Substrate and Stereochemical Control of Peptidoglycan Cross-Linking by Transpeptidation by Escherichia coli PBP1B.

Penicillin binding proteins (PBPs) catalyzing transpeptidation reactions that stabilize the peptidoglycan component of the bacterial cell wall are the targets of ß-lactams, the most clinically successful antibiotics to date. However, PBP-transpeptidation enzymology has evaded detailed analysis, because of the historical unavailability of kinetically competent assays with physiologically relevant substrates and the previously unappreciated contribution of protein cofactors to PBP activity. By re-engineering peptidoglycan synthesis, we have constructed a continuous spectrophotometric assay for transpeptidation of native or near native peptidoglycan precursors and fragments by Escherichia coli PBP1B, allowing us to (a) identify recognition elements of transpeptidase substrates, (b) reveal a novel mechanism of stereochemical editing within peptidoglycan transpeptidation, (c) assess the impact of peptidoglycan substrates on ß-lactam targeting of transpeptidation, and (d) demonstrate that both substrates have to be bound before transpeptidation occurs. The results allow characterization of high molecular weight PBPs as enzymes and not merely the targets of ß-lactam acylation.

Structural identifiability for mathematical pharmacology: models of myelosuppression.

Structural identifiability is an often overlooked, but essential, prerequisite to the experiment design stage. The application of structural identifiability analysis to models of myelosuppression is used to demonstrate the importance of its considerations. It is shown that, under certain assumptions, these models are structurally identifiable and so drug and system specific parameters can truly be separated. Further it is shown via a meta-analysis of the literature that because of this the reported system parameter estimates for the "Friberg" or "Uppsala" model are consistent in the literature.

Modeling of free fatty acid dynamics: insulin and nicotinic acid resistance under acute and chronic treatments.

Nicotinic acid (NiAc) is a potent inhibitor of adipose tissue lipolysis. Acute administration results in a rapid reduction of plasma free fatty acid (FFA) concentrations. Sustained NiAc exposure is associated with tolerance development (drug resistance) and complete adaptation (FFA returning to pretreatment levels). We conducted a meta-analysis on a rich pre-clinical data set of the NiAc-FFA interaction to establish the acute and chronic exposure-response relations from a macro perspective. The data were analyzed using a nonlinear mixed-effects framework. We also developed a new turnover model that describes the adaptation seen in plasma FFA concentrations in lean Sprague-Dawley and obese Zucker rats following acute and chronic NiAc exposure. The adaptive mechanisms within the system were described using integral control systems and dynamic efficacies in the traditional [Formula: see text] model. Insulin was incorporated in parallel with NiAc as the main endogenous co-variate of FFA dynamics. The model captured profound insulin resistance and complete drug resistance in obese rats. The efficacy of NiAc as an inhibitor of FFA release went from 1 to approximately 0 during sustained exposure in obese rats. The potency of NiAc as an inhibitor of insulin and of FFA release was estimated to be 0.338 and 0.436 [Formula: see text], respectively, in obese rats. A range of dosing regimens was analyzed and predictions made for optimizing NiAc delivery to minimize FFA exposure. Given the exposure levels of the experiments, the importance of washout periods in-between NiAc infusions was illustrated. The washout periods should be [Formula: see text]2 h longer than the infusions in order to optimize 24 h lowering of FFA in rats. However, the predicted concentration-response relationships suggests that higher AUC reductions might be attained at lower NiAc exposures.

Input estimation for drug discovery using optimal control and Markov chain Monte Carlo approaches.

