\[mmfigure1\]e shows the skewness of the distribution over time, calculated as $$\begin{aligned}
\gamma=\frac{\left<(x_1-\mu_1)^3\right>}{\sigma_{11}^3}=\frac{\left<(x_1-\mu_1)^3\right>}{\left<(x_1-\mu_1)^2\right>^{3/2}}.\end{aligned}$$ For a normal distribution the skewness is zero. The skewness is approximated well using the moment approximation method up to three moments. The kurtosis, given by $$\begin{aligned}
\gamma_2=\frac{\left<(x_1-\mu_1)^4\right>}{\sigma_{11}^4}=\frac{\left<(x_1-\mu_1)^4\right>}{\left<(x_1-\mu_1)^2\right>^{4/2}},\end{aligned}$$ indicates the thickness of the tails of the probability distribution, relating to the probability of outliers. For a normal distribution, the kurtosis is equal to 3. When four moments are used to approximate the system, the approximation of the kurtosis that we obtain from the SSA is not as close as when also higher moments, here six moments, are included in the calculation. This illustrates that agreement between lower-order moments does not guarantee that higher-order moments will also agree. This problem is likely exacerbated for more complex models, e.g. those exhibiting non-linear dynamics.

P53 system
----------

Finally, we investigate the oscillatory p53 system[@Geva2006], which consists of three proteins connected through a nonlinear feedback loop: p53, precursor of Mdm2 and Mdm2. The system can be written in terms of six propensities, $$\begin{aligned}
\begin{array}{cc}\vspace{2mm}
a_1: k_1;&\hspace{7mm} a_2: k_2x\\\vspace{2mm}
a_3: k_{3}\frac{xy}{x+k_{7}};&\hspace{7mm} a_4: k_4x\\
a_5: k_5y_0; &\hspace{7mm} a_6:k_6y,
\label{p53system}
\end{array}\end{aligned}$$ and the stoichiometry matrix $$\begin{aligned}
S=\left[ \begin{array}{cccccc}
1 & -1 & -1 & 