Fig 1.
Comparison of peak times associated with the step responses for a set of ten transfer functions in with five poles and four zeros.
It can be seen that the transfer function with zero gain (Response shown in dark blue) provides minimum peak time. The peak values have been kept constant.
Fig 2.
Workflow of the proposed methodology.
Any given protein network is first linearized, and the conditions on the A matrix are investigated, to ultimately derive admissible motifs for the desired functionality.
Fig 3.
Two node networks capable of adaptation subject to staircase disturbance.
The abbreviations LPA, NLPA, LPAFO, NLPAFO stand for linear perfect adaptation, non-linear perfect adaptation, linear perfect adaptation for once and non-linear perfect adaptation for once respectively. The network architecture and necessary values for this simulation have been provided in S1 Text.
Fig 4.
(a) shows the response of the output node for a three-node IFFLP topology. (b) shows the same for a three-node NFBLB. The oscillatory behavior can be attributed to the complex eigenvalues of the A. Similarly, (c) shows a non-oscillatory response of an NFBLB motif. (d) is the response of the output node of a network containing both the admissible network structure i. e. incoherent feedforward path and negative feedback. The network architecture and necessary values for this simulation have been provided in S1 Text.
Table 1.
Necessary mathematical conditions for adaptation.
These mathematical conditions can be translated to structural requirements assuming the monotone property of the underlying autonomous dynamical system.
Fig 5.
(a) shows the response of the output node for a five node IFFLP topology. (b) shows the same for a five node NFBLB with a hyperbolic response. The oscillatory behavior in (c) can be due to negative feedback, leading to complex eigenvalues of the underlying A matrix. (d) demonstrates the modular behavior of an NFBLB motif when connected to a downstream system. (e) is the response of the output node of an IFFLP network connected with a downstream system. Although the functionality of adaptation is not compromised, the oscillatory behavior is undoubtedly due to the negative feedback associated with the output of the IFFLP module and the downstream node. The network architecture and necessary values for this simulation have been provided in S1 Text.