to identify TF-genes, and XYZ11111111 **to identify P-genes, as they have both**
the same probability of appearing (and no overlapping of their signatures).

Note that a previous approach for extracting an output signal from this model exists [9], where a random site of the genome is used as a regulation site, but despite the results achieved, it does not offer the same degree of flexibility as the technique now presented.

Dynamic analysis **Several possibilities exist, when choosing the dynamic equation to use when calculating the concentration of P-proteins. In order to keep**
with the nature of the model, equations based on the calculation of concentration of TF-proteins were tested; the following equation was used:

$$c_{i}^{t}=c_{i}^{t-1}+\delta(e_{i}-h_{i})-\Phi(1.0)$$
i + δ(ei − hi) − Φ(1.**0) (4)**
where c t i is the concentration of the P-protein at time t, c t−1 i**its concentration**
at time t − 1, ei and hi are calculated as before at time t − 1, and Φ(1.**0) is a**
scaling factor, ensuring the sum of all P-proteins concentrations3**is 1**.0.

This equation was chosen as it seems to give P-genes similar dynamics to TF-genes, for both random genomes and DM-genomes

## 3 The Problem: Single-Pole Balancing

The potential of using gene regulatory networks as a representation for an Evolutionary Algorithm lies in their possibly rich, non-linear dynamics [9]. A famous dynamic control benchmark is the pole-balancing problem [10,11], also **known** as the inverted pendulum problem. It consists in controlling, along a finite one dimensional track, a cart on which a rigid pole is attached. The command is a bang-bang command: the user can apply a constant force to either side of the cart. The objective is to keep the pole balanced on top of the cart, **while keeping** the cart within the (limited) boundaries of the track. There are four input variables associated with this problem:
x ∈ [−2.4, 2.4] m **is the position of the cart, relative to the centre;**
θ ∈ [−12, 12] ◦**is the angle of the pole with the vertical;**
x˙ ∈ [−1, 1] m/s **is the velocity of the cart on the track;**
˙θ ∈ [−1.5, 1.5] ◦/s **is the angular velocity of the pole.**
The physical simulation of the cart and pole movements is modelled by the following equations of motion:

![0_image_0.png](0_image_0.png)

3 Concentrations of TF-proteins and P-proteins are normalised independently.