This slideshow presents the analysis of a non-linear dynamics approach to one such model, proposed by Lewis et al. in 1977, using a single element biochemical switch.

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Non-Linear Analysis of Biochemical Switches
Shaleen Kumar Gupta Visharad Bansal
CS304 Non-Linear Science Course Project
Dhirubhai Ambani Institute of Information and Communication
Technology, Gandhinagar
April 25, 2016
Shaleen Kumar Gupta, Visharad Bansal Non-Linear Analysis of Biochemical Switches
Dhirubhai Ambani Institute of Information and
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Analyzing Single Element Biological Switches
Problem Description
Figure: An example of Gene Activation
Lewis et al.[1] proposed modeling using Biochemical Switches
The Equation:
dg
dt
= k1so − k2g +
k3g2
k4
2
+ g2
(1)
The gene concentration, biochemical signal, linear decay and the
feed-forward term
Shaleen Kumar Gupta, Visharad Bansal Non-Linear Analysis of Biochemical Switches
Dhirubhai Ambani Institute of Information and
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Approach
Non-Dimensionalizing the equation
dx
dτ
= s − rx +
x2
1 + x2
(2)
Key Analysis Points
In the absence of the trigger
In the presence of the trigger
s-r variations
Shaleen Kumar Gupta, Visharad Bansal Non-Linear Analysis of Biochemical Switches
Dhirubhai Ambani Institute of Information and
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Absence of the Biochemical Trigger
s=0
The Equation:
dx
dτ
= −rx +
x2
1 + x2
(3)
Figure: Variation in Fixed Points
ro=0.5 is the bifurcation point
Figure: Bifurcation Plot
(no biochemical trigger)
Shaleen Kumar Gupta, Visharad Bansal Non-Linear Analysis of Biochemical Switches
Dhirubhai Ambani Institute of Information and
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In the presence of the Biochemical trigger
s not equal to zero
The Equation:
dx
dτ
= s − (rx −
x2
1 + x2
) (4)
Figure: Variation in Fixed Points(r < ro)
Shaleen Kumar Gupta, Visharad Bansal Non-Linear Analysis of Biochemical Switches
Dhirubhai Ambani Institute of Information and
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In the presence of the Biochemical trigger
s not equal to zero
The Equation:
dx
dτ
= s − (rx −
x2
1 + x2
) (5)
Figure: Variation in Fixed Points(r > rm)
Shaleen Kumar Gupta, Visharad Bansal Non-Linear Analysis of Biochemical Switches
Dhirubhai Ambani Institute of Information and
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Varying s-r
The s-r plane
Figure: The s-r Bifurcation Plane
Shaleen Kumar Gupta, Visharad Bansal Non-Linear Analysis of Biochemical Switches
Dhirubhai Ambani Institute of Information and
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Biological Inferences
In the absence of the biochemical trigger, the activation depends on
the initial gene concentration However, beyond a certain decay rate,
the gene gets ”switched off”
As the triggering enzyme is released, the gene gets activated. Once
activated, it is independent of the concentration of the enzyme, and
remains activated even when the trigger vanishes. This biological
phenomena is captured by the model beautifully.
Shaleen Kumar Gupta, Visharad Bansal Non-Linear Analysis of Biochemical Switches
Dhirubhai Ambani Institute of Information and
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