(541f) Caveats of Parametric Sensitivity Analysis: Dynamical Analysis of Biological Systems

The challenge of understanding complex cellular networks has spurred the use of quantitative (mathematical) modeling and analysis in biology [1]. These quantitative models represent virtual biochemistry, mimicking biochemical reactions in the cell [2]. Here, sensitivity analysis, a well established method in science and engineering literature [3], is the most frequently applied analysis due to its ease of computation and intuitive interpretation of the results. In systems biology, this analysis has become a powerful tool to investigate the manner of which model outputs or behavior depend on model parametrization [4], and has also been included in majority of off-the-shelf systems biology tools [5-8]. This analysis has been used for many purposes, such as model calibration and model identifiably, model validation and reduction, finding bottle-necking processes of the system [4].

There are generally two types of sensitivity analysis that are employed in biological modeling; local and global. Local sensitivity analysis concerns with the change in the system behaviour with respect to an infinitesimal change in the parameter from its nominal value. On the other hand, global sensitivities describe the effect of simultaneous or large variations of parameters on the system behavior [3]. Local analysis is used more often than global as it is easier to compute and interpret. The computed sensitivity coefficients give information regarding the contribution of individual parameter to the system output behavior. The magnitude of the sensitivities thus indicates the degree of importance, based on which parameter ranking can be generated.

Many biological behaviors are characterized by their dynamic response, for example in signal transduction pathways and gene regulatory networks. In this case, ODE models are often used to simulate such dynamics using time-dependent state or output variables (such as concentrations). The model parameters typically include physicochemical constants (such as those related to reaction kinetics, transport properties, etc.) as well as initial conditions, operating conditions, and geometric parameters of the systems. The physicochemical parameters are measured experimentally or estimated theoretically or chosen ad-hoc, and therefore have uncertainties. In addition, the initial and operating conditions may change in time for a variety of reasons, such as varying receptor concentration and environmental conditions like temperature. Parametric sensitivity coefficients are computed as the ratio between the differences in state variable caused by changes in the parameter values and used to investigate the impact of parameter uncertainty on model outputs.

Some examples of applications of parametric sensitivity analysis in dynamic biological models include programmed cell death [9], budding yeast cell cycle control [10], IL-6 signaling pathway [11], circadian rhythm model in Neurospora [12], coupled MAPK and PI3K signal transduction pathway [13]. The interpretations of the resulting time varying parametric sensitivities depend on the applications. In most cases, consolidated sensitivity metrics, e.g. using integrals of sensitivity magnitudes or absolute sum of sensitivities over time or the Fisher Information Matrix, are used to indicate parametric importance and to generate parameter ranking. In doing so, such metrics can mislead the modeler in understanding system dynamics. In addition, one should be cautious in inferring dynamical information from time-dependent parametric sensitivities since parametric perturbations are made on the system parameters. As these parameters are usually constants and do not have dynamics, the parametric sensitivity coefficients thus reflect an integrated effect of parameter change on the system behavior.

In this work, we show that parametric sensitivity analysis, either taken directly or as consolidated metrics, can give wrong conclusions about the bottle-necking or governing process in a dynamical system. Two examples, an artificial network and a Fas-induced programmed cell death model [14] are used to illustrate this issue. In addition, we offer a modified parametric sensitivity analyses (both global and local) that can reveal dynamical importance of parameters directly using modified local sensitivities (local analysis) and information transfer (global analysis). Application to the cell death model above revealed the importance of mitochondrial-dependent pathway, while standard parametric sensitivities pointed to the direct caspase-8 pathway as the most sensitive. Comparison of this result to pathway knock-outs in silico confirmed the control of cell death regulation by mitochondria in this model. This work highlights the pitfalls of using parametric sensitivity analysis to infer dynamical bottle-necking process.

References:

1. Kitano, H., Systems biology: a brief overview. Science, 2002. 295(5560): p. 1662--1664.

2. Szallasi, Z., J. Stelling, and V. Periwal, System Modeling in Cell Biology From Concepts to Nuts and Bolts. 2006, Cambridge, Massachusetts: The MIT Press.

3. Varma, A., M. Morbidelli, and H. Wu, Parametirc Sensitivity in Chemical Systems, ed. A. Varma, M. Morbidelli, and H. Wu. 1999: Cambridge University Press, Cambridge, UK.

4. Ingalls, B., Sensitivity analysis: from model parameters to system behaviour. Essays Biochem, 2008. 45: p. 177-93.

5. Maiwald, T. and J. Timmer, Dynamical modeling and multi-experiment fitting with PottersWheel. Bioinformatics, 2008. 24(18): p. 2037-43.

6. Hoare, A., D.G. Regan, and D.P. Wilson, Sampling and sensitivity analyses tools (SaSAT) for computational modelling. Theor Biol Med Model, 2008. 5: p. 4.

7. Klipp, E., et al., Systems biology standards--the community speaks. Nat Biotechnol, 2007. 25(4): p. 390-1.

8. Zi, Z., et al., SBML-SAT: a systems biology markup language (SBML) based sensitivity analysis tool. BMC Bioinformatics, 2008. 9: p. 342.

9. Shoemaker, J.E. and F.J. Doyle III, Identifying Fragilities in Biochemical Networks: Robust Performance Analysis of Fas Signaling-Induced Apoptosis. Biophys J, 2008.

10. A. Lovrics, I.G.Z., A. Csikász-Nagy, J. Zádor, T. Turányi, B. Novák,, Analysis of a budding yeast cell cycle model using the shapes of local sensitivity functions. International Journal of Chemical Kinetics, 2008. 40(11): p. 710-720.

11. Chu, Y., A. Jayaraman, and J. Hahn, Parameter sensitivity analysis of IL-6 signalling pathways. IET Syst Biol, 2007. 1(6): p. 342-52.

12. Jin, Y., et al., Uniform design-based sensitivity analysis of circadian rhythm model in Neurospora. Computers and Chemical Engineering, 2008. 32(8): p. 1956-1962.

13. Hu, D. and J.M. Yuan, Time-dependent sensitivity analysis of biological networks: coupled MAPK and PI3K signal transduction pathways. J Phys Chem A, 2006. 110(16): p. 5361-70.

14. Hua, F., et al., Integrated mechanistic and data-driven modelling for multivariate analysis of signalling pathways. J R Soc Interface, 2006. 3(9): p. 515--526.