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.     

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.     

Structural identifiability analysis of some highly structured families of statespace models using differential algebra

In this paper we identify biologically relevant families of models whose structural identifiability analysis could not be performed with available techniques directly. The models considered come from both the immunological and epidemiological literature.     

Qualitative theory of compartmental systems with lags

Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems. Often, these systems include time lags; in this context, continuous probability density functions (pdfs) of lags are far more important than discrete lags. There is a relatively complete theory of compartmental systems without lags, both linear and non-linear [SIAM Rev. 35 (1993) 43]. The authors extend their previous work on compartmental systems without lags to show that, for discrete lags and for a very large class of pdfs of continuous lags, compartmental systems with lags are equivalent to larger compartmental systems without lags. Consequently, the properties of compartmental systems with lags are the same as those of compartmental systems without lags. For a very large class of compartmental systems with time lags, one can show that the time lags themselves can be generated by compartmental systems without lags. Thus, such systems can be partitioned into a main system, which is the original system without the lags, plus compartmental subsystems without lags that generate the lags. The latter may be linear or non-linear and may be inserted into main systems that are linear or non-linear. The state variables of the compartmental lag subsystems are hidden variables in the formulation with explicit lags.     

Structural identifiability of a model for the acetic acid fermentation process

Modelling has proved an essential tool for addressing research into biotechnological processes, particularly with a view to their optimization and control. Parameter estimation via optimization approaches is among the major steps in the development of biotechnology models. In fact, one of the first tasks in the development process is to determine whether the parameters concerned can be unambiguously determined and provide meaningful physical conclusions as a result. The analysis process is known as 'identifiability' and presents two different aspects: structural or theoretical identifiability and practical identifiability. While structural identifiability is concerned with model structure alone, practical identifiability takes into account both the quantity and quality of experimental data. In this work, we discuss the theoretical identifiability of a new model for the acetic acid fermentation process and review existing methods for this purpose.     

The structural identifiability of susceptible–infective–recovered type epidemic models with incomplete immunity and birth targeted vaccination

This paper considers the implications of a structural identifiability analysis on a series of fundamental three-compartment epidemic model structures, derived around the general SIR (susceptible–infective–recovered) framework. The models represent various forms of incomplete immunity acquired through natural infection, or from administration of a birth targeted vaccination programme. It is shown that the addition of a vaccination campaign has a negative effect on the structural identifiability of all considered models. In particular, the actual proportion of vaccination coverage achieved, an essential parameter, cannot be uniquely estimated from even ideal prevalence data.     

Density functions of residence times for deterministic and stochastic compartmental systems

A significant consideration in modeling systems with stages is to obtain models for the individual stages that have probability density functions (pdfs) of residence times that are close to those of the real system. Consequently, the theory of residence time distributions is important for modeling. Here I show first that linear deterministic compartmental systems with constant coefficients and their corresponding stochastic analogs (stochastic compartmental systems with linear rate laws) have the same pdfs of residence times for the same initial distributions of inputs. Furthermore, these are independent of inflows. Then I show that does not hold for non-linear deterministic systems and their stochastic analogs (stochastic compartmental systems with non-linear rate laws). In fact, for given initial distributions of inputs, the pdfs of non-linear determistic systems without inflows and of their stochastic analogs, are functions of the initial amounts injected. For systems with inflows, the pdfs change as the inflows influence the occupancies of the compartments of the system; they are state-dependent pdfs.     

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows with postmilking teat disinfection

A mathematical model for the transmission of two interacting classes of mastitis causing bacterial pathogens in a herd of dairy cows is presented and applied to a specific data set. The data were derived from a field trial of a specific measure used in the control of these pathogens, where half the individuals were subjected to the control and in the others the treatment was discontinued. The resultant mathematical model (eight non-linear simultaneous ordinary differential equations) therefore incorporates heterogeneity in the host as well as the infectious agent and consequently the effects of control are intrinsic in the model structure. A structural identifiability analysis of the model is presented demonstrating that the scope of the novel method used allows application to high order non-linear systems. The results of a simultaneous estimation of six unknown system parameters are presented. Previous work has only estimated a subset of these either simultaneously or individually. Therefore not only are new estimates provided for the parameters relating to the transmission and control of the classes of pathogens under study, but also information about the relationships between them. We exploit the close link between mathematical modelling, structural identifiability analysis, and parameter estimation to obtain biological insights into the system modelled.     

