Sensitivity analysis investigates a relation between uncertain input or parameters of a model, and a property of an observable output [10
]. Sensitivity analysis has been used for various parametrization tasks of models of biological systems, including finding essential parameters for research prioritization [30
], identifying insignificant parameters for the model reduction [31
] or parameters clustering for the discovery of common functions [32
Biochemical reaction networks yield models of a nonlinear nature for which global sensitivity analysis methods (GSA) are the most suitable [29
]. GSA examines a range of input parameter values simultaneously as opposed to one-factor-at-a-time methods such as those calculating the derivatives of output with respect to parameters. Multi-parameter sensitivity analysis (MPSA) [33
] is an implementation of the GSA concept. MPSA is an instance of a Monte Carlo filtering method, which maps samples from a parameter space into behavioral and non behavioral output regions [10
]. For the examples of applications of the multi-parameter sensitivity analysis to signaling pathways see [28
]. The MPSA method works as follows:
1. Select parameters to assess.
2. Set parameters range.
3. Generate independent samples.
4. For each sample calculate the error (based on the output).
5. Classify samples as acceptable or unacceptable.
6. For each of the selected parameters compare the classified samples sets.
This procedure is depicted in Figure as a workflow.
The multi-parameter sensitivity analysis workflow with an ODE-based error function. Pink and brown boxes represent essential steps of the procedure. Remaining boxes represent workflow’s parameters and outputs. See text for details.
Calculating the error for each sample (Step 4) involves a separate analysis of the model. This is a factor that determines the running time of the MPSA procedure. We ran two variants of MPSA, differing in the way in which the error is calculated. In one variant we used ODEs simulations and in the other one we exploited the probabilistic model checking technique. We focused on kinetic parameters of two forward reactions of enzymatic reaction models (Equation (1)), i.e. k1 and k3. As an error function we took, respectively, the mean squared error of an ODE trajectory of the product P and the absolute difference of the value of the formula (3), in both cases between results for a parameters sample and for the reference values of parameters (Equation (2)). In turn, we obtained empirical cumulative distribution functions (ECDF) of acceptable and unacceptable samples, for each of the selected parameters. ECDFs were compared using the Kolmogorov-Smirnov test (KS-test) and one minus the Pearson product–moment correlation coefficient (PMCC). As a final output of the MPSA method, we got two rankings for each of the sensitivity indices: KS-test and PMCC.
Figure depicts values of the error function and ECDFs of acceptable and unacceptable samples, for parameters k1
, for both variants of the MPSA procedure. In the variant based on ODEs simulations and the error function which measures changes in the product P
trajectory, one clearly observes that parameter k3
significantly dominates parameter k1
, as far as sensitivity of the system is concerned. This is an expected result. Firstly, k3
is a rate parameter of a reaction which is directly responsible for a product creation. Secondly, from the Michaelis-Menten approximation [35
Figure 5 MPSA error surfaces, ECDFs and values of sensitivity indices for error calculated using deterministic model with the mean squared error of product trajectories (left column) and using stochastic model with the absolute difference of a value of the formula (more ...)
one can expect that, for values from Equation (2), variation of parameter k3 will be more influential, with respect to the product rate, than variation of parameter k1.
Interestingly, the results of the other variant of the MPSA procedure are significantly different; one observes that now k1 dominates k3. This may be ascribed to the particular choice of the formula (3) which calculates the average number of occurrences of the first reaction r1. Furthermore, an inspection of values of sensitivity indices given in Figure brings to light that the domination is not as definite as in the first variant of MPSA. Results demonstrate that an application of the probabilistic model checking technique may allow for revealing more subtle dependencies in the model, depending on the properties of interest.
MPSA combined with PMC may be applied as a pre-processing step which finds parameters that are insignificant for an analysis oriented on a very specific property of a model. This would provide a novel notion of a probabilistic abstraction [37
], i.e. property-specific reduction of the probabilistic model. However, for a successful application, the pre-processing should have low running time, compared to an analysis that follows. In our experiment this is not the case, as we run the exact PMC procedure, which is essentially the same one that would be ran during the further analysis. However, we conjecture that for the MPSA procedure the level of accuracy offered by PRISM is much too high. We suppose that satisfactory results may be obtained using an approximate approach, such as Monte Carlo model-checking [38
]. We plan to pursue this idea as a continuation of the work presented here.