Skip to main content
SpringerLink
Log in

Numerical Methods for Modeling Enzyme Kinetics

  • Protocol
  • First Online:
Enzyme Kinetics in Drug Metabolism

Part of the book series: Methods in Molecular Biology ((MIMB,volume 2342))

Abstract

Differential equations are used to describe time-dependent changes in enzyme kinetics and pharmacokinetics. Analytical and numerical methods can be used to solve differential equations. This chapter describes the use of numerical methods in solving differential equations and its applications in characterizing the complexities observed in enzyme kinetics. A discussion is included on the use of numerical methods to overcome limitations of explicit equations in the analysis of metabolism kinetics, reversible inhibition kinetics, and inactivation kinetics. The chapter describes the advantages of using numerical methods when Michaelis–Menten assumptions do not hold.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Protocol
USD 49.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 379.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Butcher JC, Wanner GJANM (1996) Runge-Kutta methods: some historical notes. Appl Numer Math 22(1-3):113–151

    Article  Google Scholar 

  2. Heun K (1900) Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen. Z Math Phys 45:23–38

    Google Scholar 

  3. Runge C (1895) Über die numerische Auflösung von Differentialgleichungen. Mathematische Annalen 46(2):167–178

    Article  Google Scholar 

  4. Kutta W (1901) Beitrag zur naherungsweisen integration totaler differentialgleichungen. Z Math Phys 46:435–453

    Google Scholar 

  5. Butcher JC (1996) A history of Runge-Kutta methods. Appl Numer Math 20(3):247–260

    Article  Google Scholar 

  6. Korzekwa K, Krishnamachary N, Shou M, Ogai A, Parise R, Rettie A, Gonzalez F, Tracy T (1998) Evaluation of atypical cytochrome P450 kinetics with two-substrate models: evidence that multiple substrates can simultaneously bind to cytochrome P450 active sites. Biochemistry 37(12):4137–4147

    Article  CAS  PubMed  Google Scholar 

  7. Shou M, Mei Q, Ettore JMW, Dai R, Baillie TA, Rushmore TH (1999) Sigmoidal kinetic model for two co-operative substrate-binding sites in a cytochrome P450 3A4 active site: an example of the metabolism of diazepam and its derivatives. Biochem J 340(3):845–853

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  8. Kandel SE, Lampe JN (2014) Role of protein–protein interactions in cytochrome P450-mediated drug metabolism and toxicity. Chem Res Toxicol 27(9):1474–1486

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Guengerich FP, Wilkey CJ, Phan TT (2019) Human cytochrome P450 enzymes bind drugs and other substrates mainly through conformational-selection modes. J Biol Chem 294(28):10928–10941

    Article  PubMed  PubMed Central  Google Scholar 

  10. Galetin A, Clarke SE, Houston JB (2003) Multisite kinetic analysis of interactions between prototypical CYP3A4 subgroup substrates: midazolam, testosterone, and nifedipine. Drug Metab Dispos 31(9):1108–1116

    Article  CAS  PubMed  Google Scholar 

  11. Yadav J, Korzekwa K, Nagar S (2019) Impact of lipid partitioning on the design, analysis, and interpretation of microsomal time-dependent inactivation. Drug Metab Dispos 47(7):732–742

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Nagar S, Korzekwa K (2012) Commentary: nonspecific protein binding versus membrane partitioning: it is not just semantics. Drug Metab Dispos 40(9):1649–1652

    Article  CAS  PubMed  Google Scholar 

  13. Margolis JM, Obach RS (2003) Impact of nonspecific binding to microsomes and phospholipid on the inhibition of cytochrome P4502D6: implications for relating in vitro inhibition data to in vivo drug interactions. Drug Metab Dispos 31(5):606–611

