We tested the MCMC-HFM algorithm for parameter inference in the kinetic toy model and the mathematical model of apoptosis signal transduction network. In the former, we inferred the kinetic parameters mainly related to positive feedback. As a result, MCMC-HFM could reliably infer the kinetic parameters with use of hybrid fitness measures. In the apoptosis model, we inferred the kinetic parameters which are related to the 3,4,5-Trimethoxyphenylacetic acid implicit positive feedback. As a result, MCMC-HFM could reliably infer the kinetic parameters, especially those of which values were experimentally estimated. Inferred parameter sets reproduced the function approximating the distribution of execution time of caspase-3. In the current study, the function was assumed based on the experimental result. This function can be replaced with explicit experimental data represented as a histogram in 
possible future applications in the same way as the switching time of caspase-3 activation. For inference, we define the representative parameter values of inference by the values at peak of each marginal distribution. This definition could reliably infer the kinetic parameters with use of hybrid fitness measures. Another definition of representative parameter values is the values at peak of joint distribution of all inferred parameters. We used the uniform distribution as the proposal distribution in MCMC algorithms. Actually, almost the same results shown in Figure 3, 4, 6�C13 and S1�C5 were obtained when we used the normal distribution as the proposal distribution in MCMC. These results indicate that MCMC-HFM is a useful and reliable method for parameter inference, and the results presented in the current study are reproducible. In the apoptosis model, by the credible intervals of inferred parameters, joint probability distributions and correlation coefficients between inferred parameters, we could also specify the important relationships between kinetic parameters and corresponding biochemical processes, especially for irreversibility and execution time of caspase-3 activation. In the process of parameter inference by Bayesian statistics with MCMC, we can usually obtain many parameter sets, which can be used to understand and specify important biochemical processes in the target system as shown in the current study. In the apoptosis model, inferred parameter sets reproduced well the assumed function of execution time of caspase-3, but did not well reproduced the assumed function of the switching time of caspase-3. This is not because of the restriction by two qualitative conditions, Folinic acid calcium salt pentahydrate bistability and irreversibility. Because when we performed parameter inference only with a quantitative condition, “Ts”, calculated histogram of Ts was not consistent with the assumed function. Thus, one possibility of inconsistency might be other experimentally-unknown kinetic parameters in the model are not correct. Legewie et al’s model has a number of experimentally unknown kinetic parameters. If we tried to infer all the unknown parameters in the model, the assumed function of the switching time of caspase-3 might be reproduced. Otherwise the mathematical model might need to be improved to be able to reproduce experimental results shown by Albeck et al.. We note that our calculation could not well reproduce the switching time of caspase-3 activation in the distribution level, but most of parameter sets showed acceptable switching time around ten and a few minutes compared with the experimental result. Robustness analysis of kinetic parameters in systems biology sometimes assumes the size of the parameter space as the measure of robustness. For example, the volume of the ellipsoid containing 95% of the parameters generated by Monte Carlo method was calculated and assumed as the measure of robustness.