Input estimation is employed in cases where it is desirable to recover the form of an input function which cannot be directly observed and for which there is no model for the generating process. In pharmacokinetic and pharmacodynamic modelling, input estimation in linear systems (deconvolution) is well established, while the nonlinear case is largely unexplored. In this paper, a rigorous definition of the input-estimation problem is given, and the choices involved in terms of modelling assumptions and estimation algorithms are discussed. In particular, the paper covers Maximum a Posteriori estimates using techniques from optimal control theory, and full Bayesian estimation using Markov Chain Monte Carlo (MCMC) approaches. These techniques are implemented using the optimisation software CasADi, and applied to two example problems: one where the oral absorption rate and bioavailability of the drug eflornithine are estimated using pharmacokinetic data from rats, and one where energy intake is estimated from body-mass measurements of mice exposed to monoclonal antibodies targeting the fibroblast growth factor receptor (FGFR) 1c. The results from the analysis are used to highlight the strengths and weaknesses of the methods used when applied to sparsely sampled data. The presented methods for optimal control are fast and robust, and can be recommended for use in drug discovery. The MCMC-based methods can have long running times and require more expertise from the user. The rigorous definition together with the illustrative examples and suggestions for software serve as a highly promising starting point for application of input-estimation methods to problems in drug discovery.

In vitro PK/PD modeling of tyrosine kinase inhibitors in non-small cell lung cancer cell lines.

Tyrosine kinase inhibitors (TKIs) are routinely prescribed for the treatment of non-small cell lung cancer (NSCLC). As with all medications, patients can experience adverse events due to TKIs. Unfortunately, the relationship between many TKIs and the occurrence of certain adverse events remains unclear. There are limited in vivo studies which focus on TKIs and their effects on different regulation pathways. Many in vitro studies, however, that investigate the effects of TKIs observe additional changes, such as changes in gene activations or protein expressions. These studies could potentially help to gain greater understanding of the mechanisms for TKI induced adverse events. However, in order to utilize these pathways in a pharmacokinetic/pharmacodynamic (PK/PD) framework, an in vitro PK/PD model needs to be developed, in order to characterize the effects of TKIs in NSCLC cell lines. Through the use of ordinary differential equations, cell viability data and nonlinear mixed effects modeling, an in vitro TKI PK/PD model was developed with estimated PK and PD parameter values for the TKIs alectinib, crizotinib, erlotinib, and gefitinib. The relative standard errors for the population parameters are all less than 25%. The inclusion of random effects enabled the model to predict individual parameter values which provided a closer fit to the observed response. It is hoped that this model can be extended to include in vitro data of certain pathways that may potentially be linked with adverse events and provide a better understanding of TKI-induced adverse events.

Parameter Identification for a Model of Neonatal Fc Receptor-Mediated Recycling of Endogenous Immunoglobulin G in Humans.

Salvage of endogenous immunoglobulin G (IgG) by the neonatal Fc receptor (FcRn) is implicated in many clinical areas, including therapeutic monoclonal antibody kinetics, patient monitoring in IgG multiple myeloma, and antibody-mediated transplant rejection. There is a clear clinical need for a fully parameterized model of FcRn-mediated recycling of endogenous IgG to allow for predictive modeling, with the potential for optimizing therapeutic regimens for better patient outcomes. In this paper we study a mechanism-based model incorporating nonlinear FcRn-IgG binding kinetics. The aim of this study is to determine whether parameter values can be estimated using the limited in vivo human data, available in the literature, from studies of the kinetics of radiolabeled IgG in humans. We derive mathematical descriptions of the experimental observations-timecourse data and fractional catabolic rate (FCR) data-based on the underlying physiological model. Structural identifiability analyses are performed to determine which, if any, of the parameters are unique with respect to the observations. Structurally identifiable parameters are then estimated from the data. It is found that parameter values estimated from timecourse data are not robust, suggesting that the model complexity is not supported by the available data. Based upon the structural identifiability analyses, a new expression for the FCR is derived. This expression is fitted to the FCR data to estimate unknown parameter values. Using these parameter estimates, the plasma IgG response is simulated under clinical conditions. Finally a suggestion is made for a reduced-order model based upon the newly derived expression for the FCR. The reduced-order model is used to predict the plasma IgG response, which is compared with the original four-compartment model, showing good agreement. This paper shows how techniques for compartmental model analysis-structural identifiability analysis, linearization, and reparameterization-can be used to ensure robust parameter identification.