    Article  CAS  PubMed  Google Scholar 

  14. Brown S, Muhamad N, Pedley KC, Simcock DC (2014) The kinetics of enzyme mixtures. Mol Biol Res Commun 3(1):21

    PubMed  PubMed Central  Google Scholar 

  15. Rodgers JT, Davydova NY, Paragas EM, Jones JP, Davydov DR (2018) Kinetic mechanism of time-dependent inhibition of CYP2D6 by 3, 4-methylenedioxymethamphetamine (MDMA): functional heterogeneity of the enzyme and the reversibility of its inactivation. Biochem Pharmacol 156:86–98

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Davydov DR, Fernando H, Baas BJ, Sligar SG, Halpert JR (2005) Kinetics of dithionite-dependent reduction of cytochrome P450 3A4: heterogeneity of the enzyme caused by its oligomerization. Biochemistry 44(42):13902–13913

    Article  CAS  PubMed  Google Scholar 

  17. Johnson KA, Goody RS (2011) The original Michaelis constant: translation of the 1913 Michaelis–Menten paper. Biochemistry 50(39):8264–8269

    Article  CAS  PubMed  Google Scholar 

  18. Briggs GE, Haldane JBS (1925) A note on the kinetics of enzyme action. Biochem Js 19(2):338

    Article  CAS  Google Scholar 

  19. Seibert E, Tracy TS (2014) Fundamentals of enzyme kinetics. In: Enzyme kinetics in drug metabolism. Springer, pp 9–22

    Chapter  Google Scholar 

  20. Abbasi A, Paragas EM, Joswig-Jones CA, Rodgers JT, Jones JP (2019) Time course of aldehyde oxidase and why it is nonlinear. Drug Metab Dispos 47(5):473–483

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Yadav J, Paragas E, Korzekwa K, Nagar S (2020) Time-dependent enzyme inactivation: numerical analyses of in vitro data and prediction of drug-drug interactions. Pharmacol Ther 206:107449

    Article  CAS  PubMed  Google Scholar 

  22. Korzekwa K, Tweedie D, Argikar UA, Whitcher-Johnstone A, Bell L, Bickford S, Nagar S (2014) A numerical method for analysis of in vitro time-dependent inhibition data. Part 2. Application to experimental data. Drug Metab Dispos 42(9):1587–1595

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  23. Nagar S, Jones JP, Korzekwa K (2014) A numerical method for analysis of in vitro time-dependent inhibition data. Part 1. Theoretical considerations. Drug Metab Dispos 42(9):1575–1586

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  24. Tracy TS (2006) Atypical cytochrome P450 kinetics. Drugs R D 7(6):349–363

    Article  CAS  PubMed  Google Scholar 

  25. Hutzler JM, Tracy TS (2002) Atypical kinetic profiles in drug metabolism reactions. Drug Metab Dispos 30(4):355–362

    Article  CAS  PubMed  Google Scholar 

  26. Atkins WM (2005) Non-Michaelis-Menten kinetics in cytochrome P450-catalyzed reactions. Annu Rev Pharmacol Toxicol 45:291–310

    Article  CAS  PubMed  Google Scholar 

  27. Denisov IG, Sligar SG (2012) A novel type of allosteric regulation: functional cooperativity in monomeric proteins. Arch Biochem Biophys 519(2):91–102

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  28. Leow JWH, Chan ECY (2019) Atypical Michaelis-Menten kinetics in cytochrome P450 enzymes: A focus on substrate inhibition. Biochem Pharmacol 169:113615

    Article  CAS  PubMed  Google Scholar 

  29. Ekroos M, Sjögren T (2006) Structural basis for ligand promiscuity in cytochrome P450 3A4. Proc Natl Acad Sci U S A 103(37):13682–13687

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  30. Foti RS, Honaker M, Nath A, Pearson JT, Buttrick B, Isoherranen N, Atkins WM (2011) Catalytic versus inhibitory promiscuity in cytochrome P450s: implications for evolution of new function. Biochemistry 50(13):2387–2393