Dinuclear nickel(II) complexes of unsymmetric compartmental ligands bearing a terdentate linear arm and a terdentate dipodal arm

The synthesis and crystal structures of two dinuclear nickel(II) complexes of unsymmetric compartmental ligands derived from phenol and bearing a terdentate linear arm and a terdentate dipodal arm are reported. The binding of the terminal donor atom of the terdentate linear arm appears to be dependent on the nature of the accompanying counter-anion. In the presence of the non-coordinating tetrafluoroborate anion a terminal alcohol is coordinated, whereas in the presence of added isothiocyanate ion a terminal amine is not coordinated but the anion is.     

Predicting abundances of plants and pollinators using a simple compartmental mutualistic model

Key gaps to be filled in population and community ecology are predicting the strength of species interactions and linking pattern with process to understand species coexistence and their relative abundances. In the case of mutualistic webs, like plant–pollinator networks, advances in understanding species abundances are currently limited, mainly owing to the lack of methodological tools to deal with the intrinsic complexity of mutualisms. Here, we propose an aggregation method leading to a simple compartmental mutualistic population model that captures both qualitatively and quantitatively the size-segregated populations observed in a Mediterranean community of nectar-producing plant species and nectar-searching animal species. We analyse the issue of optimal aggregation level and its connection with the trade-off between realism and overparametrization. We show that aggregation of both plants and pollinators into five size classes or compartments leads to a robust model with only two tunable parameters. Moreover, if, in each compartment, (i) the interaction coefficients fulfil the condition of weak mutualism and (ii) the mutualism is facultative for at least one party of the compartment, then the interactions between different compartments are sufficient to guarantee global stability of the equilibrium population.     

A dichotomy for a class of cyclic delay systems

Two complementary analyses of a cyclic negative feedback system with delay are considered in this paper. The first analysis applies the work by Sontag, Angeli, Enciso and others regarding monotone control systems under negative feedback, and it implies the global attractiveness towards an equilibrium for arbitrary delays. The second one concerns the existence of a Hopf bifurcation with respect to the delay parameter, and it implies the existence of nonconstant periodic solutions for special delay values. A key idea is the use of the Schwarzian derivative, and its application for the study of Hill function nonlinearities. The positive feedback case is also addressed.     

Complete parameter bounds and quasiidentifiability conditions for a class of unidentifiable linear systems

Lack of unique structural identifiability for parameters of dynamic system models is a very common situation with practical experimental schemes, particularly when studying biological systems. However, for well-structured (e.g., multicompartmental) models, it is often possible to localize unidentifiable parameters between finite limits (“interval identifiability”), using the same data base, and under certain conditions these limits nearly coincide. Two new results in this area are presented: (1) The smallest ranges on all unidentifiable rate constants and pool sizes of the most general n-compartment mammillary system are derived, in an easy-to-program algorithmic form, for the common case of input forcing and output measurements in the central pool only. From these results we see why elimination rate constants (“leaks”) are difficult to distinguish from zero, whereas exchange rate constants between pools, and pool sizes, may be bounded very tightly in certain circumstances. (2) The notion of quasiidentifiability, or sufficient identifiability for practical purposes, is introduced to quantify these circumstances. Each of the rate constants between central and peripheral pools, and all pool sizes, are quasiidentifiable if the magnitude of the ratio of the coefficient to the eigenvalue of the slowest mode is very much greater than the largest coefficient in the sum-of-exponentials response function. Also quasiidentifiability is a necessary condition for applicability of noncompartmental analysis to estimate pool sizes and residence times of mammillary systems with “leaky” noncentral pools.     

Dynamic modeling of complex biological systems: a link between metabolic and macroscopic description

In this study, a class of dynamic models based on metabolic reaction pathways is analyzed, showing that systems with complex intracellular reaction networks can be represented by macroscopic reactions relating extracellular components only. Based on rigorous assumptions, the model reduction procedure is systematic and allows an equivalent 'input-output' representation of the system to be derived. The procedure is illustrated with a few examples.     

Estimation of the rate and extent of chiral inversion using linear systems analysis

An alternative method based on linear systems analysis is presented for the analysis of concentration-time data for the enantiomers of the 2-arylpropionic acids. This approach uses deconvolution to estimate the rate and extent of chiral inversion with respect to time, assuming linear pharmacokinetics and time invariance, without the need for complicated modelling procedures. Application to data for the chiral inversion of ibuprofen in the rat indicates that this approach provides a valid alternative to previous procedures for the analysis of chiral inversion data. © 1995 Wiley-Liss, Inc.     

