PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26DerivingKinetic Parameters &Rate Equations forMulti-Substrate SystemsDetermination of Kinetic Parameters in Multi-Substrate Systems* With one substrate variable and the other(s) fixed:-- generate kinetics plots that can be analyzed using any of the graphical orcomputational tools that we discussed with single substrate enzymes.* BUT, V & Km parameters derived are apparent values only. What do they mean?-- change in conc. of fixed substrate(s) will change apparent V & Km for variable one.* To get true values, we have to extend our definitions of V & Km a bit from those we usedfor single substrate systems. FOR MULTIPLE SUBSTRATES:(1) Maximal velocity, V, is defined as the reaction velocity which occurs when all substrates are at saturation levels. (2) Each substrate will have its own Michaelis constant which is defined as the concentration of that substrate which gives a velocity of half the maximal velocity when all other substrates are present at saturation levels. * Analysis methods are also extensions of methods used for single substrate systems.Use of Lineweaver-Burk Plot to Estimate “True” Multi-Substrate Kinetic ParametersPrimary LB plot for sequential enzymeA + B <=> P + QA is variable, B is fixed1/Vapp for differentfixed [B]’s-1/KA, app for differentfixed [B]’sSecondary Plot-* Likewise, for primary plot with B variable and A fixed, a secondary plot of 1/Vapp vs 1/[A]:(1) gives -1/KA,true from the horizontal intercept;(2) also gives 1/Vapp from the vertical intercept, which should equal the value above.,trueDeriving Rate Equations:King-Altman MethodOrdered Sequential BiBi Mechanism:All rate constants must be first-order;e.g. the second-order rate constant k+1must be represented by a pseudo-first-order constant by including theconcentration of A: k+1aNow find every pattern that: (1) consists only of lines from the master pattern;(2) connects every enzyme species; and(3) contains no closed loops.YES:NO:A master pattern is drawn representing the skeleton of the scheme; here a square:Next, for each enzyme species, draw arrows oneach pattern, leading to the species considered,regardless of starting point. Thus for E: Then a sum of products of rate constants is written, such that each product contains therate constants corresponding to the arrows. So, from the patterns leading to E, the sumof products is:k-1k-2k-3p + k-1k-2k+4 + k-1k+3k+4 + k+2k+3k+4bThis sum is then the numerator of an expression representing the fraction of the total enzymeconcentration e0 present as the species in question. So for all four species we have:The denominator  is the sum of all 4 numerators, i.e. the sum of all 16 products obtained from the pattern.The rate of the reaction is then the sum of the rates of steps that generate one particularproduct, minus the sum of the rates of steps that consume the same product.In this example, there is only one step that generates P: (EAB + EAQ) --> EQ + P,and only one step that consumes P: EQ + P --> (EAB + EPQ), so we haveGENERAL RULE FOR NUMERATOR: -- Positive term is the product of total enzyme conc., all substrate concentrations for the forward rxn, and all rate constants for a complete cycle in the forward direction. -- Negative term is the product of total enzyme conc., all substrate concentrations for thereverse rxn, and all rate constants for a complete cycle in the reverse direction.“For most purposes it is more important to know the form of the steady-state rate equation than to know its detailed expression interms of rate constants.” -- A. Cornish-Bowden COEFFICIENT FORM (Ordered Sequential BiBi)Modifications to the King-Altman Method[E]/e0 = (k-1 + k+2)/(k-1 + k+2 + k+1s + k-2p)[ES]/e0 = (k+1s + k-2p)/(k-1 + k+2 + k+1s + k-2p)M-Mequationv = dp/dt = k+2[ES] - k-2[E]p = k+2e0(k+1s + k-2p)/(k-1 + k+2 + k+1s + k-2p)= k+2e0s/((k-1 + k+2)/k+1 + s)= Vs/(Km + s)p = 0 due toinitial velocitiesThis can greatly simplify the derivation for more complicatedmechanisms such as random sequential:As shown, this master pattern requires 12 patterns, but if the parallel paths betweenE and ES and between EX and EXS are added, the master pattern becomes a square, which requires only 4 patterns!Recall Coefficient Form of Rate Equation for Ordered Sequential BiBi13 coefficients defined interms of only 8 rate constants--------Must be inter-related!Cleland devised a system for defining these coefficients in terms of measurablekinetic parameters. Thereby the rate equation for this mechanism becomes:From King-Altman Coefficient Form, Can Write Rate Equations for Other Kinetic Mechanisms in Terms of Kinetic Parameters, TooPing-Pong BiBiRandom Sequential BiBi-- simpler because it assumes all steps except (EAB <=> EPQ) are at equilibrium.Max velocities forforward & reverse rxnsSubstrate Michaelisconstants forforward & reverse rxnsInhibition constants forforward & reverse rxnsOrdered Sequential BiBi Ping-Pong BiBiFor Ordered Sequential BiBi, Can Calculate Individual Rate Constants by RearrangingDefinitions of Kinetic Parameters:Doesn’t work for Ping-Pong!INITIAL VELOCITIES (p = q = 0) reduce steady-state rate equation for Ordered Sequential BiBi…… to this form:In limiting case where a & b areboth very large, v = V.When b is very large:KmA is the limiting Michaelis constant for A when B is saturating. SimilarlyKmB is limiting when A is saturating.When b is very small (but not zero):KiA is the limiting Michaelis constant for A when B approaches zero, and is alsothe true dissociation constant for EA.Ordered Sequential BiBi -- Initial Velocities:IN GENERAL: for a = variable and b = fixed (normal conc’s, b not very high or very low):Terms that do not contain a are constant!Same form as Michaelis-Mentenequation--Plots of Vapp or Vapp/Kapp vs. bgive rectangular hyperbolasAnalysis same as single substrate M-MPrimary Plot Using Hanes Plot:Increasing bSecondary PlotsPing-PongBiBiInitial velocities (p = q = 0):No constant term in denominator!a = variable, b = fixed (normal)Only one secondaryplot is necessary--b/Vapp vs. b (Hanes)like sequential.Increasing bProduct Inhibition-- Ex: Ordered Sequential BiBiIf only one product added, then:(1) Negative (reverse) term in numerator drops out.(2) All terms containing missing product drop out of