrations.","use_when":"You know an enzyme's Vmax and Km and need the rate at a given substrate concentration, or want to tabulate the saturation curve.","do_not_use_when":"The enzyme shows cooperativity (Hill equation), inhibition, or substrate depletion over time (integrated rate law), or you need to fit Vmax and Km from data (use linear-regression on a Lineweaver–Burk transform).","inputs":[{"name":"vmax","label":"Vmax","type":"number","required":true,"max":1000000000000,"exclusive_min":0,"description":"Maximum reaction velocity at saturating substrate, in any rate unit (e.g. µmol/min or µM/s); the rate output uses the same unit.","example":100},{"name":"km","label":"Km","type":"number","required":true,"max":1000000000000,"exclusive_min":0,"description":"Michaelis constant: substrate concentration at half Vmax, in the same concentration unit as substrate_concentration.","example":5},{"name":"substrate_concentration","label":"Substrate concentration [S]","type":"number","required":true,"min":0,"max":1000000000000,"description":"Substrate concentration in the same unit as Km.","example":5},{"name":"substrate_list","label":"Substrate concentrations for a table","type":"number_list","required":false,"description":"Optional list of up to 20 substrate concentrations (same unit as Km) to tabulate rates.","example":[1,2,5,10,20,50]}],"outputs":[{"name":"reaction_rate","label":"Reaction rate v","type":"number","decimals":4,"description":"Vmax × [S] / (Km + [S]), in the unit of vmax."},{"name":"fraction_of_vmax_percent","label":"Fraction of Vmax","type":"number","unit":"%","decimals":2,"description":"100 × [S] / (Km + [S])."},{"name":"substrate_for_half_vmax","label":"[S] for 50 % Vmax","type":"number","decimals":4,"description":"Equals Km."},{"name":"substrate_for_90_percent_vmax","label":"[S] for 90 % Vmax","type":"number","decimals":4,"description":"9 × Km."},{"name":"reciprocal_substrate","label":"1/[S] (Lineweaver-Burk x)","type":"number","decimals":6,"description":"1 / substrate_concentration (when [S] > 0)."},{"name":"reciprocal_rate","label":"1/v (Lineweaver-Burk y)","type":"number","decimals":6,"description":"1 / reaction_rate (when [S] > 0)."},{"name":"table","label":"Rate table","type":"list","decimals":4,"description":"Rows of substrate, rate and percent_of_vmax for each value of substrate_list."}],"input_schema":{"type":"object","properties":{"vmax":{"description":"Maximum reaction velocity at saturating substrate, in any rate unit (e.g. µmol/min or µM/s); the rate output uses the same unit.","type":"number","maximum":1000000000000,"exclusiveMinimum":0,"examples":[100]},"km":{"description":"Michaelis constant: substrate concentration at half Vmax, in the same concentration unit as substrate_concentration.","type":"number","maximum":1000000000000,"exclusiveMinimum":0,"examples":[5]},"substrate_concentration":{"description":"Substrate concentration in the same unit as Km.","type":"number","minimum":0,"maximum":1000000000000,"examples":[5]},"substrate_list":{"description":"Optional list of up to 20 substrate concentrations (same unit as Km) to tabulate rates.","type":"array","items":{"type":"number"},"examples":[[1,2,5,10,20,50]]}},"additionalProperties":false,"required":["vmax","km","substrate_concentration"]},"output_schema":{"type":"object","properties":{"reaction_rate":{"description":"Vmax × [S] / (Km + [S]), in the unit of vmax.","type":"number"},"fraction_of_vmax_percent":{"description":"100 × [S] / (Km + [S]). Unit: %.","type":"number","x-unit":"%"},"substrate_for_half_vmax":{"description":"Equals Km.","type":"number"},"substrate_for_90_percent_vmax":{"description":"9 × Km.","type":"number"},"reciprocal_substrate":{"description":"1 / substrate_concentration (when [S] > 0).","type":"number"},"reciprocal_rate":{"description":"1 / reaction_rate (when [S] > 0).","type":"number"},"table":{"description":"Rows of substrate, rate and percent_of_vmax for each value of substrate_list.","type":"array"}}},"formula":"v = Vmax × [S] / (Km + [S]); fraction = [S] / (Km + [S]); [S] at 50 % Vmax = Km; [S] at 90 % Vmax = 9 Km; Lineweaver-Burk: 1/v = (Km / Vmax) × 1/[S] + 1/Vmax","method":"Michaelis–Menten (1913) steady-state model for a single-substrate enzyme without cooperativity or inhibition; v is the initial velocity before significant substrate depletion or product inhibition. Reaching 99 % of Vmax needs [S] = 99 Km.","sources":[{"name":"Wikipedia – Michaelis–Menten kinetics","url":"https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics","type":"reference","retrieved_at":"2026-09-24"},{"name":"Johnson KA & Goody RS (2011) The original Michaelis constant: translation of the 1913 Michaelis-Menten paper, Biochemistry 50(39):8264-8269","url":"https://pubmed.ncbi.nlm.nih.gov/21888353/","type":"peer_reviewed","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"Vmax 100, Km 5, [S] 5 with table","inputs":{"vmax":100,"km":5,"substrate_concentration":5,"substrate_list":[1,2,5,10,20,50]},"expected":{"reaction_rate":50,"fraction_of_vmax_percent":50,"substrate_for_half_vmax":5,"substrate_for_90_percent_vmax":45,"table":[{"substrate":1,"rate":16.6667},{"substrate":2,"rate":28.5714},{"substrate":5,"rate":50},{"substrate":10,"rate":66.6667},{"substrate":20,"rate":80},{"substrate":50,"rate":90.9091}]},"url":"https://tttkmbb.com/api/v1/calculate/michaelis-menten?vmax=100&km=5&substrate_concentration=5&substrate_list=1%2C2%2C5%2C10%2C20%2C50"},{"name":"Vmax 100, Km 5, [S] 20","inputs":{"vmax":100,"km":5,"substrate_concentration":20},"expected":{"reaction_rate":80,"fraction_of_vmax_percent":80,"reciprocal_substrate":0.05,"reciprocal_rate":0.0125},"url":"https://tttkmbb.com/api/v1/calculate/michaelis-menten?vmax=100&km=5&substrate_concentration=20"}],"faq":[{"q":"What does Km mean?","a":"The substrate concentration at which the rate is half of Vmax; a lower Km means the enzyme reaches half-saturation at lower substrate concentration (often read as higher apparent affinity)."},{"q":"Which units should I use?","a":"Any, as long as Km and [S] share a concentration unit; the rate comes out in the unit of Vmax (µmol/min, µM/s, absorbance/min)."},{"q":"How do I get Vmax and Km from experimental data?","a":"Fit v against [S] by non-linear regression, or use the Lineweaver–Burk transform (1/v vs 1/[S]: slope Km/Vmax, intercept 1/Vmax) with the linear-regression calculator, keeping in mind that the double-reciprocal plot amplifies errors at low [S]."}],"tags":["michaelis menten","enzyme kinetics","vmax","km","reaction rate","lineweaver burk"],"related":[{"calculator_id":"arrhenius-equation","reason":"Temperature dependence of the rate constant."},{"calculator_id":"molarity","reason":"Prepare substrate solutions at the required concentration."},{"calculator_id":"linear-regression","reason":"Fit Vmax and Km from a Lineweaver-Burk plot."}],"links":{"html":"https://tttkmbb.com/science/michaelis-menten","markdown":"https://tttkmbb.com/science/michaelis-menten.md","json":"https://tttkmbb.com/science/michaelis-menten.json","api":"https://tttkmbb.com/api/v1/calculate/michaelis-menten","schema":"https://tttkmbb.com/api/v1/calculators/michaelis-menten","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T03:46:19Z","next_actions":[{"tool":"run_calculator","calculator_id":"michaelis-menten","reason":"Run Michaelis-Menten Enzyme Kinetics Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/science/michaelis-menten.md"}}