    Article  CAS  PubMed  Google Scholar 

  31. Davydov DR, Halpert JR (2008) Allosteric P450 mechanisms: multiple binding sites, multiple conformers or both? Expert Opin Drug Metab Toxicol 4(12):1523–1535

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  32. Kapelyukh Y, Paine MJ, Maréchal J-D, Sutcliffe MJ, Wolf CR, Roberts GC (2008) Multiple substrate binding by cytochrome P450 3A4: estimation of the number of bound substrate molecules. Drug Metab Disposs 36(10):2136–2144

    Article  CAS  Google Scholar 

  33. Korzekwa K (2014) Enzyme kinetics of oxidative metabolism: cytochromes P450. In: Enzyme kinetics in drug metabolism. Springer, pp 149–166

    Chapter  Google Scholar 

  34. Fisher HF, Gates RE, Cross DG (1970) A ligand exclusion theory of allosteric effects. Nature 228(5268):247–249

    Article  CAS  PubMed  Google Scholar 

  35. Pearson JT, Hill JJ, Swank J, Isoherranen N, Kunze KL, Atkins WM (2006) Surface plasmon resonance analysis of antifungal azoles binding to CYP3A4 with kinetic resolution of multiple binding orientations. Biochemistry 45(20):6341–6353

    Article  CAS  PubMed  Google Scholar 

  36. Denisov IG, Frank DJ, Sligar SG (2009) Cooperative properties of cytochromes P450. Pharmacol Ther 124(2):151–167

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  37. Davydov DR, Davydova NY, Sineva EV, Kufareva I, Halpert JR (2013) Pivotal role of P450–P450 interactions in CYP3A4 allostery: the case of α-naphthoflavone. Biochem J 453(2):219–230

    Article  CAS  PubMed  Google Scholar 

  38. Davydov DR, Davydova NY, Sineva EV, Halpert JR (2015) Interactions among cytochromes P450 in microsomal membranes oligomerization of cytochromes P450 3A4, 3A5, and 2E1 and its functional consequences. J Biol Chem 290(6):3850–3864

    Article  CAS  PubMed  Google Scholar 

  39. Stone AN, Mackenzie PI, Galetin A, Houston JB, Miners JO (2003) Isoform selectivity and kinetics of morphine 3-and 6-glucuronidation by human UDP-glucuronosyltransferases: evidence for atypical glucuronidation kinetics by UGT2B7. Drug Metab Dispos 31(9):1086–1089

    Article  CAS  PubMed  Google Scholar 

  40. Peng Y, Zhang X, Zhu Y, Wu H, Gu S, Chang Q, Zhou Y, Wang G, Sun J (2018) Atypical kinetics and albumin effect of glucuronidation of 5-n-butyl-4-{4-[2-(1H-tetrazole-5-yl)-1H-pyrrol-1-yl] phenylmethyl}-2, 4-dihydro-2-(2, 6-dichlorophenyl)-3H-1, 2, 4-triazol-3-one, a novel nonpeptide angiotensin type 1 receptor antagonist, in liver microsomes and UDP-glucuronosyl-transferase. Molecules 23(3):688

    Article  PubMed Central  CAS  Google Scholar 

  41. Basu NK, Kole L, Kubota S, Owens IS (2004) Human UDP-glucuronosyltransferases show atypical metabolism of mycophenolic acid and inhibition by curcumin. Drug Metab Dispos 32(7):768–773

    Article  CAS  PubMed  Google Scholar 

  42. James MO (2014) Enzyme kinetics of conjugating enzymes: PAPS sulfotransferase. In: Enzyme kinetics in drug metabolism. Springer, pp 187–201

    Chapter  Google Scholar 

  43. Liu X, Huang J, Sun Y, Zhan K, Zhang Z, Hong M (2013) Identification of multiple binding sites for substrate transport in bovine organic anion transporting polypeptide 1a2. Drug Metab Dispos 41(3):602–607

