Understanding dynamics using sensitivity analysis: caveat and solution
© Perumal and Gunawan; licensee BioMed Central Ltd. 2011
Received: 24 September 2010
Accepted: 15 March 2011
Published: 15 March 2011
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© Perumal and Gunawan; licensee BioMed Central Ltd. 2011
Received: 24 September 2010
Accepted: 15 March 2011
Published: 15 March 2011
Parametric sensitivity analysis (PSA) has become one of the most commonly used tools in computational systems biology, in which the sensitivity coefficients are used to study the parametric dependence of biological models. As many of these models describe dynamical behaviour of biological systems, the PSA has subsequently been used to elucidate important cellular processes that regulate this dynamics. However, in this paper, we show that the PSA coefficients are not suitable in inferring the mechanisms by which dynamical behaviour arises and in fact it can even lead to incorrect conclusions.
A careful interpretation of parametric perturbations used in the PSA is presented here to explain the issue of using this analysis in inferring dynamics. In short, the PSA coefficients quantify the integrated change in the system behaviour due to persistent parametric perturbations, and thus the dynamical information of when a parameter perturbation matters is lost. To get around this issue, we present a new sensitivity analysis based on impulse perturbations on system parameters, which is named impulse parametric sensitivity analysis (iPSA). The inability of PSA and the efficacy of iPSA in revealing mechanistic information of a dynamical system are illustrated using two examples involving switch activation.
The interpretation of the PSA coefficients of dynamical systems should take into account the persistent nature of parametric perturbations involved in the derivation of this analysis. The application of PSA to identify the controlling mechanism of dynamical behaviour can be misleading. By using impulse perturbations, introduced at different times, the iPSA provides the necessary information to understand how dynamics is achieved, i.e. which parameters are essential and when they become important.
Parametric sensitivity analysis (PSA) has become a must have tool in the computational systems biologists' arsenal. In most applications of this analysis, one computes sensitivity coefficients or metrics, which generally reflect the ratios between the change in a biological model output and the perturbation on system parameters that cause this change. Depending on the magnitude of the perturbations, sensitivity analyses can be classified into local (infinitesimal perturbation) and global (finite perturbation). Regardless of these classes, the interpretation of the sensitivity metrics is intuitive; parameters with large sensitivity magnitude are deemed to be important and hence considered to be the controlling factors in the system functional regulation. Consequently, one of the common uses of PSA in systems biology is to infer the importance of cellular processes or pathways and to provide mechanistic explanations for biological behaviour [1–5].
On a separate note, dynamics is a prominent feature of many important biological processes (e.g., oscillations in cell cycle and circadian rhythm [6, 7], switching behaviour in programmed cell death , and adaptation in chemotaxis ). Cellular homeostatic regulation, despite the name, relies on an active dynamical response, in which orchestrated events take place in response to internal and external stimuli. Thus, understanding cellular dynamics has become a prime concern in systems biology, in which mathematical modelling coupled with quantitative analysis have been used to gain insights on the mechanisms that give rise to and control the dynamic behaviour[1–5]. These insights can provide the molecular targets for altering system dynamic behaviour, such as in finding treatment for diseases or in (re)engineering of cellular systems.
While there are many choices of mathematical frameworks for dynamic modelling, ordinary differential equations (ODEs) are the most commonly used modelling paradigm in systems biology and have been used to describe a wide range of biological systems. In addition, ODEs are amenable to many standard quantitative and theoretical analyses, including sensitivity analysis and bifurcation analysis, for which many off-the-shelf software packages exist that provide an integrated and user-friendly computational platform for model simulations and analyses (e.g., MATLAB  and XPPAUT ). The PSA of ODE models can be readily done using software packages such as SimBiology toolbox of MATLAB , PottersWheel , Gepasi , Copasi , JDesigner/Jarnac , JSim , BioSens , SBML-SAT , and SensSB . These and other software for sensitivity analysis have been summarized in the review articles by Alves. et al.  and Klipp. et al. .
