Making models match measurements: model optimization for morphogen patterning networks

Mathematical modeling of developmental signaling networks has played an increasingly important role in the identification of regulatory mechanisms by providing a sandbox for hypothesis testing and experiment design. Whether these models consist of an equation with a few parameters or dozens of equations with hundreds of parameters, a prerequisite to model-based discovery is to bring simulated behavior into agreement with observed data via parameter estimation. These parameters provide insight into the system (e.g., enzymatic rate constants describe enzyme properties). Depending on the nature of the model fit desired - from qualitative (relative spatial positions of phosphorylation) to quantitative (exact agreement of spatial position and concentration of gene products) - different measures of data-model mismatch are used to estimate different parameter values, which contain different levels of usable information and/or uncertainty. To facilitate the adoption of modeling as a tool for discovery alongside other tools such as genetics, immunostaining, and biochemistry, careful consideration needs to be given to how well a model fits the available data, what the optimized parameter values mean in a biological context, and how the uncertainty in model parameters and predictions plays into experiment design. The core discussion herein pertains to the quantification of model-to-data agreement, which constitutes the first measure of a model's performance and future utility to the problem at hand. Integration of this experimental data and the appropriate choice of objective measures of data-model agreement will continue to drive modeling forward as a tool that contributes to experimental discovery. The Drosophila melanogaster gap gene system, in which model parameters are optimized against in situ immunofluorescence intensities, demonstrates the importance of error quantification, which is applicable to a wide array of developmental modeling studies.

Experiment design through dynamical characterisation of non-linear systems biology models utilising sparse grids

The sparse grid-based experiment design algorithm sequentially selects an experimental design point to discriminate between hypotheses for given experimental conditions. Sparse grids efficiently screen the global uncertain parameter space to identify acceptable parameter subspaces. Clustering the located acceptable parameter vectors by the similarity of the simulated model trajectories characterises the data-compatible model dynamics. The experiment design algorithm capitalizes on the diversity of the experimentally distinguishable system output dynamics to select the design point that best discerns between competing model-structure and parameter-encoded hypotheses. As opposed to designing the experiments to explicitly reduce uncertainty in the model parameters, this approach selects design points to differentiate between dynamical behaviours. This approach further differs from other experimental design methods in that it simultaneously addresses both parameter- and structural-based uncertainty that is applicable to some ill-posed problems where the number of uncertain parameters exceeds the amount of data, places very few requirements on the model type, available data and a priori parameter estimates, and is performed over the global uncertain parameter space. The experiment design algorithm is demonstrated on a mitogen-activated protein kinase cascade model. The results show that system dynamics are highly uncertain with limited experimental data. Nevertheless, the algorithm requires only three additional experimental data points to simultaneously discriminate between possible model structures and acceptable parameter values. This sparse grid-based experiment design process provides a systematic and computationally efficient exploration over the entire uncertain parameter space of potential model structures to resolve the uncertainty in the non-linear systems biology model dynamics.

Identification of age-structured models: cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.

Systematic method for determining intravenous drug treatment strategies aiding the humoral immune response

This paper delineates a systematic method for determining "optimal" intravenous drug delivery strategies for patients having illnesses that primarily evoke a humoral immune response and are treatable by antibiotics. The method derives from a nonlinear, distributed predator-prey model that captures the dominant antigen and antibody interaction. This model is developed from relevant physiology, past predator-prey-type modeling work, available data, and pertinent parameter identification. Embedding this predator-prey model into a larger class of uncertain systems, by a finite dimensional approximation and a transformation to a linear fractional representation, enables the application of robust control based on linear matrix inequality optimization techniques. The optimization problem is solved by minimizing an upper bound on a measure of the total drug delivered subject to patient recovery (stability to healthy equilibrium state). Specifically, the paper addresses the treatment of Haemophilus influenzae through modeling, controller development, and simulations of infected adult patients subjected to typical and proposed intravenous antibiotic treatments. Through simulations the proposed intravenous drug strategy shortens patient recovery time, lowers peak drug concentrations and decreases the total drug administered when compared to standard antibiotic strategies.