## Deterministic Model

##
The biological networks are highly nonlinear and exhibit interesting phenotypical behaviour for certain operating conditions. One of such behaviour is the limit cycle in the
mathematical sense, which shows a sustained oscillations of protein levels in the cell. It
is quite interesting as well important to look for topologies which can produce such oscillations for different amplitude, oscillation and shape. Here, we have used a theoretical framework to begin with, for identifying topology based on following theorem.

Then, if the Jacobian of f and x has no repeated eigenvalues and has any eigenvalue
with positive real parts, then the system must have a consistent periodic orbit.

To design a squarewave oscillator, we used the theorem to idenify the biological system
which can satisfy such condition. One of classical example is Repressillator (Elowitz et.
al.) or 5n1 ring oscillator (Murray et. al.). This kind of oscillator is based on negative
feedback with delay and able to produce stable limit cycle computationally and as well as
experimentally. However, these oscillators are more of a phaselag oscillator matching the
sinusoidal umbrella behavior. As philosophy behind our work is to design
square wave, we exploit the system parameters to produce relaxation oscillations. The
relaxation oscillator typically works on the principle of level of concentration, where once
the level is reached it relaxes there for some additional time and falls back to another
level and resides there for some till it jumps back (slowly). The time evolution of such
trajectories portray a square wave-ish in state-space.

The dynamical model of the five node oscillator can be written as;

where i ∈ [0 = 5, 1, 2, 3, 4, 5], x_{mi} is the mRNA concentration level, x_{pi} is the protein
transcription level, β_{m0} is the leaky expression, β_{mi} is the production rate of mRNA, β_{pi}
is the production rate of protein, γ_{mi} is the degradation/dilution of mRNA and γ_{pi} is
the degradation/dilution of protein for ith protein. The simulation results of the model
presented in Fig. below. It is evident the such system can exhibit a oscillation resembling a
squarewave.

As the dynamical model comprises of two time-scale, one can use the singular perturbation analysis to reduce the model in to smaller one, i.e. 5th order, as discussed earlier.
The simpler version of the model, where multiple constants product are clubbed into one,
can be reproduced as follows,

where α_{i} can be considered as the protein production rate constant and γ as the degradation constant. The simulation of 5th order model presented in figure is almost identical to the simulation for full order model. Both of the model can exhibit sustained square wave like response of arbitrary initial conditions.

Then, if the Jacobian of f and x has no repeated eigenvalues and has any eigenvalue with positive real parts, then the system must have a consistent periodic orbit.

To design a squarewave oscillator, we used the theorem to idenify the biological system which can satisfy such condition. One of classical example is Repressillator (Elowitz et. al.) or 5n1 ring oscillator (Murray et. al.). This kind of oscillator is based on negative feedback with delay and able to produce stable limit cycle computationally and as well as experimentally. However, these oscillators are more of a phaselag oscillator matching the sinusoidal umbrella behavior. As philosophy behind our work is to design square wave, we exploit the system parameters to produce relaxation oscillations. The relaxation oscillator typically works on the principle of level of concentration, where once the level is reached it relaxes there for some additional time and falls back to another level and resides there for some till it jumps back (slowly). The time evolution of such trajectories portray a square wave-ish in state-space.

The dynamical model of the five node oscillator can be written as;

where i ∈ [0 = 5, 1, 2, 3, 4, 5], x

_{mi}is the mRNA concentration level, x

_{pi}is the protein transcription level, β

_{m0}is the leaky expression, β

_{mi}is the production rate of mRNA, β

_{pi}is the production rate of protein, γ

_{mi}is the degradation/dilution of mRNA and γ

_{pi}is the degradation/dilution of protein for ith protein. The simulation results of the model presented in Fig. below. It is evident the such system can exhibit a oscillation resembling a squarewave.

As the dynamical model comprises of two time-scale, one can use the singular perturbation analysis to reduce the model in to smaller one, i.e. 5th order, as discussed earlier. The simpler version of the model, where multiple constants product are clubbed into one, can be reproduced as follows,

where α

_{i}can be considered as the protein production rate constant and γ as the degradation constant. The simulation of 5th order model presented in figure is almost identical to the simulation for full order model. Both of the model can exhibit sustained square wave like response of arbitrary initial conditions.