Time depending inhibition with atypical kinetics, what is one to do? Ken Korzekwa Temple University School of Pharmacy

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1 Time depending inhibition with atypical kinetics, what is one to do? Ken Korzekwa Temple University School of harmacy

2 Acknowledgements TUS Swati Nagar Jaydeep Yadav harma - Donald Tweedie - Andrea Whitcher-Johnstone - Leslie Bell - Shari Bickford - Upendra Argikar WSU Jeff Jones Jim Brozik Carlo Barnaba NIH Grants 5R01GM R01GM

3 Are Numerical Solutions from Differential Equations Today like Nonlinear Least-Squares in 1979? 3

4 Michaelis-Menten Kinetics k 3 E S k 1 S k 2 ES E'[t] = -k 1 E[t] S[t] + k 2 ES[t] + k 3 ES[t], ES'[t] = k 1 E[t] S[t] - (k 2 + k 3 ) ES[t], S'[t] = -k 1 E[t] S[t] + k 2 ES[t], '[t] = k 3 ES[t] v = V max S/(K m +S), where Vmax = k 3 Et Km = (k 2 +k 3 )/k 1 4

5 Fitting the Michaelis Menten Equation Assumptions: Steady state to Kinetic Data Initial Rates (low substrate consumption) Can be corrected using the logarithmic mean: (S 0 -S)/Ln(S 0 /S) or work directly from the differential equations. 5

6 Excess Substrate Consumption aram Sim Std Km kcat

7 Excess Substrate Consumption aram Sim Std Km kcat S corr

8 Michaelis-Menten Kinetics Numerical Solution k3 E S k1 S k2 ES For a hyperbolic saturation curve, we can solve for two parameters. Assume a value for k1 ( M -1 sec -1 ) Solve for k 2 (affinity) and k 3 (k cat ) K m = (k 2 +k 3 )/k 1 8

9 Numerical MM Fitting 9

10 Excess Substrate Consumption aram Sim Std Km kcat S corr Num Soln

11 Numerical Method for Enzyme Kinetics Advantages No assumptions steady state initial rates etc. Completely flexible Any number of parameters can be introduced. rovided that you have sufficient data Any change at any time can be incorporated into the model. Statistics are available. 11

12 Numerical Method for Enzyme Kinetics Disadvantages Nice equations are not available Must be derived if needed Equations often provide insight to internal relationships Results need to be evaluated carefully arameter errors Correlation must be considered (e.g. V max /K m ) 12

13 CY Two Binding Site Model Figure 2. Models for two substrates binding simultaneously to the CY active site, (A) Substrates are randomly oriented within the active site. (B) substrates bind to specific sites within the active site. Reference: K. Korzekwa, Cytochrome 450 Kinetics, in Enzyme Kinetics in Drug Metabolism: Fundamentals and Applications. Nagar, Argikar, Tweedie, Eds, Springer

14 Complex Enzyme Kinetics- ESS velocity A Hyperbolic K 1 m1 =10, K m2 =10, V m1 =10, V m2 =10 0 H velocity Biphasic K m1 =10, K m2 =1000, V m1 =10, V m2 = [substrate] [substrate] velocity D Sigmoidal K m1 =10, K m2 =10, V m1 =10, V m2 = [substrate] B velocity Substrate Inhibition K m1 =10, K m2 =10, V m1 =10, V m2 = [substrate] 14

15 ESI Model 15

16 Metabolism - Non-Michaelis-Menten Kinetics S ΜM I ΜM

17 Kinetic Rate Equations vs Numerical Solutions of Differential Equations Equations for many kinetic schemes have been derived. e.g. Segal,IH, Enzyme Kinetics: Behavior and analysis of rapid equilibrium and steady-state enzyme systems, Nonlinear least squares solutions to appropriate kinetic rate equations gives results identical to numerical solutions from differential equations. The numerical method can be used when standard equations are not available or when fewer assumptions are required. Time dependent inhibition is an example where rate equations are available only for MM kinetics. 17

