Applications of Monte Carlo Simulation in Modelling of Biochemical Processes

The biochemical models describing complex and dynamic metabolic systems are typically multi-parametric and non-linear, thus the identification of their parameters requires non-linear regression analysis of the experimental data. The stochastic nature of the experimental samples poses the necessity to estimate not only the values fitting best to the model, but also the distribution of the parameters, and to test statistical hypotheses about the values of these parameters. In such situations the application of analytical models for parameter distributions is totally inappropriate because their assumptions are not applicable for intrinsically non-linear regressions. That is why, Monte Carlo simulations are a powerful tool to model biochemical processes. The classification of Monte Carlo approaches is not unified, so here we comply with the interpretation given in (Press et al., 1992), where the general Monte Carlo approach is to construct parallel virtual worlds, in which the experimental estimates will play the role of true parameters, if the way in which the true parameters generate a sample is known. Bootstrap is a modification of Monte Carlo, which uses very few premises imposed on the data, and does not need to know the mechanism by which the true parameters generate experimental samples. Instead, resampling with replacement from the experimental sample is used to construct synthetic samples. As far as confidence intervals (CI) are concerned, literature offers multiple types, but each of them belongs to one of the two main groups: root (Politis, 1998) and percentile intervals (Efron & Tibshirani, 1993). The difference in the philosophy of those two CI types is substantial for the biochemical interpretation of results. The difference here is explained with the difference between classical statistics (where the parameters are fixed unknown quantities) and Bayesian statistics (where the parameters are random variables with unknown distributions), and also with the philosophical differences between objectivity and subjectivity of scientific research. The main conclusion is that root confidence intervals are confidence intervals of the investigated parameters, whereas percentile confidence intervals refer to the estimates of the investigated parameters. Our first application of Monte Carlo and Bootstrap simulation procedures is with a simulation platform for training students in medical biochemistry (Tenekedjiev & Kolev, 2002). In this system, students search for estimates and confidence intervals of parameters of a given biochemical system for different enzyme-substrate pairs. The platform applies Monte Carlo simulation on two stages. Initially, a Monte Carlo procedure is applied to emulate a biochemical experimental measurement setting along with given enzyme kinetic reactions as realistically as possible. The system is in position to simulate continuous enzyme assay (used for adjustment of the “experimental” conditions) and end-point enzyme assay “measurements” (suitable for parameter identification). We use an ordinary differential equation (ODE) as basis of the generation of pseudo-experimental data. The pseudo-real nature of the generated data is ensured by the random incorporation of three types of errors for each repetition of the experiments. The Briggs-Haldane steady-state model is fitted to the pseudo-measured and end-point assay data obtained by the system. The kinetic parameters can be calculated by χ-minimization. The task is simplified by the existence of a good initial guess from a linearized Lineweaver-Burk model. The two-dimensional root confidence regions of the parameters can be calculated by either Monte Carlo or Bootstrap, following similar procedures. The best point estimate is identified using trimmed mean over the flipped parameters taking only the values from the identified root confidence region. In the majority of biochemical reactions, parameters are unknown in very wide intervals, and may have different numerical order. Finding the root confidence regions (intervals) includes parameter flipping, which often generates results with an incorrect sign. That is why, in a second example (Tanka-Salamon et al., 2008) we propose a multiplicative modification for the estimation of root confidence regions and the best estimate of the parameters, which ensures that all estimates will have a physical meaning. The main assumption is that the ratio between the true parameter value and the optimal parameter value derived from the true data sample has the same distribution as the ratio between the optimal parameter value derived from the true data sample, and the optimal synthetic parameter value derived from the synthetic data sample. The assumption is equivalent to performing classical Bootstrap over the logarithms of the estimated parameters. This method is applied in a real experimental set-up for the estimation of root confidence regions of kinetic constants and root best estimates in amidolytic activity of plasmin under the influence of three fatty acids. By doing so, the inhibition effect of the three fatty acids can be proven and quantified. The measured data have the form of continuous reaction progress curves with several replicas. The product concentrations are predicted by three different models with increasing complexity. We model the instability of the inhibited enzyme and represent the resulting continuous assay model with concomitant inactivation of the enzyme as a system of two stiff ODE. From there, we derive the closed form of the progress curve. The four-dimensional root confidence regions are acquired by Monte Carlo simulation in every data point in each of the progress curves using an analytical model of the measured standard deviation, similarly to the first example.