Network Motifs Capable of Decoding Transcription Factor Dynamics

Transcription factors (TFs) can encode the information of upstream signal in terms of its temporal activation dynamics. However, it remains unclear how different types of TF dynamics are decoded by downstream signalling networks. In this work, we studied all three-node transcriptional networks for their ability to distinguish two types of TF dynamics: amplitude modulation (AM), where the TF is activated with a constant amplitude, and frequency modulation (FM), where the TF activity displays an oscillatory behavior. We found two sets of network topologies: one set can differentially respond to AM TF signal but not to FM; the other set to FM signal but not to AM. Interestingly, there is little overlap between the two sets. We identified the prevalent topological features in each set and gave a mechanistic explanation as to why they can differentially respond to only one type of TF signal. We also found that some network topologies have a weak (not robust) ability to differentially respond to both AM and FM input signals by using different values of parameters for AM and FM cases. Our results provide a novel network mechanism for decoding different TF dynamics.

of the previous cycle. If the peak-peak difference is smaller than 10 -4 (and the 3 trough-trough difference is smaller than 10 -4 ), it is considered to have reached a 36 steady state. We used the integral average of the current cycle as the output.

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Previous studies show that inhibition in transcription gene regulation usually has 38 a stronger effect than (overrides) activation 6,7 . In "strong inhibition" rule, we sum up 39 all the activation terms and multiply with equal weights ( = 1/ where n is the 40 number of activation terms) to ensure proper normalization.

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Analytical solution for two-node network topology with normalized equation 42 For a two-node network topology with positive regulation from TF to output node, 43 we can write the normalized differential equation as: 45 where x is the output level,  is the half-life of the protein, TF is the concentration of 46 transcription factor, K is the half-activation threshold, and n is the Hill coefficient.
With AM input, we set the input level as 1TF and obtain the steady-state output 48 by setting the left-hand-side to zero: With FM input, input alternates between 0 and 2TF (Fig. 2B). We assume that 51 after a long time, the FM output approaches a "stable" oscillation regardless of the 52 initial conditions with the period of 2T. When the input level starts with 2TF, and the 53 initial value of output x(0)=a, the output level after the half-period time T: After time T, the input level becomes 0. Under this condition, the initial output is 56 x(T) and the equation is The solution of this equation can be obtained after the remaining half-period time 59 T: (S10)

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If we simply calculate the average of the crest and the trough of the output protein 66 level, we have the average output From Eqs. (S10) and (S11), we have Similarly, we can obtain the output of the network topology with direct inhibitory 71 regulation only. With AM input signal, the steady-state output is The average of steady-state output with an FM input is Analytical solution for two-node network topology without normalized equation 76 For two-node network topology with positive regulation from TF to output node, 77 the gene expression can be described as follow ordinary differential equation: where the parameter vx is the maximum production rate, and  is the half-life of the 80 protein, TF is the concentration of transcription factor, K is the half-activation 81 threshold, and n is the Hill coefficient.

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With AM input, we set the input level as 1TF and obtain the steady-state output: With FM input, input alternates between 0 and 2TF. The steady-state average of 85 the crest and the trough of the output protein level is Similarly, for two-node network topology with direct inhibitory regulation only, 88 the gene expression can be described as follow equation: With AM input, the steady-state output is With FM input, the average of steady-state output is Comparing with Eqs. (S13)-(S14), Eqs. (S19)-(S20) have a common constant 95 factor. Thus, whether the equation (1) was normalized or not, it had no effect on the For a two-node network topology with positive regulation from TF to output node, 101 the normalized differential equation can be written as: 103 where x is the output level,  is the half-life of the protein, TF is the concentration of 104 transcription factor, K is the half-activation threshold, and n is the Hill coefficient.
For the oscillatory input with different number pulse (Fig. S10A), it is easy to 106 obtain the analytical expression between the number of pulse and the output level