Time-frequency analysis of yeast metabolic responses to environmental change
We initially examined two yeast systems, the oleate (OLE
and the galactose (GAL
core transcriptional networks in Saccharomyces cerevisiae
, which both respond to carbon source switching, but do so with very different network topologies (). The analysis of the spectral response of these systems to time varying inputs of simulated oleate and galactose concentrations makes it possible to study how the circuit structure and parameters in these two systems affect their ability to balance noise suppression with responsiveness. To this end, the time-frequency characteristics of the network input and output signals were extracted from their spectrograms, which are calculated using the short-time Fourier transform (STFT) 
. The spectrogram illustrates how the frequency content of a signal varies with time. The value Xi,j
of each element in the spectrogram indicates the power of the signal at a particular frequency (fi
) and at a particular time (tj
) ( and S1
and Text S1
The generalized time-frequency analysis of the dynamical properties of molecular networks.
Characteristics of the signal and the network response can be quantified by integration across all frequency bands. Two characteristics of the networks, noise suppression
(low-pass filtering) and responsiveness
(detail preservation), can be inferred from the spectrograms. First, as a measure of circuit noise suppression, the spectrogram coefficients for each frequency band were summed over time and the mean frequency of the signal (μ) was calculated. The noise suppression characteristic (ξ) is defined as a ratio of mean frequencies of the stimulus and the system response (ξ
, Figure S1
). A greater ξ corresponds to a system with a greater ability to filter high-frequency input fluctuations. Second, in each frequency band, the relationship between the total variation of the signal power at the output and input of the system serves as a measure of responsiveness. Specifically, the total variation of the spectrogram coefficients within each frequency band (Vi
, Figure S1I
) was calculated and the system responsiveness (ρ) is defined as the inverse divergence between distributions of normalized input (Viin*
) and output (Viout*
) variations across all frequency bands. A greater ρ corresponds to a more responsive system.
To investigate the different features of the GAL and OLE networks, their noise suppression (ξ) and responsiveness (ρ) characteristics were calculated based on their model-predicted responses to simulated random, noisy time-varying stimuli. The TFA analysis revealed that the two networks have distinct noise suppression and responsiveness properties. The OLE network effectively filters high-frequency fluctuations of the stimulus, thus acting as a low-pass filter. At the same time, it is relatively unresponsive to transient stimulus variations. By comparison, the GAL network is highly responsive but does not filter high frequency fluctuations as effectively (). Thus, each system exhibits a different noise suppression-responsiveness trade-off, suggesting that these properties have selective advantages in different contexts.
Feed-forward loops of the OLE network endow the system with ability to filter high-frequency fluctuations of the stimulus whereas feedback loops of the GAL network confer responsiveness to the environmental changes.
The shift from glucose to oleate involves a substantial commitment to build and maintain new organelles (peroxisomes) that are responsible for metabolizing the new carbon source (fatty acids) 
, a switch from fermentative to non-fermentative metabolism (requiring mitochondrial respiration), as well as the coordination of additional responses to the stress associated with exposure to fatty acids 
. Therefore, the nature of the oleate response demands that the system be capable of filtering high frequency fluctuations of the environmental stimulus, which may otherwise inappropriately commit the cell to significant morphological and metabolic reorganization. By contrast, the switch from glucose to galactose requires relatively few enzymes and transporters to convert galactose into glucose-1P for glycolysis 
. Thus, while the ability of the cell to be highly responsive to galactose appears to come at the expense of noise suppression, such noise suppression can be sacrificed to a greater extent than during the oleate response.
A major difference between these networks lies in their topologies. The GAL
network is comprised of dual positive and negative feedback loops (FBLs) whereas the OLE
network is comprised of a positive FBL and two (positive and negative) feed-forward loops (FFLs) (). By removing coherent positive and negative FFLs (Adr1p and Oaf3p nodes) and leaving only the positive FBL (on PIP2
), the ensemble of the calculated TFA statistics resembles that of the GAL
network (). To investigate how topology of the OLE
network contributes to noise suppression and responsiveness of the system, different configurations of the OLE
network were explored. Density distributions of the ξ and ρ for adr1
Δ-, “no positive feedback”-, “no positive feedback”-adr1
Δ- and “no positive feedback”-oaf3
networks were calculated (Figures S4
and Table S4
). The “no positive feedback”-OLE
model represents the OLE
network where Pip2p does not upregulate its own gene PIP2
but upregulates only its target genes.