18 Time-Dependent Inhibition When a drug binds irreversibily to a drug metabolizing enzyme Enzyme activity is lost over time Can be identified by preincubating the drug with the enzyme and measuring activity Time dependent inhibition is a major cause of drug-drug interactions Cytochrome 450 enzymes Aldehyde oxidase 18

19 TDI with MM Kinetics Standard kinetic scheme for TDI Typically analyzed by the replot method Measure rate of inactivation (k obs ) at different Inhibitor concentrations lot k obs vs inhibitor concentration Calculate k inact and K I from the hyperbola k4 k6 EI E* I k5 k1 E ES k2 S k3 19

20 Standard Method to Determine TDI 20

21 Standard Method to Determine TDI k inact K I 21

22 TDI K I vs Competitive K i E k 1 k 2 EI k 3 EI* k 5 k 4 I k 2 /k 1 = 10 mm red: rate limiting EI* blue: rate limiting E* E* k 2 /k 1 = 1 mm red: rapid equil Kinetics blue: slow k2 TDI competitive TDI competitive 22

23 TDI K I vs Competitive K i For MM kinetics, TDI K I = Competitive K i A competitive experiment will give the same Ki as a TDI experiment, provided enzyme loss is low in the competitive experiment. 50% loss of enzyme in a competitive experiment results in a 25% divergence between K I and K i The numerical method accounts for enzyme loss and provides the correct K I. When K i and K I are different, non-mm kinetics is likely. K i can be estimated from the zero preincubation time points. 23

24 Direct Simulation of TDI k6 EI E* I k4 k5 k1 E ES k2 S k3 The MM TDI model was simulated 500 times each with random errors of 2.5%, 5%, and 10%. The MM TDI model was fit to each datasets to obtain K I and k inact. 24

25 robability Distribution of K I and k inact Estimates Distribution is shown for 500 runs at each condition 10% error 5% error 2.5% error robability Density Blue: replot method Red: numeric method robability Density robability Density Ki ΜM Ki ΜM Ki ΜM K I robability Density robability Density robability Density Kinact min Kinact min Kinact min 1 Nagar, S., Jones, J.. & Korzekwa, K., Drug Metab Dispos 42, (2014). k inact 25

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27 A) MM EI k6 E* k4 I k5 E k1 ES k2 S k3 D) artial inactivation EI k6 E* k4 I k5 k2 k1 E E*S k7 k1 ES k2 S k3 TDI Kinetic Schemes k4 E B) EII EII I k7 k9 k8 k6 EI E* I k5 k1 ES k2 S k3 C) Quasi-irreversible k6 EI E* I k4 k5 k7 k1 E ES k2 S k3 E) Enzyme loss k7 k6 EI E* I k7 k4 k5 k1 E k2 S k3 k7 ES 27

28 TDI with EII kinetics K m1 =10 mm K m2 =1000 mm V m1 =0.025 min -1 V m2 =0.25 min -1 K m1 =10 mm K m2 =100 mm V m1 =0.025 min -1 V m2 = min -1 K m1 =20 mm K m2 =20 mm V m1 = min -1 V m2 =0.05 min -1 28

29 Simulations with MM and EII Kinetics Simulations were performed for MM, biphasic, inhibitor inhibition, and sigmoidal kinetics. 100 simulations were performed for each kinetic scheme with errors of 2.5%, 5%, and 10%. The correct model was identified (lowest AIC) by the numerical method 100% of the time for 5% error, and % of the time for 10% error The correct model could not be identified even at 2.5% error with the modified replot method. EII I k7 k9 k8 k6 EI E* k4 E I k5 EII Kinetics k1 ES k2 S k3 29

30 Quasi-Irreversible inhibition Numerical Simulation K I (k 5/ k 4 ) 10 mm k inact (k 6 ) min -1 k rev (k 7 ) min -1 Replot K I Replot Method 2.0 mm K inact min -1 30

