In this paper, we address the system identification problem in the context of biological modelling. We present and demonstrate a methodology for (i) assessing the possibility of inferring the unknown quantities in a dynamic model and (ii) effectively estimating them from output data. We introduce the term Full Input-State-Parameter Observability (FISPO) analysis to refer to the simultaneous assessment of state, input and parameter observability (note that parameter observability is also known as identifiability). This type of analysis has often remained elusive in the presence of unmeasured inputs. The method proposed in this paper can be applied to a general class of nonlinear ordinary differential equations models. We apply this approach to three models from the recent literature. First, we determine whether it is theoretically possible to infer the states, parameters and inputs, taking only the model equations into account. When this analysis detects deficiencies, we reformulate the model to make it fully observable. Then we move to numerical scenarios and apply an optimization-based technique to estimate the states, parameters and inputs. The results demonstrate the feasibility of an integrated strategy for (i) analysing the theoretical possibility of determining the states, parameters and inputs to a system and (ii) solving the practical problem of actually estimating their values.

Motivation: Optimality principles have been used to explain many biological processes and systems. However, the functions being optimized are in general unknown a priori. Here we present an inverse optimal control (IOC) framework for modeling dynamics in systems biology. The objective is to identify the underlying optimality principle from observed time-series data and simultaneously estimate unmeasured time-dependent inputs and time-invariant model parameters. As a special case, we also consider the problem of optimal simultaneous estimation of inputs and parameters from noisy data. After presenting a general statement of the IOC problem, and discussing special cases of interest, we outline numerical strategies which are scalable and robust. Results: We discuss the existence, relevance and implications of identifiability issues in the above problems. We present a robust computational approach based on regularized cost functions and the use of suitable direct numerical methods based on the control-vector parameterization approach. To avoid convergence to local solutions, we make use of hybrid global-local methods. We illustrate the performance and capabilities of this approach with several challenging case studies, including simulated and real data. We pay particular attention to the computational scalability of our approach (with the objective of considering large numbers of inputs and states). We provide a software implementation of both the methods and the case studies.

Supplementary data are available at Bioinformatics online.

Background Optimality principles have been used to explain the structure and behavior of living matter at different levels of organization, from basic phenomena at the molecular level, up to complex dynamics in whole populations. Most of these studies have assumed a single-criteria approach. Such optimality principles have been justified from an evolutionary perspective. In the context of the cell, previous studies have shown how dynamics of gene expression in small metabolic models can be explained assuming that cells have developed optimal adaptation strategies. Most of these works have considered rather simplified representations, such as small linear pathways, or reduced networks with a single branching point, and a single objective for the optimality criteria. Results Here we consider the extension of this approach to more realistic scenarios, i.e. biochemical pathways of arbitrary size and structure. We first show that exploiting optimality principles for these networks poses great challenges due to the complexity of the associated optimal control problems. Second, in order to surmount such challenges, we present a computational framework which has been designed with scalability and efficiency in mind, including mechanisms to avoid the most common pitfalls. Third, we illustrate its performance with several case studies considering the central carbon metabolism of S. cerevisiae and B. subtilis. In particular, we consider metabolic dynamics during nutrient shift experiments. Conclusions We show how multi-objective optimal control can be used to predict temporal profiles of enzyme activation and metabolite concentrations in complex metabolic pathways. Further, we also show how to consider general cost/benefit trade-offs. In this study we have considered metabolic pathways, but this computational framework can also be applied to analyze the dynamics of other complex pathways, such as signal transduction or gene regulatory networks.