Three novel approaches to structural identifiability analysis in mixed-effects models.

BACKGROUND AND OBJECTIVE: Structural identifiability is a concept that considers whether the structure of a model together with a set of input-output relations uniquely determines the model parameters. In the mathematical modelling of biological systems, structural identifiability is an important concept since biological interpretations are typically made from the parameter estimates. For a system defined by ordinary differential equations, several methods have been developed to analyse whether the model is structurally identifiable or otherwise. Another well-used modelling framework, which is particularly useful when the experimental data are sparsely sampled and the population variance is of interest, is mixed-effects modelling. However, established identifiability analysis techniques for ordinary differential equations are not directly applicable to such models. METHODS: In this paper, we present and apply three different methods that can be used to study structural identifiability in mixed-effects models. The first method, called the repeated measurement approach, is based on applying a set of previously established statistical theorems. The second method, called the augmented system approach, is based on augmenting the mixed-effects model to an extended state-space form. The third method, called the Laplace transform mixed-effects extension, is based on considering the moment invariants of the systems transfer function as functions of random variables. RESULTS: To illustrate, compare and contrast the application of the three methods, they are applied to a set of mixed-effects models. CONCLUSIONS: Three structural identifiability analysis methods applicable to mixed-effects models have been presented in this paper. As method development of structural identifiability techniques for mixed-effects models has been given very little attention, despite mixed-effects models being widely used, the methods presented in this paper provides a way of handling structural identifiability in mixed-effects models previously not possible.

Exploration of the intercellular heterogeneity of topotecan uptake into human breast cancer cells through compartmental modelling.

A mathematical multi-cell model for the in vitro kinetics of the anti-cancer agent topotecan (TPT) following administration into a culture medium containing a population of human breast cancer cells (MCF-7 cell line) is described. This non-linear compartmental model is an extension of an earlier single-cell type model and has been validated using experimental data obtained using two-photon laser scanning microscopy (TPLSM). A structural identifiability analysis is performed prior to parameter estimation to test whether the unknown parameters within the model are uniquely determined by the model outputs. The full model has 43 compartments, with 107 unknown parameters, and it was found that the structural identifiability result could not be established even when using the latest version of the symbolic computation software Mathematica. However, by assuming that a priori knowledge is available for certain parameters, it was possible to reduce the number of parameters to 81, and it was found that this (Stage Two) model was globally (uniquely) structurally identifiable. The identifiability analysis demonstrated how valuable symbolic computation is in this context, as the analysis is far too lengthy and difficult to be performed by hand.

Extending existing structural identifiability analysis methods to mixed-effects models.

The concept of structural identifiability for state-space models is expanded to cover mixed-effects state-space models. Two methods applicable for the analytical study of the structural identifiability of mixed-effects models are presented. The two methods are based on previously established techniques for non-mixed-effects models; namely the Taylor series expansion and the input-output form approach. By generating an exhaustive summary, and by assuming an infinite number of subjects, functions of random variables can be derived which in turn determine the distribution of the system's observation function(s). By considering the uniqueness of the analytical statistical moments of the derived functions of the random variables, the structural identifiability of the corresponding mixed-effects model can be determined. The two methods are applied to a set of examples of mixed-effects models to illustrate how they work in practice.

Input Estimation for Extended-Release Formulations Exemplified with Exenatide.

