## 2 Materials and Methods
### Dilation Functions  
Dilation functions (DFs) are reversible transformations of a \(D\)-dimensional real space with the aim of \"expanding\" and \"compressing\" the regions of the search space in order to facilitate the optimization process and to improve the performances obtained by any optimization method (e.g., PSO). Formally, DFs are defined as mappings in the unitary interval \(\mathcal{W}\): \([0,1]\rightarrow[0,1]\); these mappings can be re-scaled to any arbitrary interval by means of linear transformations. Such functions are applied individually to each dimension of the search space, and it is also possible to apply different DFs to different dimensions. In this work, we employed DFs generated by two different methods: (a) DFs based on control points [6]; (b) DFs automatically determined through evolutionary strategies [25].  
A control point \(\chi\) is a couple such that \(\chi\in(0,1)\times(0,1)\). A DF is defined by a set of \(Q+2\) control points \(\mathbf{\chi}=\{(0,0),\chi_{0},\ldots,\chi_{Q-1},(1,1)\}\), where the control points \(\chi_{i}\) (blue square dots in Figure 2), \(i=0,\ldots,Q-1\), are in ascending order (i.e., \(\chi_{i-1}\leq\chi_{i}\leq\chi_{i+1}\)) and the points \((0,0)\), \((1,1)\) are called boundary control points (which are represented by gray crosses in Figure 2). The control points map their values into the respective new values according to their definition (such values are represented in Figure 2 by red and yellow  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Molecular Species** & **Amount** \\ \hline \(S\) (substrate) & 200 \\ \(E\) (enzyme) & 100 \\ \(ES\) (enzyme–substrate complex) & 0 \\ \(P\) (product) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Initial state of the MM model.  
points). In order to map any given value of the unitary interval, a linear interpolation between the control points \(\in\chi\) is required. In such a way, the original value is applied to the linear function that interpolates the control points and the point is mapped into the \"dilated\" value (green points in Figure 2).  
In Figure 3, an example of DF based on control points is provided; in particular, an anti-log function is represented. The selection of this DF is determined by the log-uniform distribution followed by the stochastic parameters in biochemical models (i.e., they present a uniform distribution when a logarithmic scale is considered). Due to this consideration, in [26], we proved that it is possible to improve the performances of PE using particle swarm optimization (PSO) by means of a particle initialization according to a log-uniform distribution. The anti-log DF can be used to initialize the particles in regions of the search space characterized by fitness values of lower orders of magnitude.  
In general, the update rules of the positions of the particles performed in a linear fashion prevent the exploration of such regions. The use of the anti-log DF during the optimization process--and for the initialization of the particles--can overcome this problem by allowing the exploration of the search space in a logarithmic fashion by dilating the regions in the lowest orders of magnitudes. The DF is defined by the control points determined according to Equation (3):  
\[\chi_{i}=(i/Q,\exp(-8(1-i/Q))), \tag{3}\]  
for \(i=1,\ldots,Q\), with \(Q=9\).  
Figure 2: Mapping of parameters with control points.