Algorithmic criteria for the validity of quasi-steady state and partial equilibrium models: the Michaelis-Menten reaction mechanism.

We present "on the fly" algorithmic criteria for the accuracy and stability (non-stiffness) of reduced models constructed with the quasi-steady state and partial equilibrium approximations. The criteria comprise those introduced in Goussis (Combust Theor Model 16:869-926, 2012) that addressed the case where each fast time scale is due to one reaction and a new one that addresses the case where a fast time scale is due to more than one reactions. The development of these criteria is based on the ability to approximate accurately the fast and slow subspaces of the tangent space. Their validity is assessed on the basis of the Michaelis-Menten reaction mechanism, for which extensive literature is available regarding the validity of the existing various reduced models. The criteria predict correctly the regions in both the parameter and phase spaces where each of these models is valid. The findings are supported by numerical computations at indicative points in the parameter space. Due to their algorithmic character, these criteria can be readily employed for the reduction of large and complex mathematical models.

Algorithmic multiscale analysis for the FcRn mediated regulation of antibody PK in human.

A demonstration is provided on how algorithmic asymptotic analysis of multi-scale pharmacokinetics (PK) systems can provide (1) system level understanding and (2) predictions on the response of the model when parameters vary. Being algorithmic, this type of analysis is not hindered by the size or complexity of the model and requires no input from the investigator. The algorithm identifies the constraints that are generated by the fast part of the model and the components of the slow part of the model that drive the system within these constraints. The demonstration is based on a typical monoclonal antibody PK model. It is shown that the findings produced by the traditional methodologies, which require significant input by the investigator, can be produced algorithmically and more accurately. Moreover, additional insights are provided by the algorithm, which cannot be obtained by the traditional methodologies; notably, the dual influence of certain reactions depending on whether their fast or slow component dominates. The analysis reveals that the importance of physiological processes in determining the systemic exposure of monoclonal antibodies (mAb) varies with time. The analysis also confirms that the rate of mAb uptake by the cells, the binding affinity of mAb to neonatal Fc receptor (FcRn), and the intracellular degradation rate of mAb are the most sensitive parameters in determining systemic exposure of mAbs. The algorithmic framework for analysis introduced and the resulting novel insights can be used to engineer antibodies with desired PK properties.

Computational singular perturbation analysis of brain lactate metabolism.

Lactate in the brain is considered an important fuel and signalling molecule for neuronal activity, especially during neuronal activation. Whether lactate is shuttled from astrocytes to neurons or from neurons to astrocytes leads to the contradictory Astrocyte to Neuron Lactate Shuttle (ANLS) or Neuron to Astrocyte Lactate Shuttle (NALS) hypotheses, both of which are supported by extensive, but indirect, experimental evidence. This work explores the conditions favouring development of ANLS or NALS phenomenon on the basis of a model that can simulate both by employing the two parameter sets proposed by Simpson et al. (J Cereb. Blood Flow Metab., 27:1766, 2007) and Mangia et al. (J of Neurochemistry, 109:55, 2009). As most mathematical models governing brain metabolism processes, this model is multi-scale in character due to the wide range of time scales characterizing its dynamics. Therefore, we utilize the Computational Singular Perturbation (CSP) algorithm, which has been used extensively in multi-scale systems of reactive flows and biological systems, to identify components of the system that (i) generate the characteristic time scale and the fast/slow dynamics, (ii) participate to the expressions that approximate the surfaces of equilibria that develop in phase space and (iii) control the evolution of the process within the established surfaces of equilibria. It is shown that a decisive factor on whether the ANLS or NALS configuration will develop during neuronal activation is whether the lactate transport between astrocytes and interstitium contributes to the fast dynamics or not. When it does, lactate is mainly generated in astrocytes and the ANLS hypothesis is realised, while when it doesn't, lactate is mainly generated in neurons and the NALS hypothesis is realised. This scenario was tested in exercise conditions.

A new Michaelis-Menten equation valid everywhere multi-scale dynamics prevails.