Estimating the in vivo absorption profile of a drug is essential when developing extended-release medications. Such estimates can be obtained by measuring plasma concentrations over time and inferring the absorption from a model of the drug's pharmacokinetics. Of particular interest is to predict the bioavailability-the fraction of the drug that is absorbed and enters the systemic circulation. This paper presents a framework for addressing this class of estimation problems and gives advice on the choice of method. In parametric methods, a model is constructed for the absorption process, which can be difficult when the absorption has a complicated profile. Here, we place emphasis on non-parametric methods that avoid making strong assumptions about the absorption. A modern estimation method that can address very general input-estimation problems has previously been presented. In this method, the absorption profile is modeled as a stochastic process, which is estimated using Markov chain Monte Carlo techniques. The applicability of this method for extended-release formulation development is evaluated by analyzing a dataset of Bydureon, an injectable extended-release suspension formulation of exenatide, a GLP-1 receptor agonist for treating diabetes. This drug is known to have non-linear pharmacokinetics. Its plasma concentration profile exhibits multiple peaks, something that can make parametric modeling challenging, but poses no major difficulties for non-parametric methods. The method is also validated on synthetic data, exploring the effects of sampling and noise on the accuracy of the estimates.

Analysis of a Compartmental Model of Endogenous Immunoglobulin G Metabolism with Application to Multiple Myeloma.

Immunoglobulin G (IgG) metabolism has received much attention in the literature for two reasons: (i) IgG homeostasis is regulated by the neonatal Fc receptor (FcRn), by a pH-dependent and saturable recycling process, which presents an interesting biological system; (ii) the IgG-FcRn interaction may be exploitable as a means for extending the plasma half-life of therapeutic monoclonal antibodies, which are primarily IgG-based. A less-studied problem is the importance of endogenous IgG metabolism in IgG multiple myeloma. In multiple myeloma, quantification of serum monoclonal immunoglobulin plays an important role in diagnosis, monitoring and response assessment. In order to investigate the dynamics of IgG in this setting, a mathematical model characterizing the metabolism of endogenous IgG in humans is required. A number of authors have proposed a two-compartment nonlinear model of IgG metabolism in which saturable recycling is described using Michaelis-Menten kinetics; however it may be difficult to estimate the model parameters from the limited experimental data that are available. The purpose of this study is to analyse the model alongside the available data from experiments in humans and estimate the model parameters. In order to achieve this aim we linearize the model and use several methods of model and parameter validation: stability analysis, structural identifiability analysis, and sensitivity analysis based on traditional sensitivity functions and generalized sensitivity functions. We find that all model parameters are identifiable, structurally and taking into account parameter correlations, when several types of model output are used for parameter estimation. Based on these analyses we estimate parameter values from the limited available data and compare them with previously published parameter values. Finally we show how the model can be applied in future studies of treatment effectiveness in IgG multiple myeloma with simulations of serum monoclonal IgG responses during treatment.

The tumor kinetics of multiple myeloma following autologous stem cell transplantation as assessed by measuring serum-free light chains.

In multiple myeloma, changes in serum-free immunoglobulin light chains (FLC) are a more rapid indicator of treatment response than intact immunoglobulin due to their shorter serum half-life. The present study analysed the changes in serum FLC after autologous peripheral blood stem cell transplantation (PBSCT) in 19 patients. The majority of myeloma patients (18 of 19) undergoing PBSCT had a rapid fall in FLC concentrations. In all 11 of 19 patients with raised tumor FLC, it fell within 48 h following high-dose melphalan. In patients with monoclonal intact immunoglobulin, the tumor FLC fell quicker (median half-life 4.3 days) than the monoclonal intact immunoglobulin (median half-life 14 days). FLC recovery occurred after (13 of 19) or around the time of neutrophil engraftment (6 of 19). With a median follow up of 220 days post-transplant, 16 of 19 patients have a normal FLC ratio and 3 of 19 have an elevated tumor FLC/abnormal ratio. FLC assays provided a sensitive monitor of changes in tumor and non-tumor plasma cells after PBSCT. This assay is potentially valuable as a marker of chemosensitivity, as an indicator of residual tumor and indicated time to lymphocyte engraftment. Further follow-up is required to ascertain whether differences in the kinetics of FLC responses have any prognostic clinical utility.

The mechanism distinguishability problem in biochemical kinetics: the single-enzyme, single-substrate reaction as a case study.

