IntroductionRate EquationsSummary1/26 College of the RedwoodsMath 55, Differential EquationsMichaelis-Menten Enzyme KineticsThe Jigman and The SauceMane-mail: [email protected]@yahoo.com2/26 IntroductionFigure 1: Triophosphate EnzymeWhat is Enzyme Kinetics?• Kinetics is the study of rates of chemical reactions• Enzymes are little molecular machines that carry outreactions in cells• Enzyme kinetics is the study of rates of chemical reactions thatinvolve enzymes3/26 Michaelis-Menten Equation• The Michaelis-Menten Equation is a differential equation used tomodel the rate at which enzymatic reactions occur• This model allows scientist to predict how fast a reaction will takeplace based on the concentrations of the chemicals being reacted.Figure 2: A Model of an Enzyme4/26 Typical Enzymatic ReactionsE0+ Sk1−−*)−−k−1E1E1k2−→ E0+ PS the concentration of the substrate(the unreacted molecules)P the concentration of product(the reacted molecules)E0the concentration of the unoccupied enzymesE1the concentration of occupied enzymes.k1, k−1, k2the rate constants5/26 Conditions for Michaelis-Menten Modelling• In order to model an enzymatic reaction, some conditions must bemaintained:– Temperature, ionic strength, pH, and other physicalconditions that might affect the rate must remain constant6/26 Conditions for Michaelis-Menten Modelling• In order to model an enzymatic reaction, some conditions must bemaintained:– Temperature, ionic strength, pH, and other physicalconditions that might affect the rate must remain constant– Each enzyme can act on only one other molecule at a time7/26 Conditions for Michaelis-Menten Modelling• In order to model an enzymatic reaction, some conditions must bemaintained:– Temperature, ionic strength, pH, and other physicalconditions that might affect the rate must remain constant– Each enzyme can act on only one other molecule at a time– The enzyme must remain unchanged during the course of thereaction.8/26 Conditions for Michaelis-Menten Modelling• In order to model an enzymatic reaction, some conditions must bemaintained:– Temperature, ionic strength, pH, and other physicalconditions that might affect the rate must remain constant– Each enzyme can act on only one other molecule at a time– The enzyme must remain unchanged during the course of thereaction.– The concentration of substrate must be much higher than theconcentration of enzyme9/26 Rate EquationsE0+ Sk1−−*)−−k−1E1(1)E1k2−→ E0+ P (2)• The rate at which reaction (1) occurs is derived as follows:– The number of possible contacts between S and E0is directlyproportional to SE0.– The numb er of successful contacts over a certain amount of timeis proportional to the number of possible contacts.– Thus, the rate of reaction is directly proportional to SE0:Rate1= k1SE0.where k1is the rate constant.10/26 E0+ Sk1−−*)−−k−1E1(1)E1k2−→ E0+ P (2)• The rate at which the reverse of reaction (1) occurs is derived asfollows:– A certain proportion of E1will release S over a certain amountof time before the reaction is carried out.– The rate of the reverse reaction is directly proportional to E1:Rate−1= k−1E1where k−1is the rate constant.11/26 E0+ Sk1−−*)−−k−1E1(1)E1k2−→ E0+ P (2)• The rate at which reaction (2) occurs is derived as follows:– A certain proportion of E1will produce P over a certain amountof time.– The rate of production of P is directly proportional to E1:Rate2= k2E1where k2is the rate constant.12/26 Specific Rates of Reactions for each CompoundE0+ Sk1−−*)−−k−1E1(1)E1k2−→ E0+ P (2)Rate1= k1SE0, Rate−1= k−1E1, and Rate2= k2E1The rate equations associated with each reaction determines the rateof change of S, E0, E1, and P . Realizing this, we can write the follow-ing:dSdt= −Rate1+ Rate−1= −k1SE0+ k−1E113/26 Specific Rates of Reactions for each CompoundE0+ Sk1−−*)−−k−1E1(1)E1k2−→ E0+ P (2)Rate1= k1SE0, Rate−1= k−1E1, and Rate2= k2E1The rate equations associated with each reaction determines the rateof change of S, E0, E1, and P . Realizing this, we can write the follow-ing:dE0dt= −Rate1+ Rate−1+ Rate2= −k1SE0+ k−1E1+ k2E114/26 Specific Rates of Reactions for each CompoundE0+ Sk1−−*)−−k−1E1(1)E1k2−→ E0+ P (2)Rate1= k1SE0, Rate−1= k−1E1, and Rate2= k2E1The rate equations associated with each reaction determines the rateof change of S, E0, E1, and P . Realizing this, we can write the follow-ing:dE1dt= Rate1− Rate−1− Rate2= k1SE0− k−1E1− k2E115/26 Specific Rates of Reactions for each CompoundE0+ Sk1−−*)−−k−1E1(1)E1k2−→ E0+ P (2)Rate1= k1SE0, Rate−1= k−1E1, and Rate2= k2E1The rate equations associated with each reaction determines the rateof change of S, E0, E1, and P . Realizing this, we can write the follow-ing:dPdt= Rate2= k2E116/26 Thus the system of differential equations modelling the proces s is:dSdt= −k1SE0+ k−1E1dE0dt= −k1SE0+ k−1E1+ k2E1dE1dt= k1SE0− k−1E1− k2E1dPdt= k2E1The rate constants can be difficult or impossible to determine. For thepurpose of seeing the behavior of the system, we give them the thevalues k1= 10, k−1= 1, and k2= 5, with initial conditions S = 1.0and an E0= 0.08.17/26 0 2 4 6 800.20.40.60.81Substrate/ Enzyme Reaction ModelTimeConcentrationSubstrate, SUnoccupied Enzyme, E0Occupied Enzyme, E1Product, P18/26 Reducing the Four Equations to T woBy adding equations and doing some algebraic manipulation, we findthatdSdt= −k1SET+ (k−1+ k1S)E1dE1dt= k1SET− (k1S + k−1+ k2)E1Figure 3: A model of an enzyme19/26 The Quasi-Steady-State AssumptionAs long as ET S then we can assume that dE1/dt ≈ 0.dSdt= −k1SET+ (k−1+ k1S)E10 ≈ k1SET− (k1S + k−1+ k2)E120/26 The Quasi-Steady-State AssumptionAs long as ET S then we can assume that dE1/dt ≈ 0.dSdt= −k1SET+ (k−1+ k1S)E10 ≈ k1SET− (k1S + k−1+ k2)E1Solve dE1/dt for E1E1=k1SET(k−1+ k2+ k1S)21/26 The Quasi-Steady-State AssumptionAs long as ET S then we can assume that dE1/dt ≈ 0.dSdt= −k1SET+