The Michaelis-Menten reaction scheme is among the most influential models in the field of biochemistry, since it led to a very popular expression for the rate of an enzyme-catalysed reaction. After the realisation that this expression is valid in a limited region of the parameter space, two additional expressions were later introduced. The range of validity of these three expressions has been studied thoroughly, since the significance of a reliable rate is not based only on the accuracy of its predictive abilities but also on the physical insight that is acquired in the process of its construction. Here a new expression for the rate is introduced that is valid in practically the full parameter space and reduces to the expressions in the literature, when considering appropriate limits. The new expression is produced by employing algorithmic tools for asymptotic analysis, so that its construction is not hindered by the dimensional form and the complexity of the full model or by a wide parameter range of interest. These tools can be employed for the derivation of enzyme rate expressions of much more complex kinetics mechanisms.

Asymptotic analysis of a TMDD model: when a reaction contributes to the destruction of its product.

The multi-scale dynamics of a two-compartment with first order absorption Target-Mediated Drug Disposition (TMDD) pharmacokinetics model is analysed, using the Computational Singular Perturbation (CSP) algorithm. It is shown that the process evolves along two Slow Invariant Manifolds (SIMs), on which the most intense components of the model are equilibrated, so that the less intensive are the driving ones. The CSP tools allow for the identification of the components of the TMDD model that (i) constrain the evolution of the process on the SIMs, (ii) drive the system along the SIMs and (iii) generate the fast time scales. Among others, such diagnostics identify (i) the factors that determine the start and the duration of the period in which the ligand-receptor complex acts and (ii) the processes that determine its degradation rate. The counterintuitive influence of the process that transfers the ligand from the tissue to the main compartment, as it is manifested during the final stage of the process, is studied in detail.

Asymptotic Analysis of a Target-Mediated Drug Disposition Model: Algorithmic and Traditional Approaches.

A detailed analysis is reported on a multiscale pharmacokinetic model, simulating the interaction of a drug with its target, the binding of the compounds and the outcome of their interaction. The analysis is based on the algorithmic computational singular perturbation (CSP) methodology. Among others, the analysis concludes that the partial equilibrium approximation and the quasi-steady-state approximation (PEA and QSSA) are valid in two distinct stages in the evolution of the process. Similar conclusions are reached from the algorithmic criteria for the validity of the QSSA and PEA. The reactions in the pharmacokinetic model that (i) generate the fast time scales, (ii) generate the constraints in which the system evolves and (iii) drive the system at various phases are identified, with the use of algorithmic CSP tools. These identifications are very important for the improvement of the model and for the identification of ways to control the evolution of the process. Regarding the qualitative understanding of the process, the present analysis systematises the findings in the literature and provides some new insights.

Glycolysis in Saccharomyces cerevisiae: algorithmic exploration of robustness and origin of oscillations.

The glycolysis pathway in saccharomyces cerevisiae is considered, modeled by a dynamical system possessing a normally hyperbolic, exponentially attractive invariant manifold, where it exhibits limit cycle behavior. The fast dissipative action simplifies considerably the exploration of the system's robustness, since its dynamical properties are mainly determined by the slow dynamics characterizing the motion along the limit cycle on the slow manifold. This manifold expresses a number of equilibrations among components of the cellular mechanism that have a non-negligible projection in the fast subspace, while the motion along the slow manifold is due to components that have a non-negligible projection in the slow subspace. The characteristic time scale of the limit cycle can be directly altered by perturbing components whose projection in the slow subspace contributes to its generation. The same effect can be obtained indirectly by perturbing components whose projection in the fast subspace participates in the generated equilibrations, since the slow manifold will thus be displaced and the slow dynamics must adjust. Along the limit cycle, the characteristic time scale exhibits successively a dissipative and an explosive nature (leading towards or away from a fixed point, respectively). Depending on their individual contribution to the dissipative or explosive nature of the characteristic time scale, the components of the cellular mechanism can be classified as either dissipative or explosive ones. Since dissipative/explosive components tend to diminish/intensify the oscillatory behavior, one would expect that strengthening a dissipative/explosive component will diminish/intensify the oscillations. However, it is shown that strengthening dissipative (explosive) components might lead the system to amplified oscillations (fixed point). By employing the Computational Singular Perturbation method, it is demonstrated that such a behavior is due to the constraints imposed by the slow manifold.