Sensitivity analysis of ODE models is well established in the science and engineering literature [23–32]. In systems biology, PSA has found wide applications, such as for model calibration and identifiability, model validation and reduction, identification of bottlenecking processes, elucidation of mechanisms of complex cellular behaviour, and investigation of cellular robustness [30, 33]. A few notable examples of PSA applications in dynamic biological models include programmed cell death [34–39], budding yeast cell cycle control , IL-6 signalling pathway , circadian rhythm models [7, 40, 41], and coupled MAPK and PI3K signal transduction pathway . In many applications, PSA is used to generate parameter ranking based on the magnitude of sensitivity coefficients, either taken at a specific time or using consolidated sensitivity metrics, such as time-integral or average or norm of sensitivity coefficients [34, 43, 44]. The parameter ranking is subsequently used to conclude about the mechanism or property (such as robustness) of the biological system behaviour [1–5].
In this article, we show that the dynamical aspects of cellular functional regulation cannot be inferred from the sensitivity coefficients of PSA, neither directly nor as consolidated sensitivity metrics. More importantly, the corresponding parameter rankings from PSA can give erroneous inference about the controlling mechanisms. Briefly, the reason stems from the fact that in PSA, perturbations are introduced on system parameters, which are time-invariant or static. In other words, these parametric perturbations are persistent and their effects on the system behaviour are integrated over time. Therefore, while PSA can indicate which parameter perturbations are important, it does not point to when these perturbations matter. This problem is illustrated using local PSA of two examples: a synthetic network and a model of programmed cell death . Although the illustration here was done using local sensitivity analysis, the same issue generally applies to global PSA.
To overcome this issue, a new parametric sensitivity analysis is developed in this work. This analysis differs from the classical PSA in the manner of which perturbations are introduced on model parameters, specifically using impulses, and thus is named impulse parametric sensitivity analysis (iPSA). By analyzing the consequence of impulse parameter perturbations introduced at different times, the iPSA provides time-varying, mechanistic explanation on how system dynamics is carried out. The new insights from the iPSA are demonstrated using the same two examples mentioned above.
In this example, in silico knock-out experiments were performed by removing each pathway individually in order to assess the dominance of one pathway over the other in x 6 activation. Both full network and knock-out (KO) simulations were performed under a stimulus of x 1(t 0 = 0) = 1. As illustrated in Figure 1(b), while the initial x 6 activation in the indirect pathway KO remained the same as that of the original model, the switch-like activation was much less pronounced. On the other hand, the original switching behaviour was preserved in the direct pathway KO, but the switching time was delayed due to a slower initial activation. Taken together, these KO simulations suggested that the x 6 activation is mainly accomplished through the indirect pathway, while the direct pathway contributes mainly to the initial x 6 activation.
where x i is the i -th state in an ODE model and p j is the j -th kinetic parameter of an ODE model (for a detailed description of sensitivity coefficient derivation, see Methods). These sensitivity coefficients describe the change in system output (state trajectory) at time t with respect to (an infinitesimal) perturbation on the system parameter values at time τ. Here, the PSA was performed for the same stimulus x 1(0) = 1 with τ = 0 and the sensitivity coefficients were computed for the time range of 0-15 time units. The term local sensitivity analysis refers to the fact that the results will depend on the nominal parameter values around which the derivatives in Eq. 1 are calculated.
Here, one sees two terms in the integrand that contribute to the sensitivity coefficients at time t : (1) the first is related to the (integrated) sensitivities that are carried over from the initial perturbation time τ and (2) the second accounts for the instantaneous rate changes due to the parametric perturbations that still persist at time Thus, in the PSA, a large sensitivity magnitude of S i, j(t, τ) indicates the importance of the j -th parameter in time window of τ and t, during which the perturbation is applied to the system. Hence, the use of these coefficients to infer the dynamical importance of parameters is inappropriate and can even be misleading.
For the reason above, the PSA of the simple network model gave an incorrect conclusion regarding direct versus indirect pathway activating x 6. As seen in the in silico KO experiments, the direct pathway regulates the initial activation of x 6, while the actual switching is carried out by the indirect pathway. In the PSA of this model (τ = 0), the early importance of the direct pathway and also the reaction r 1 persisted beyond the initial times in the sensitivity coefficients due to the aforementioned integrated effect. In this case, the importance of the indirect pathway was not apparent from the parameter sensitivity rankings in the background of large (integrated) sensitivities with respect to r 1 and the direct pathway. Correspondingly, the use of any time-consolidated sensitivity metrics will only worsen this problem.
