Path Integration and the Structural Sensitivity Problem in Partially Specified Biological Models

Biological systems are known to be inherently uncertain and complex, and it is often difficult to justify rigorously functional forms used in a mathematical model. Generally, one chooses some simple function having qualitatively correct behavior without concern for the precise choice of function; the assumption has been that qualitatively similar functions will produce qualitatively similar behavior from the model. However, it has been shown that, in some cases, the qualitative predictions from a model are sensitive to the particular functional form used; this property is known as structural sensitivity. A promising tool for quantification of structural sensitivity is the use of partially specified models, in which some number of functions in equations are not defined explicitly. However, in this approach we cannot rely on the traditional parameter-based framework, where all functional forms are well defined but contain parameters to be determined. The main difficulties are that, mathematically, we need to deal with an infinite-dimensional function space and that a well-defined measure on that space is needed. We propose a novel framework to reveal the structural sensitivity and quantify uncertainty in partially specified biological models based on ordinary differential equations backed up by empirical data. Our method uses path integration, previously introduced in theoretical physics. As an insightful example, we explore structural sensitivity of a well-known tritrophic food chain model in which the functional response of the predator is uncertain, using available experimental data on protozoan predation on bacteria. For the mentioned model, we compare the novel framework with the classical methods of parameter-based sensitivity analysis and demonstrate distinct outcomes for the two approaches. Finally, we discuss a further extension of the proposed framework.