On linear models and parameter identifiability in experimental biological systems

A simple method for identifying parameter correlations in partially observed linear dynamic models

Background

Parameter estimation represents one of the most significant challenges in systems biology. This is because biological models commonly contain a large number of parameters among which there may be functional interrelationships, thus leading to the problem of non-identifiability. Although identifiability analysis has been extensively studied by analytical as well as numerical approaches, systematic methods for remedying practically non-identifiable models have rarely been investigated.

Results

We propose a simple method for identifying pairwise correlations and higher order interrelationships of parameters in partially observed linear dynamic models. This is made by derivation of the output sensitivity matrix and analysis of the linear dependencies of its columns. Consequently, analytical relations between the identifiability of the model parameters and the initial conditions as well as the input functions can be achieved. In the case of structural non-identifiability, identifiable combinations can be obtained by solving the resulting homogenous linear equations. In the case of practical non-identifiability, experiment conditions (i.e. initial condition and constant control signals) can be provided which are necessary for remedying the non-identifiability and unique parameter estimation. It is noted that the approach does not consider noisy data. In this way, the practical non-identifiability issue, which is popular for linear biological models, can be remedied. Several linear compartment models including an insulin receptor dynamics model are taken to illustrate the application of the proposed approach.

Conclusions

Both structural and practical identifiability of partially observed linear dynamic models can be clarified by the proposed method. The result of this method provides important information for experimental design to remedy the practical non-identifiability if applicable. The derivation of the method is straightforward and thus the algorithm can be easily implemented into a software packet.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0234-3) contains supplementary material, which is available to authorized users.

A Priori Identifiability of Target-Mediated Drug Disposition Models and Approximations

A priori identifiability of mathematical models assures that for a given input/output experiment, the parameter set has one unique solution within a defined space, independent of the experimental design. Many biologic therapeutics exhibit target-mediated drug disposition (TMDD), and use of the full compartmental model describing this system is well documented. In practice, estimation of the full parameter set for TMDD models, given real-world clinical data, is characterized by convergence difficulties and unstable solutions. Still, the formal assessment of the a priori identifiability of these systems has yet to be reported. The exact arithmetic rank (EAR) approach was used to test the a priori identifiability of a TMDD model as well as model approximations. The full TMDD and quasi-equilibrium/rapid binding (QE/RB), quasi-steady state (QSS), and Michaelis-Menten (MM) approximations were fully identifiable, a priori, regardless of whether observations were taken from a single or multiple compartments. The results of these identifiability analyses indicated that the difficulty with TMDD model convergence, a posteriori, lies in the experimental design, not in the mathematical identifiability in the lack of samples from several compartments. Experiments can be tailored to resolve these structurally non-identifiable parameters, notwithstanding practical implementation challenges. This work highlights the importance of identifiability analyses, specifically how they can influence experimental design and selection of the appropriate model structure to describe a dynamic biological system.

Dynamic imaging of the recycling endosomal network in macrophages

Recycling endosomes (REs) form an extensive and complex network of subcompartmentalized vesicular and tubular elements that connect with the cell surface and other endosomes in macrophages. As surveillance and defense cells of the innate immune system, macrophages are highly dependent on REs for their active and voluminous cell surface turnover and endocytic, exocytic, and recycling of membrane and cargo. Here we set out three approaches for imaging and analyzing REs in macrophages, based on the expression of fluorescently labeled RE-associated proteins and the uptake of fluorescent cargo. Subcompartments of the REs are identified by co-expression and co-localization analysis of RE associated Rab GTPases. Transferrin is a well-known cargo marker as it recycles through REs and it is compared here to other cargo, revealing how different endocytic routes intersect with REs. We show how the movement of transferrin through REs can be modeled and quantified in live cells. Finally, since phagosomes are a signature organelle for macrophages, and REs fuse with the maturing phagosome, we show imaging of REs with phagosomes using a genetically encoded pH-sensitive SNARE-based probe. Together these approaches provide multiple ways to comprehensively analyze REs and the important roles they play in these immune cells and more broadly in other cell types.