31 artial Inactivation Numerical Simulation K I (k 5/ k 4 ) 10 mm k inact (k 6 ) min -1 k cat (k 3 ) 10 min -1 k partial (k 7 ) 4 min -1 Replot K I Replot Method 0.75 mm K inact min -1 31

32 Enzyme Loss Numerical Simulation K I (k 5/ k 4 ) 10 mm k inact (k 6 ) min -1 k loss (k 7 ) min -1 Replot K I Replot Method 10.1 mm K inact min -1 32

33 A) MM EI k6 E* k4 I k5 E k1 ES k2 S k3 D) artial inactivation EI k6 E* k4 I k5 k2 k1 E E*S k7 k1 ES k2 S k3 TDI Kinetic Schemes k4 E B) EII EII I k7 k9 k8 k6 EI E* I k5 k1 ES k2 S k3 C) Quasi-irreversible k6 EI E* I k4 k5 k7 k1 E ES k2 S k3 E) Enzyme loss k7 k6 EI E* I k7 k4 k5 k1 E k2 S k3 k7 ES 33

34 MDZ-TAO data Model: Quasi-irreversible Korzekwa, K., Tweedie, D., Argikar, U. A., Whitcher- Johnstone, A., et al., Drug Metab Dispos 42, (2014). 34

35 MDZ-TAO data Model: artial inactivation 35

36 arameter estimates Quasi artial Standard replot K I 0.93 (0.11) 2.1 (0.35) 0.3 (0.03) k inact 0.25 (0.02) 0.38 (0.04) 0.06 (0.002) AICc NA 36

37 Inhibition of CY2C8 by Gemfibrozil Glucuronide Quasi-irreversible k 6 EI E* k 4 k 5 I k 7 E k 1 ES k 2 k 3 S AICc = -302 artial inactivation k 6 EI E* k 4 k 5 I k 2 E*S k 1 k 7 E k 2 k 1 S ES AICc = -302 k 3 37

38 Inhibition of CY2B6 by EII-artial inactivation EII k7 k 8 I k 9 EI k 6 E* k I 4 k 5 E k 1 k 2 S k 2 k 1 E*S ES k 10 k 3 38

39 MM vs EII The data cannot be fit to a single Ki. The data Can be fit to an EII model or using the standard replot method. 39

40 Inhibition of CY2D6 with MDMA Data from Heydari et al., Drug Metab Dispos 32:1213, MM k 6 EI E* k 4 k 5 I E k 1 k 2 k 3 ES S Quasi-irreversible K I = 15.4 ± 3.5 k inact = 0.23 ± 0.03 AIC = -159 k 6 EI E* k 4 k 5 I k 7 E k 2 k 3 k 1 S ES K I = 27.6 ± 7.5 k inact = 0.54 ± 0.15 k rev = 0.09 ± 0.02 AIC =

41 Numerical Method for TDI Analysis General rocedure The solvent control data (no inhibitor) are used to evaluate if enzyme loss. An initial estimate of K I is obtained from the highest primary incubation time data. This value was used to calculate the initial estimate for inhibitor release. The competitive inhibition constant, Ki, can be estimated from the zero primary incubation time data. When K I K i an EII model should be tested. An estimate of k inact can be obtained from the observed half-life at the highest inhibitor concentration. An estimate of the rate of product formation from can be obtained at zero inhibitor concentration and zero primary incubation time. A plot of the log percent remaining activity versus primary incubation time should be inspected to determine if curvature is observed. 41

42 Summary For MM kinetics, much better estimates of KI can be obtained with the numerical method compared with the standard replot method. Even IC50 shift data can provide meaningful estimates of TDI kinetic parameters. The replot method can be modified to fit non-mm data, but normal experimental error precludes this approach. The numerical method consistently predicts the correct non-mm model at errors of 10% or less, whereas the replot method cannot identify the correct kinetic model at experimental errors of 2.5% or greater. EII formation, quasi-irreversible inhibition and partial inactivation can only be modeled with the numerical method. Numerical methods provides tools for exploration and incorporation of complex schemes into any kinetic analyses. 42

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