Team:IIT Delhi/Deterministic Model
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 begind with for identifying topology based on following theorem.
Theorem:1 Consider a system ̇x = f(x), which is of ring in nature, and f is a monotone
function and in the form
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 satisfies 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 in computation and as well as
experimentally. However, these oscillators are more of a phaselag oscillator matching the
sinusoidal umbrella behavior. As the the philosophy behind our work is to design towards
square wave, we exploit the system parameters to produce relaxation oscillations. The
relaxation oscillator typically works on the principle to 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 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], xmi is the mRNA concentration level, xpi is the protein
transcription level, βm0 is the leaky expression, βmi is the production rate of mRNA, βpi
is the production rate of protein, γmi is the degradation/dilution of mRNA and γpi is
the degradation/dilution of protein for ith protein. The simulation results of the model
presented in Fig. . 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 αi can be considered as the protein production rate constant and γ as the degra-
dation constant. The simulation of 5th order comes model presented in FigX 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.
Theorem:1 Consider a system ̇x = f(x), which is of ring in nature, and f is a monotone function and in the form
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 satisfies 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 in computation and as well as experimentally. However, these oscillators are more of a phaselag oscillator matching the sinusoidal umbrella behavior. As the the philosophy behind our work is to design towards square wave, we exploit the system parameters to produce relaxation oscillations. The relaxation oscillator typically works on the principle to 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 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], xmi is the mRNA concentration level, xpi is the protein transcription level, βm0 is the leaky expression, βmi is the production rate of mRNA, βpi is the production rate of protein, γmi is the degradation/dilution of mRNA and γpi is the degradation/dilution of protein for ith protein. The simulation results of the model presented in Fig. . 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 αi can be considered as the protein production rate constant and γ as the degra- dation constant. The simulation of 5th order comes model presented in FigX 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.