Structural insights into conserved l-arabinose metabolic enzymes reveal the substrate binding site of a thermophilic l-arabinose isomerase

Structural genomics demonstrates that despite low levels of structural similarity of proteins comprising a metabolic pathway, their substrate binding regions are likely to be conserved. Herein based on the 3D-structures of the α/β-fold proteins involved in the ara operon, we attempted to predict the substrate binding residues of thermophilic Geobacillus stearothermophilusl-arabinose isomerase (GSAI) with no 3D-structure available. Comparison of the structures of l-arabinose catabolic enzymes revealed a conserved feature to form the substrate-binding modules, which can be extended to predict the substrate binding site of GSAI (i.e., D195, E261 and E333). Moreover, these data implicated that proteins in the l-arabinose metabolic pathway might retain their substrate binding niches as the modular structure through conserved molecular evolution even with totally different structural scaffolds.     

Asymptotic and explicit formulae for a non-markovian model with linear transition rule

Asymptotic and explicit formulae are obtained for the n-step stochastic kernel of a special class of non-MARKOVIAN models with linear transition rule which occur in learning theory, adaption theory, control theory, and biological research.     

Development and validation of a high-throughput method for the quantitative analysis of d-amphetamine in rat blood using liquid chromatography/MS on a hybrid triple quadrupole-linear ion trap mass spectrometer and its application to a pharmacokinetic study

Amphetamines are a group of sympathomimetic drugs that exhibit strong central nervous system stimulant effects. d-Amphetamine ((+)-alpha-methylphenetylamine) is the parent drug in this class to which all others are structurally related. In drug discovery, d-amphetamine is extensively used either for the exploration of novel mechanisms involving the catecholaminergic system, or for the validation of new behavioural animal models. Due to this extensive use of d-amphetamine in drug research and its interest in toxicologic–forensic investigation, a specific and high-throughput method, with minimal sample preparation, is necessary for routine analysis of d-amphetamine in biological samples. We propose here a sensitive, specific and high-throughput bioanalytical method for the quantitative determination of d-amphetamine in rat blood using MS³ scan mode on a hybrid triple quadrupole-linear ion trap mass spectrometer (LC–MS/MS/MS). Blood samples, following dilution with water, were prepared by fully automated protein precipitation with acetonitrile containing an internal standard. The chromatographic separation was achieved on a Waters XTerra C18 column (2.1 mm × 30 mm, 3.5 μm) using gradient elution at a flow rate of 1.0 mL/min over a 2 min run time. An Applied Biosystems API4000 QTRAP™ mass spectrometer equipped with turbo ion-spray ionization source was operated simultaneously in MS³ scan mode for the d-amphetamine and in multiple reaction monitoring (MRM) for the internal standard. The MS/MS/MS ion transition monitored was m/z 136.1 → 119.1 → 91.1 for the quantitation of d-amphetamine and for the internal standard (rolipram) the MS/MS ion transition monitored was m/z 276.1 → 208.2. The linear dynamic range was established over the concentration range 0.5–1000 ng/mL (r² = 0.9991). The method was rugged and sensitive with a lower limit of quantification (LLOQ) of 0.5 ng/mL. All the validation data, such as accuracy, precision, and inter-day repeatability, were within the required limits. This method was successfully applied to evaluate the pharmacokinetics of d-amphetamine in rat. On a more general extent, this work demonstrated that the selectivity of the fragmentation pathway (MS³) can be used as alternative approach to significantly improve detection capability in complex situation (e.g., small molecules in complex matrices) rather than increasing time for sample preparation and chromatographic separation.     

Validation and implementation of a method for determination of bryostatin 1 in human plasma by using liquid chromatography/tandem mass spectrometry

A new sensitive and specific method using liquid chromatography/tandem mass spectrometry for determination of bryostatin 1 was developed and validated. Sample pretreatment involved a double liquid-liquid extraction step with a mixture of acetonitrile/n-butyl chloride (1/4, v/v). Separation of the compound of interest, including the internal standard paclitaxel, was achieved on a Waters X-Terra C18 (50 x 2.1 mm i.d., 3.5 microm) analytical column with acetonitrile/water mobile phase (80:20, v/v) containing 0.1% formic acid using isocratic flow at 0.15 mL/min for 13 min. The analytes of interest were monitored by tandem mass spectrometry with electrospray positive ionization. The linear calibration curves were generated over the range of 50-2000 pg/mL with values for the coefficient of determination of >0.99. The values for both within-day and between-day precision and accuracy were <15%. This method was used to characterize the plasma pharmacokinetics of bryostatin 1 at doses of 20 microg/m2) to optimize treatment with this agent.