    Article  CAS  PubMed  Google Scholar 

  44. Shirasaka Y, Mori T, Shichiri M, Nakanishi T, Tamai I (2011) Functional pleiotropy of organic anion transporting polypeptide OATP2B1 due to multiple binding sites. Drug Metab Pharmacokinet 27(3):360–364

    Article  PubMed  CAS  Google Scholar 

  45. Grimm SW, Einolf HJ, Hall SD, He K, Lim H-K, Ling K-HJ LC, Nomeir AA, Seibert E, Skordos KW (2009) The conduct of in vitro studies to address time-dependent inhibition of drug-metabolizing enzymes: a perspective of the pharmaceutical research and manufacturers of America. Drug Metab Dispos 37(7):1355–1370

    Article  CAS  PubMed  Google Scholar 

  46. Escribano J, Tudela J, Garcia-Carmona F, Garcia-Canovas F (1989) A kinetic study of the suicide inactivation of an enzyme measured through coupling reactions. Application to the suicide inactivation of tyrosinase. Biochem J 262(2):597–603

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  47. Waley SG (1980) Kinetics of suicide substrates. Biochem J 185(3):771–773

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  48. Waley SG (1985) Kinetics of suicide substrates. Practical procedures for determining parameters. Biochem J 227(3):843–849

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  49. McConn DJ, Lin YS, Allen K, Kunze KL, Thummel KE (2004) Differences in the inhibition of cytochromes P450 3A4 and 3A5 by metabolite-inhibitor complex-forming drugs. Drug Metab Dispos 32(10):1083–1091

    Article  CAS  PubMed  Google Scholar 

  50. Zhang X, Quinney SK, Gorski JC, Jones DR, Hall SD (2009) Semiphysiologically based pharmacokinetic models for the inhibition of midazolam clearance by diltiazem and its major metabolite. Drug Metab Dispos 37(8):1587–1597

    Article  CAS  PubMed  Google Scholar 

  51. Eliasson E, Mkrtchian S, Halpert JR, Ingelman-Sundberg M (1994) Substrate-regulated, cAMP-dependent phosphorylation, denaturation, and degradation of glucocorticoid-inducible rat liver cytochrome P450 3A1. J Biol Chem 269(28):18378–18383

    Article  CAS  PubMed  Google Scholar 

  52. Eliasson E, Johansson I, Ingelman-Sundberg M (1988) Ligand-dependent maintenance of ethanol-inducible cytochrome P-450 in primary rat hepatocyte cell cultures. Biochem Biophys Res Commun 150(1):436–443

    Article  CAS  PubMed  Google Scholar 

  53. Eliasson E, Johansson I, Ingelman-Sundberg M (1990) Substrate-, hormone-, and cAMP-regulated cytochrome P450 degradation. Proc Natl Acad Sci U S A 87(8):3225–3229

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  54. Eliasson E, Mkrtchian S, Ingelman-Sundberg M (1992) Hormone-and substrate-regulated intracellular degradation of cytochrome P450 (2E1) involving MgATP-activated rapid proteolysis in the endoplasmic reticulum membranes. J Biol Chem 267(22):15765–15769

    Article  CAS  PubMed  Google Scholar 

  55. Johansson I, Eliasson E, Ingelman-Sundberg M (1991) Hormone controlled phosphorylation and degradation of CYP2B1 and CYP2E1 in isolated rat hepatocytes. Biochem Biophys Res Commun 174(1):37–42

    Article  CAS  PubMed  Google Scholar 

  56. Yang J, Atkins WM, Isoherranen N, Paine MF, Thummel KE (2012) Evidence of CYP3A allosterism in vivo: analysis of interaction between fluconazole and midazolam. Clin Pharmacol Ther 91(3):442–449

    Article  CAS  PubMed  Google Scholar 

  57. Houston JB, Galetin A (2005) Modelling atypical CYP3A4 kinetics: principles and pragmatism. Arch Biochem Biophys 433(2):351–360

