Are improper kinetic models hampering drug development?
Bottom Line:
The over complication of inhibitory models stems from the common use of the inhibitory term (1 + [I]/Ki ), an equilibrium binding term that does not distinguish between inhibitor binding and inhibitory effect.These equations are overly complex and are seldom compared to each other, which has destroyed their usefulness resulting in the proliferation and regulatory acceptance of simpler models such as IC50s for drug characterization.The empirical kinetic method described here provides an improved way of probing disease mechanisms, expanding the investigation of possible therapeutic interventions.
Affiliation: Department of Chemistry, Carleton University , Ottawa, ON , Canada.
ABSTRACT
Reproducibility of biological data is a significant problem in research today. One potential contributor to this, which has received little attention, is the over complication of enzyme kinetic inhibition models. The over complication of inhibitory models stems from the common use of the inhibitory term (1 + [I]/Ki ), an equilibrium binding term that does not distinguish between inhibitor binding and inhibitory effect. Since its initial appearance in the literature, around a century ago, the perceived mechanistic methods used in its production have spurred countless inhibitory equations. These equations are overly complex and are seldom compared to each other, which has destroyed their usefulness resulting in the proliferation and regulatory acceptance of simpler models such as IC50s for drug characterization. However, empirical analysis of inhibitory data recognizing the clear distinctions between inhibitor binding and inhibitory effect can produce simple logical inhibition models. In contrast to the common divergent practice of generating new inhibitory models for every inhibitory situation that presents itself. The empirical approach to inhibition modeling presented here is broadly applicable allowing easy comparison and rational analysis of drug interactions. To demonstrate this, a simple kinetic model of DAPT, a compound that both activates and inhibits γ-secretase is examined using excel. The empirical kinetic method described here provides an improved way of probing disease mechanisms, expanding the investigation of possible therapeutic interventions. No MeSH data available. |
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Mentions: A comparison of the work involved in producing the conventional mechanistic model (Eq. (10)) (Svedruzic, Popovic & Sendula-Jengic, 2013) vs. the empirical approach described here also highlights the utility of this approach. Equation (5) and the expansions used to describe the interactions of DAPT with γ-secretase were produced, analyzed and fit to the data (Supplemental Information 1) in a few hours using the solver feature of Excel (Kemmer & Keller, 2010). The classical mechanistic approach used to describe the same kinetic process involved the generation of a complex reaction schematic with 14 enzyme, substrate and inhibitor interactions (Svedruzic, Popovic & Sendula-Jengic, 2013). This reaction scheme was used to define around 25 disassociation constants and three rate constants. The equation derived from this structure was constructed from a connection matrix which was then fed into Mathematica to produce a simplified version that ultimately only had five kinetic constants, (Eq. (10)). While Eq. (10) does contain fewer parameters than the 17 constants used in Eq. (5), a comparison of the predicted values produced using Eq. (10) with the observed experimental data suggests that Eq. (10) does not fit the data very well (Fig. 2). Refitting the parameters of Eq. (10) only marginally improved the model’s ability to fit the observed data (Fig. 4). (10)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}\usepackage{upgreek}\usepackage{mathrsfs}\setlength{\oddsidemargin}{-69pt}\begin{document}}{}\begin{eqnarray*} \displaystyle v=\frac{{V}_{1}[S]\left(\frac{1}{1+\frac{[S]}{{K}_{s i}}}\right)\left(\frac{1}{1+\frac{[I]}{{K}_{i i}}}\right)+{V}_{2}[S]\left(\frac{1}{1+\frac{[I]}{{K}_{i i}}}\right)\left(\frac{1}{1+\frac{[I]}{{K}_{i a}}}\right)}{[S]\left(\frac{1}{1+\frac{[S]}{{K}_{s i}}}\right)+{K}_{0.5 s}\left(\frac{1}{1+\frac{[I]}{{K}_{i i}}}\right)\left(\frac{1}{1+\frac{[I]}{{K}_{i a}}+\frac{[I]}{{K}_{s i}}}\right)}&&\displaystyle \end{eqnarray*}\end{document}v=V1S11+SKsi11+IKii+V2S11+IKii11+IKiaS11+SKsi+K0.5s11+IKii11+IKia+IKsi Boxplots and correlation plots were used to evaluate the fit associated with each model. Ideally a correlation plot of calculated vs. observed data should produce a slope of one (Fig. 4) and an R2 as close to one as possible, providing a visualization of the model’s fit. The residual boxplot provided a similar representation, where improvements in fitting of the models were evaluated based on decreases in spread and increased symmetric distribution around zero. The correlation plot produced by Eq. (10) (Fig. 4A) suggested that it was able to approximate the data fairly well. However, the boxplot produced a negative asymmetric distribution of the residuals (Fig. 5A). Refitting the kinetic parameters associated with Eq. (10) improved the slope of the correlation plot (Fig. 4B) and also improved the symmetric distribution of the residuals around zero (Fig. 5B). Equation (5) however improved both the slope and the R2 value for the correlation (Fig. 4C). A marked improvement in the symmetry and spread of the residual values was also observed (Fig. 5C). However, as previously mentioned Eq. (10) only relies on five kinetic parameters while Eq. (5), when expanded to describe DAPT interactions, contains 17. Thus, an increase in kinetic parameters might be viewed as over fitting of the data as models of greater complexity are known to produce improved fitting (Burnham & Anderson, 2002). To evaluate whether the improvement in fitting provided by Eq. (5) resulted from over fitting, Eqs. (5) and (10) were compared using the bayesian information criterion (BIC). BIC was developed to specifically penalize increasing complexity in model selection where a difference greater than ten is considered strong evidence against the higher value (Burnham & Anderson, 2002; Faraway, 2004). Not surprisingly when the BIC was used to compare Eq. (10) with the results produced by refitting the kinetic constants of Eq. (10) (Figs. 4A–4B), a significant improvement in the BIC value was observed (published values BIC = 597, refit kinetic parameters BIC 515). The decrease in BIC was attributed to a reduction in residuals without any increase in the complexity of Eq. (10). Examination of the BIC value produced by Eq. (5) (BIC = 448) also suggested a significant improvement over the fit achieved with Eq. (10) even though the number of parameters had increased by 12. |
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Affiliation: Department of Chemistry, Carleton University , Ottawa, ON , Canada.
No MeSH data available.