A theoretical analysis of the distinguishability problem of two rival models of the single enzyme-single substrate reaction, the Michaelis-Menten and Henri mechanisms, is presented. We also outline a general approach for analysing the structural indistinguishability between two mechanisms. The approach involves constructing, if possible, a smooth mapping between the two candidate models. Evans et al. [N.D. Evans, M.J. Chappell, M.J. Chapman, K.R. Godfrey, Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems, Automatica 40 (2004) 1947-1953] have shown that if, in addition, either of the mechanisms satisfies a particular criterion then such a transformation always exists when the models are indistinguishable from their experimentally observable outputs. The approach is applied to the single enzyme-single substrate reaction mechanism. In principle, mechanisms can be distinguished using this analysis, but we show that our ability to distinguish mechanistic models depends both on the precise measurements made, and on our knowledge of the system prior to performing the kinetics experiments.

PKPD modelling of PR and QRS intervals in conscious dogs using standard safety pharmacology data.

INTRODUCTION: Pharmacokinetic-pharmacodynamic (PKPD) modelling can improve safety assessment, but few PKPD models describing drug-induced QRS and PR prolongations have been published. This investigation aims to develop and evaluate PKPD models for describing QRS and PR effects in routine safety studies. METHODS: Exposure and telemetry data from safety pharmacology studies in conscious beagle dogs were acquired. Mixed effects baseline and PK-QRS/PR models were developed for the anti-arrhythmic compounds AZD1305, flecainide, quinidine and verapamil and the anti-muscarinic compounds AZD8683 and AZD9164. RR interval correction and circadian rhythms were investigated for predicting baseline variability. Individual PK predictions were used to drive the pharmacological effects evaluating linear and non-linear direct and effect compartment models. RESULTS: Conduction slowing induced by the tested anti-arrhythmics was direct and proportional at low exposures, whilst time delays and non-linear effects were evident for the tested anti-muscarinics. AZD1305, flecainide and quinidine induced QRS widening with 4.2, 10 and 5.6% µM(-1) unbound drug. AZD1305 and flecainide also prolonged PR with 13.5 and 11.5% µM(-1). PR prolongations induced by the anti-muscarinics and verapamil were best described by Emax models with maximal effects ranging from 55 to 95%. RR interval correction and circadian rhythm improved PR but not QRS modelling. However, circadian rhythm had minor impact on estimated drug effects. DISCUSSION: Baseline and drug-induced effects on QRS and PR intervals can be effectively described with PKPD models using routine data, providing quantitative safety information to support drug discovery and development.

Dose-response-time modelling: Second-generation turnover model with integral feedback control.

This study presents a dose-response-time (DRT) analysis based on a large preclinical biomarker dataset on the interaction between nicotinic acid (NiAc) and free fatty acids (FFA). Data were collected from studies that examined different rates, routes, and modes of NiAc provocations on the FFA time course. All information regarding the exposure to NiAc was excluded in order to demonstrate the utility of a DRT model. Special emphasis was placed on the selection process of the biophase model. An inhibitory Imax-model, driven by the biophase amount, acted on the turnover rate of FFA. A second generation NiAc/FFA model, which encompasses integral (slow buildup of tolerance - an extension of the previously used NiAc/FFA turnover models) and moderator (rapid and oscillatory) feedback control, was simultaneously fitted to all time courses in normal rats. The integral feedback control managed to capture an observed 90% adaptation (i.e., almost a full return to baseline) when 10 days constant-rate infusion protocols of NiAc were used. The half-life of the adaptation process had a 90% prediction interval between 3.5-12 in the present population. The pharmacodynamic parameter estimates were highly consistent when compared to an exposure-driven analysis, partly validating the DRT modelling approach and suggesting the potential of DRT analysis in areas where exposure data are not attainable. Finally, new numerical algorithms, which rely on sensitivity equations to robustly and efficiently compute the gradients in the parameter optimization, were successfully used for the mixed-effects approach in the parameter estimation.