    Article  CAS  PubMed  Google Scholar 

  58. Kenworthy KE, Clarke SE, Andrews J, Houston JB (2001) Multisite kinetic models for CYP3A4: simultaneous activation and inhibition of diazepam and testosterone metabolism. Drug Metab Dispos 29(12):1644–1651

    CAS  PubMed  Google Scholar 

  59. Houston JB, Kenworthy KE (2000) In vitro-in vivo scaling of CYP kinetic data not consistent with the classical Michaelis-Menten model. Drug Metab Dispos 28(3):246–254

    CAS  PubMed  Google Scholar 

  60. Barnaba C, Yadav J, Nagar S, Korzekwa K, Jones JP (2016) Mechanism-based inhibition of CYP3A4 by podophyllotoxin: aging of an intermediate is important for in vitro/in vivo correlations. Mol Pharm 13(8):2833–2843

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  61. Fersht A, Fersht UA, W. H. Freeman Company (1999) Measurements and magnitude of individual rate constants. In: Structure and mechanism in protein science: a guide to enzyme catalysis and protein folding. W. H. Freeman, New York, pp 132–168

    Google Scholar 

  62. Antosiewicz J, McCammon JA (1995) Electrostatic and hydrodynamic orientational steering effects in enzyme-substrate association. Biophys J 69(1):57–65

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  63. Wade RC, Gabdoulline RR, Lüdemann SK, Lounnas V (1998) Electrostatic steering and ionic tethering in enzyme–ligand binding: Insights from simulations. Proc Natl Acad Sci U S A 95(11):5942–5949

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  64. Kingsley LJ, Lill MA (2015) Substrate tunnels in enzymes: structure–function relationships and computational methodology. Proteins 83(4):599–611

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  65. Isin EM, Guengerich FP (2006) Kinetics and thermodynamics of ligand binding by cytochrome P450 3A4. J Biol Chem 281(14):9127–9136

    Article  CAS  PubMed  Google Scholar 

  66. Sevrioukova IF, Poulos TL (2015) Current approaches for investigating and predicting cytochrome P450 3A4-ligand interactions. Adv Exp Med Biol 851:83–105

    Article  CAS  PubMed  PubMed Central  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pharmacokinetics, Pharmacodynamics and Drug Metabolism, Merck Research Laboratories, Boston, MA, USA

    Jaydeep Yadav

  2. Department of Pharmaceutical Sciences, Temple University School of Pharmacy, Philadelphia, PA, USA

    Ken Korzekwa & Swati Nagar

Authors
  1. Jaydeep Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ken Korzekwa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Swati Nagar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jaydeep Yadav .

Editor information

Editors and Affiliations

  1. Department of Pharmaceutical Sciences, Temple University School of Pharmacy, Philadelphia, PA, USA

    Swati Nagar

  2. Translational Medicine, Novartis Institutes for Biomedical Research, Inc., Cambridge, MA, USA

    Upendra A. Argikar

  3. Independent, Harleysville, PA, USA

    Donald Tweedie

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Science+Business Media, LLC, part of Springer Nature

About this protocol

Check for updates. Verify currency and authenticity via CrossMark

Cite this protocol

Yadav, J., Korzekwa, K., Nagar, S. (2021). Numerical Methods for Modeling Enzyme Kinetics. In: Nagar, S., Argikar, U.A., Tweedie, D. (eds) Enzyme Kinetics in Drug Metabolism. Methods in Molecular Biology, vol 2342. Humana, New York, NY. https://doi.org/10.1007/978-1-0716-1554-6_6

Download citation

  • DOI: https://doi.org/10.1007/978-1-0716-1554-6_6

  • Published:

  • Publisher Name: Humana, New York, NY

  • Print ISBN: 978-1-0716-1553-9

  • Online ISBN: 978-1-0716-1554-6

  • eBook Packages: Springer Protocols

Publish with us

Policies and ethics

Search

Navigation