number of time points taken into account. The larger error for the mean when using the deterministic approach is due to small differences in the first part of the trajectory, the decreasing part, where the contributions of the higher order central moments are largest. [The larger errors calculated for the third central moment are due to the fluctuations that are still present in the third central moment trajectory calculated based on 100,000 SSA runs. Increasing the number of SSA runs would reduce this effect.]{}

  $\epsilon [\%]$    deterministic     2m       3m       4m       5m       6m    
  ----------------- --------------- -------- -------- -------- -------- -------- --
  $\mu_1$                0.300       0.0545   0.0546   0.0546   0.0546   0.0546  
  $M_{x_1^2}$                        0.961    0.803    0.804    0.801    0.803   
  $M_{x_1^3}$                                  18.3     18.1     18.2     18.2   
  $M_{x_1^4}$                                           2.39     1.33     1.83   

  : \[table1\]Error between mean, second and third central moment calculated with SSA and approximation methods for the dimerization system.

Michaelis-Menten enzyme kinetics
--------------------------------

We next look at Michaelis-Menten enzyme kinetics, where an enzyme, $E$, and substrate, $S$, first bind to form a complex, $SE$; following this, the complex can dissociate into the original components $S$ and $E$, or $S$ can be converted into the product, $P$, $$\begin{aligned}
S+E \xrightarrow{k_1} SE\nonumber\\
SE \xrightarrow{k_2} S+E\\
SE \xrightarrow{k_3} P+E\nonumber\end{aligned}$$ The system 