The distributions of the TFA characteristics for the adr1
Δ- and oaf3
models reveal that both Adr1p and Oaf3p individually increase the noise suppression and decrease the responsiveness of the OLE
network (Figures S6D, S6E
, S7D and S7E
and Table S4
). Interestingly, the oaf3
model has an even more narrowed noise suppression distribution with a lower mean ξ value than the adr1
model (Table S4
). The distributions of the ξ and ρ for the “no positive feedback”-OLE
show that the Pip2p positive feedback decreases noise suppression and increases responsiveness of the OLE
network (Figures S6F
and Table S4
). The responsiveness/noise suppression “TFA clouds” for the “no positive feedback”-adr1
and “no positive feedback”-oaf3
models (Figures S4G and H
) are similar to the “TFA clouds” for the adr1
Δ- and oaf3
models. This demonstrates that Adr1p and Oaf3p have more dominant contributions to the noise suppression and responsiveness characteristics than the Pip2p positive feedback. Overall these results reflect the nonlinear relationships between regulators in this regulatory network and suggest that the positive and negative FFLs of the OLE
network serve to filter high frequency environmental fluctuations.
To examine how the noise suppression and responsiveness TFA characteristics depend on the type of random time-varying stimuli, the distributions of the ξ and ρ were calculated separately for each of the stimulus types (Figures S5
, and S7
and Table S4
). The distribution of the noise suppression characteristic for the random sinusoidal stimuli is shifted toward higher values of ξ compared to the random “block” and “saw” stimuli regardless of the network type. This suggests that all of the biomolecular systems investigated here have a greater ability to suppress the noise of smoothed (random “sinusoidal”) rather than more abrupt (random “block” or “saw”) stimuli. The distribution of the responsiveness characteristic for the random sinusoidal stimuli is shifted toward lower values of ρ for the WT-, “no positive feedback”-OLE
models and higher for WT-GAL
Δ-, “no positive feedback”-oaf3
Δ- and adr1
models compared to the random “block” and “saw” signals. The results indicate that the time-frequency characteristics of a biomolecular network does indeed depend on the nature of the stimulus, further supporting the approach of exploring the network responses to a large ensemble of random time-varying stimuli.
To understand the responsiveness and noise suppression properties of networks, typified by the interlinked positive and negative FBLs of the GAL network and the interlinked negative and positive FFLs of the OLE network, the parameters corresponding to the strengths of the FFLs and FBLs were systematically altered. The strength of each loop was independently varied (2,642 parameter sets in total) and each parameter set was explored with 100 randomized model inputs. The noise suppression and responsiveness characteristics of the networks as a function of network parameters were determined by TFA and displayed as heat maps (). The resulting “portraits” expose fundamental differences inherent to each of the networks and demonstrate how network dynamics can be predicted for these and evolutionarily conserved networks.
The noise suppression (ξ) and the responsiveness (ρ) of the networks are sensitive to both to the topologies and rate parameters.
For the OLE
network, TFA revealed that network behavior is characterized by appropriately tuned opposing positive and negative FFLs. The network is maximally stable along the arc-like front shown in . Increasing the strengths of the positive and/or negative FFLs above this front results in non-physiological responses, characterized by a reversed directionality of the output relative to the input signals (Figure S8
). Decreasing the strengths of these FFLs below the front results in lowered noise suppression at the expense of increased system responsiveness (). As might be expected for a nonlinear system, an increase in noise suppression may not be reflected by a corresponding decrease in responsiveness of the same magnitude and vice versa. While the arc-shaped front represents the range of parameters where the noise suppression and responsiveness characteristics are similar to the wild-type (WT) state (Figures S10A and S10B
and Table S5
), there are other trade-offs as parameters vary along this arc; for example, the amplitude of the response changes as a function of the strengths of the positive and negative FFLs (Figure S9
By contrast, the dual negative and positive FBL in the GAL
network is highly responsive over a broad range of parameters extending along the diagonal from the left bottom corner of the heat map (). The strengths of positive and negative FBLs can be varied over an extensive parameter range, while maintaining near WT responsiveness and noise suppression, indicating a remarkable level of robustness of the system (Figures S10C and S10D
and Table S5
). Indeed, while the GAL
network does have the capacity to act as a low-pass filter 
, a significant deviation from WT parameters would be required for this system to reach the effectiveness of the OLE
network in terms of low-pass filtering (). Based on these simulations, decreasing the strength of the negative FBL with the fixed strength of the positive FBL, would increase the noise suppression of the network and decrease its responsiveness ().
