Mathematical Modeling of G Protein-Coupled Receptor Function: What Can We Learn from Empirical and Mechanistic Models?

Abstract

Empirical and mechanistic models differ in their approaches to the analysis of pharmacological effect. Whereas the parameters of the former are not physical constants those of the latter embody the nature, often complex, of biology. Empirical models are exclusively used for curve fitting, merely to characterize the shape of the E/[A] curves. Mechanistic models, on the contrary, enable the examination of mechanistic hypotheses by parameter simulation. Regretfully, the many parameters that mechanistic models may include can represent a great difficulty for curve fitting, representing, thus, a challenge for computational method development. In the present study some empirical and mechanistic models are shown and the connections, which may appear in a number of cases between them, are analyzed from the curves they yield. It may be concluded that systematic and careful curve shape analysis can be extremely useful for the understanding of receptor function, ligand classification and drug discovery, thus providing a common language for the communication between pharmacologists and medicinal chemists.

Appendix

1.1 Fitting Mathematical Models to Experimental Data

In a pharmacological assay, the input agonist concentration, [A], and its associated output effect, E, are recorded under particular experimental conditions. Experimental data are given by a list of varying effects, E_i, i = 1,…,N, obtained using different agonist concentrations, [A]_i, i = 1,…,N. A mathematical model of pharmacological effect is a mathematical equation E = f([A]), in which E represents the pharmacological effect, [A] is the concentration of the drug and f is a mathematical function containing a number of parameters.

The problem of fitting a mathematical model to a given experimental data list consists in finding the model parameters that produce the function E = f([A]) best approximating the experimental pairs ([A]_i , E_i), i = 1,…N in the sense that the model values are as close as possible to the experimental measurements. In other words, the differences |f([A]_i)-E_i| are as small as possible. In mathematical terms this is expressed in the following sum of squares which exclusively depends on the values of the model parameters:

$$ \mathrm{Energy}(\mathrm{Parameters})={{\sum\limits_{{\mathrm{i}=1}}^{\mathrm{N}} {\left| {\mathrm{f}\left( {{{{\left[ \mathrm{A} \right]}}_{\mathrm{i}}}} \right)-{{\mathrm{E}}_{\mathrm{i}}}} \right|^{2}}}} $$

(8.14)

which should be minimum for the optimal model parametric values (least squares fitting). We would like to note that such set of optimal parameters produce a curve that visually fits the profile of the scatter plot given by the experimental pairs ([A]_i, E_i), i = 1,…N.

In the case of pharmacological models, the dependency between the concentration and the effect given by E = f([A]) is non-linear with respect the model parameters and, thus, Eq. 8.14 is known as non-linear regression.

1.2 Traditional Gradient-Based Nonlinear Regression Algorithms Versus Stochastic Approaches

Given that the best set of parameter estimates must be a minimum of the energy given by Eq. 8.14, they can be found by means of optimization techniques. There are two main families of optimization strategies: local and global.

Local approaches search for the closest local minimum given an initial seed point. They mainly rely on the gradient of the function to be minimized and update the initial guess using the direction of this gradient because is the direction of maximum decrease of the energy (gradient descent). Gradient descent methods are efficient minimizers as far as the energy function can be differentiated with respect the parameters (which is always the case in a least squares fitting of pharmacological models) and they provide an optimal solution as far as the energy function is convex (i.e. has a unique local minimum). In the case of multiple minima (which is a typical case for mechanistic pharmacological models), the gradient descent strongly depends on the initial seed. This limitation can be partially overcome by running the gradient descent algorithm using different initial seed points. This solution has two main shortcomings. First it increases the computational time of the minimization process and, second, given that there is no a-priory knowledge on the number and distribution of the local minima, there is not a clear strategy for choosing an initial set of seeds guaranteeing that the optimal solution (global minimum) will be reached.

Global approaches are a way of searching for global optimal solutions (global minimum) without the need of a special definition of the initial set of seeds. For special search spaces and cost functions, there is a solid mathematical theory ensuring convergence of some global algorithms (simulated annealing (Ashyraliyev et al. 2009)). Although, there is no proof of convergence for the general case. Evolutionary Algorithms (EAs) have proven their ability for optimizing non-analytic multi-modal functions in a wide range of real life problems, such as parameter estimation (Ravikumar and Panigrahi 2010), pattern and text recognition (Rizki et al. 2002) and image processing (Cagnoni et al. 2008). EAs are a class of stochastic optimization methods that simulate the process of natural evolution. Unlike gradient descent methods that evolve a single initial value each time, EAs maintain a population of possible solutions that evolve according to rules of selection and other operators, such as recombination and mutation (Holland 1975).

Each individual in the population receives a measure of its fitness in the environment. Selection focuses attention on high fitness individuals, thus, exploiting the available fitness information. Selection determines which individuals are chosen to produce offsprings. Recombination and mutation perturb those individuals, providing general heuristics for exploration. Recombination produces new individuals in combining the information contained in the parents and offsprings are mutated by small perturbations with low probability. Finally, survival step decides who survives (among parents and offsprings) to form the new population. This process iterates until stop criterion occurs. Figure 8.8 outlines the scheme of a standard EA.

Fig. 8.8

Scheme of the mechanisms ruling an Evolutionary Algorithm (EA). EAs maintain a population of possible solutions and their fitness on the search space. In each iteration, selection determines which individuals are chosen to produce new solutions (exploitation). Recombination and mutation combine and perturb the individuals (exploration). Finally, the survival step decides who survives in the new population. This process iterates until the stop criterion occurs