Parameter Identifiability of Fundamental Pharmacodynamic Models.

Issues of parameter identifiability of routinely used pharmacodynamics models are considered in this paper. The structural identifiability of 16 commonly applied pharmacodynamic model structures was analyzed analytically, using the input-output approach. Both fixed-effects versions (non-population, no between-subject variability) and mixed-effects versions (population, including between-subject variability) of each model structure were analyzed. All models were found to be structurally globally identifiable under conditions of fixing either one of two particular parameters. Furthermore, an example was constructed to illustrate the importance of sufficient data quality and show that structural identifiability is a prerequisite, but not a guarantee, for successful parameter estimation and practical parameter identifiability. This analysis was performed by generating artificial data of varying quality to a structurally identifiable model with known true parameter values, followed by re-estimation of the parameter values. In addition, to show the benefit of including structural identifiability as part of model development, a case study was performed applying an unidentifiable model to real experimental data. This case study shows how performing such an analysis prior to parameter estimation can improve the parameter estimation process and model performance. Finally, an unidentifiable model was fitted to simulated data using multiple initial parameter values, resulting in highly different estimated uncertainties. This example shows that although the standard errors of the parameter estimates often indicate a structural identifiability issue, reasonably "good" standard errors may sometimes mask unidentifiability issues.

The structural identifiability of the susceptible infected recovered model with seasonal forcing.

In this paper, it is shown that the SIR epidemic model, with the force of infection subject to seasonal variation, and a proportion of either the prevalence or the incidence measured, is unidentifiable unless certain key system parameters are known, or measurable. This means that an uncountable number of different parameter vectors can, theoretically, give rise to the same idealised output data. Any subsequent parameter estimation from real data must be viewed with little confidence as a result. The approach adopted for the structural identifiability analysis utilises the existence of an infinitely differentiable transformation that connects the state trajectories corresponding to parameter vectors that give rise to identical output data. When this approach proves computationally intractable, it is possible to use the converse idea that the existence of a coordinate transformation between states for particular parameter vectors implies indistinguishability between these vectors from the corresponding model outputs.

Structural identifiability for a class of non-linear compartmental systems using linear/non-linear splitting and symbolic computation.

Under certain controllability and observability restrictions, two different parameterisations for a non-linear compartmental model can only have the same input-output behaviour if they differ by a locally diffeomorphic change of basis for the state space. With further restrictions, it is possible to gain valuable information with respect to identifiability via a linear analysis. Examples are presented where non-linear identifiability analyses are substantially simplified by means of an initial linear analysis. For complex models, with four or more compartments, this linear analysis can prove lengthy to perform by hand and so symbolic computation has been employed to aid this procedure.

A mathematical model for the in vitro kinetics of the anti-cancer agent topotecan.

In this paper a compartmental modelling approach is applied to provide a mathematical description of the activity of the anti-cancer agent topotecan, and delivery to its nuclear DNA target following administration. The activity of topotecan in defined buffers is first modelled using a linear two compartment model that then forms the basis of a cell based model for drug activity in live cell experiments. An identifiability analysis is performed before parameter estimation to ensure that the model output (i.e., continuous, perfect and noise-free data) uniquely determines the parameters. Parameter estimation is performed using experimental data which offers concentrations of active and inactive forms of topotecan from high performance liquid chromatography methods.

Modelling of tumour growth and cytotoxic effect of docetaxel in xenografts.

One of the major sources of information on physiological and pathophysiological effects in pre-clinical oncology studies is the xenografted tumour animal model. However, measurement of tumour volume over time potentially masks a range of biological changes that the xenograft is undergoing. In this paper a mechanistic model of tumour growth in xenografts is presented that can be used to investigate the mode of drug action with respect to phenotypic changes. The model encapsulates key histological biomarkers and spatial constraints. The unknown model parameters are first shown to be uniquely identifiable from the proposed experimental studies, and then estimated from the resulting data using the anti-cancer agent docetaxel.