To investigate how the TFA portraits depend on the type of random time-varying stimuli, the heat maps presented in were split into three separate heat maps, each of which represents an averaged ξ/ρ over 33/34/33 random “block”/sinusoidal/“saw” stimuli. The separated heat maps for the OLE
) and the GAL
) models show similar patterns within each model for different types of stimuli; however, the ranges of ξ and ρ values (as the strengths of FFLs and FBLs are changed) differ depending on type of stimulus. For example, the difference between maximum and minimum ξ and ρ values of the “sinusoidal” heat maps is greater compared to the “block” or “saw” stimuli regardless of the network. This analysis highlights that TFA portraits (in this case, the projection onto the plane of positive and negative FFL/FBL strengths) tend to be robust to changes in stimulus type in terms of the patterns of ξ and ρ changes in the parameter space (as shown in Figures S11
The LPS-induced regulatory network in macrophages
To investigate the extent to which overall network architecture (versus biochemical parameters) defines the dynamical properties of a system, we examined the LPS and Toll-like receptor 4 (TLR4)-induced regulatory circuit from mouse macrophages by TFA, which, like the OLE
network, is characterized by overlapping positive and negative coherent FFLs. In this regulatory network NF-κB, ATF3 and C/EBPδ transcription factors coordinate the expression of cytokine encoding target genes in response to LPS 
(). The network is also interesting because it must be tightly regulated to respond vigorously to the presence of a pathogen, but at the same time must remain in check to avoid uncontrolled inflammatory responses. Analogously to the OLE
models, the behavior of the LPS-induced network model was initially investigated in the wild-type state in the presence of 3000 random and noisy stimuli (Figures S4C
and Table S4
). The WT-LPS model has a similar responsiveness/noise suppression distribution as the OLE
model, i.e. biased toward more noise suppression. The distribution of the noise suppression and responsiveness TFA characteristics for different stimulus types are presented in Figures S5C
and Table S4
LPS-induced regulatory network driven by NF-κB, ATF3 and C/EBPδ transcription factors lies at the boundary of responsiveness and noise suppression.
Similarly to the OLE
model, the strength of each FFL of the LPS-induced network was independently varied and each parameter set was explored with 100 randomized model inputs. The heat maps of the TFA characteristics resulting from this simulation were significantly different from those of the OLE
network emphasizing that the network architecture itself is not sufficient to define the dynamical characteristics of the system. Indeed, while the OLE
network appears tuned toward greater noise suppression, the LPS network appears to be tuned to lie at the boundary of responsiveness and noise suppression (, S10E, S10F
). This is perhaps not surprising considering that macrophages must be finely tuned to respond to the presence of a foreign substance, yet if cellular responses vary dramatically with the character of the signal, variations in cytokine release has the potential to lead to inappropriate inflammatory responses.
The ability to be poised at the boundary of responsiveness and stability is a hallmark of systems operating in a critical regime between order and disorder. A recent study of mouse macrophages stimulated with a variety of pathogen associated molecular patterns provided evidence that macrophages gene expression dynamics are indeed critical 
, supporting the conclusions drawn from the TFA analysis.
Deviation from the parameters that define the wild-type network has a dramatic effect on the network behavior. Increasing the strength of LPS/TLR4/NFkB activation from the WT state increases the noise suppression of the network, but at the cost of reducing responsiveness. Similarly, decreasing the strength of the activating arm increases responsiveness, but at the cost of reducing noise suppression. Changing the strength of ATF3 repression leads to an opposite pattern with less dramatic changes in network behavior. Thus, altering the strength of either the positive or negative FFLs leads to networks that are predicted to change the finely tuned balance between noise suppression and responsiveness that is critical to a controlled inflammatory response.