Background

Identification of both the structure (the form of the equations) and the parameters of ordinary differential equations (ODEs) is a challenging problem. Existing research in systems biology typically considers restricted model spaces, and algorithms with very high computational requirements.

ODEion is a software tool for efficient structural identification of ODEs in a more general framework. The aim is to identify a dynamic model as a system of ODEs, from sparse and noisy time-series data. For each variable, the generic form of the ODE is composed of a sum of arbitrary functions that can be nonlinear in both variables and parameters.

The problem we consider differs from a standard regression problem in two main ways. We need to find a best subset of terms in the ODEs, taken from a very large set of possible terms implicitly defined by the user. Then, to make the most of sparse and noisy data, we cannot simply estimate the derivative in the LHS of the ODE directly from data, but the model fit needs to be evaluated by simulating the ODEs.

Related software include well known statistical software such as R, SAS and SPSS, which typically have tools for non-linear and stepwise regression. We also mention the Eureqa system for finding mathematical relationships in data. Potentially, such systems could be used for similar tasks, especially if the derivatives can be sufficiently well estimated from data. This is however not the typical situation in systems biology where data is sparse and noisy, and these tools have not been designed to provide the functionality or performance of our current software. Furthermore, some experimental software codes can be requested from researchers in the field, applicable to specialized models such as S-systems. This leaves out models with e.g. chemical rate reactions which are possible with our software. To our knowledge there is no similar general software for structural identification of ODEs.

Specifying the identification problem

To define an identification problem with unknown structure and parameters, we require the following information:

• Time series data from one or several experiments.
• A model space describing the allowed forms of the right hand side of the ODEs.
• An initial model specifying a known prior information about parts of the structure.
• An error function specifying how well any given model fits with the data. This function must include some notion of model complexity.

For more information about how to specify an identification problem, see the the documentation.

Software functionality

There are four basic functionalities:

• Verify syntax and consistency of a problem. Potential errors and reasonable solutions to these are reported.
• Visualize data from an experiment of a problem and adjusting parameters of the smoothing spline interpolation.
• Generate a Fortran program designed to solve a specific problem. The resulting program is self-contained, and can be compiled and run immediately, or saved for future execution.
• Generate a Systems Biology Markup Language (SBML) file of the solution model, allowing visualization, simulation and further analysis on a range of softwares. Output is also given in a simple human readable format.