As the problem with local PSA mentioned above is rooted from the persistent parameter perturbations, which is also done in global PSA , a new sensitivity analysis is formulated here that introduces impulse perturbations to model parameters as shown in Figure 3(c-d). The corresponding impulse sensitivity coefficients iS i, j(t, τ) reflect the change in the i -th state x i at time t due to an impulse perturbation on the j -th parameter p j at time τ (see Methods for the derivation and definition). Since impulse perturbations on parameters cause an immediate state changes at the perturbation time (see Methods), the inference of dynamical parametric importance can be obtained from impulse sensitivities by varying the time of perturbation. However, as with the local PSA, impulse perturbations are also local in nature and thus the impulse sensitivities will depend on the nominal parameter values. The global equivalent of iPSA can be formulated using pulse perturbations and is currently under development. Finally, from time-varying impulse perturbations, the iPSA can give answer to the following questions about state dynamics: which are the important parameters and when do they become important?
The two examples above illustrate the problem of using the classical PSA in identifying the controlling mechanisms of a dynamical system. Of course, this does not mean that the PSA of dynamical models is incorrect, but rather the interpretation of the sensitivity coefficients should be carefully managed. In particular, a large sensitivity magnitude with respect to a parameter suggests the importance of this parameter in the time period between the perturbation time τ and the state observation time t. In contrast, the iPSA is developed with dynamics in mind, where the impact of a single perturbation on the system is realized only at the perturbation time and subsequently they are delivered at varying perturbation times. By doing so, the iPSA coefficients can elucidate the way system dynamics x (t) is achieved, by indicating which and when parameters or processes are essential. Because of the persistent nature of perturbations used in the PSA, it is still not possible to reproduce the conclusions of the iPSA by varying the time of perturbation (see Additional file 1 : Supplementary Figures S6 and S7).
While classical parametric sensitivity analysis provides a powerful tool to understand the parametric dependence of biological behaviour, its suitability in inferring mechanisms of dynamic behaviour has not been properly addressed. The two case studies here illustrated the caveat of using local PSA for such purpose. The issue mainly arose from the information needed to do this inference, where one needs to know not only which parameters are critical, but also when they matter. However, the persistent parametric perturbations in standard PSA are incapable of providing this information as the sensitivity coefficients represent an integrated effect. A new sensitivity analysis, called impulse parametric sensitivity analysis (iPSA), was developed with dynamical systems in mind. In particular, the iPSA makes use of local impulse perturbations introduced at different times to produce the necessary information for understanding dynamics. The application of iPSA to the case studies was able to correctly pinpoint the mechanisms responsible for dynamical system behaviour, while local PSA failed in these cases. Since the discrepancy between PSA and iPSA arises from a fundamental difference in the manner of which parametric perturbations are realized (i.e. persistent vs. impulse), the same caveat and solution can be generalized to the global PSA, in which the perturbations are no longer infinitesimal.
In biological models, the state x ∈ ℝ n is typically the concentration vector of biomolecular species, such as mRNAs and proteins, while the function f is the constitutive, often nonlinear, rate equation. The right hand side of the ODE captures the generation and consumption of biomolecules due to a variety of processes in the cell (e.g. transcription, translation, phosphorylation and dephosphorylation, etc), the rates of which depend on a set of kinetic parameters that are consolidated in the vector . Since the initial conditions x 0 can be treated in the same way as model parameters, the aggregate vector p ∈ ℝm + nis used here to denote the combined parameters and initial conditions, i.e. .
but in the PSA, the perturbation time τ is commonly taken to be the initial time t 0. Hence, the argument τ is typically dropped out of 6 and the sensitivity coefficients only carry a single time dependence on the observation time t [1–3, 5, 30, 44, 49, 50]. The higher order sensitivity coefficients in the Taylor series expansion are less commonly computed, and hence the focus of the current work is only on the first-order sensitivities. Because the magnitude of perturbations are infinitesimally small, the sensitivity coefficients will depend on the nominal or baseline parameter values, and thus the classical PSA is considered a local analysis.
where the indices i and j again denote the i -th state and j -th parameter, and S inf, S FIM, S int and are the sensitivity metrics based on infinite norm , Fisher information matrix , time integral  and sensitivity magnitude at a particular time, respectively.
Ordinary Differential Equations
Parametric Sensitivity Analysis
Impulse Parametric Sensitivity Analysis.
TMP is supported by the Singapore Millennium Foundation scholarship.
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