Article Information

• PDF (385 kb)
• Full Text with Large Figures
• Supporting Material
• Cited by in Scopus (4)

PubMed

• Articles by Cheemeng Tan
• Articles by Lingchong You

Related Articles

• Concerted Simulations Reveal How Peroxidase Compound III For...

  Concerted Simulations Reveal How Peroxidase Compound III Formation Results in Cellular Oscillations Biophysical Journal, Volume 85, Issue 3, 1 September 2003, Pages 1421-1428 Razif R. Gabdoulline, Ursula Kummer, Lars F. Olsen and Rebecca C. Wade Abstract A major problem in mathematical modeling of the dynamics of complex biological systems is the frequent lack of knowledge of kinetic parameters. Here, we apply Brownian dynamics simulations, based on protein three-dimensional structures, to estimate a previously undetermined kinetic parameter, which is then used in biochemical network simulations. The peroxidase-oxidase reaction involves many elementary steps and displays oscillatory dynamics important for immune response. Brownian dynamics simulations were performed for three different peroxidases to estimate the rate constant for one of the elementary steps crucial for oscillations in the peroxidase-oxidase reaction, the association of superoxide with peroxidase. Computed second-order rate constants agree well with available experimental data and permit prediction of rate constants at physiological conditions. The simulations show that electrostatic interactions the rate of superoxide association with myeloperoxidase, bringing it into the range necessary for oscillatory behavior in activated neutrophils. Such negative electrostatic steering of enzyme-substrate association presents a novel control mechanism and lies in sharp contrast to the electrostatically-steered fast association of superoxide and Cu/Zn superoxide dismutase, which is also simulated here. The results demonstrate the potential of an integrated and concerted application of structure-based simulations and biochemical network simulations in cellular systems biology. Abstract | Full Text | PDF (335 kb)

• The Fundamental Organization of Cardiac Mitochondria as a Ne...

  The Fundamental Organization of Cardiac Mitochondria as a Network of Coupled Oscillators Biophysical Journal, Volume 91, Issue 11, 1 December 2006, Pages 4317-4327 Miguel Antonio Aon, Sonia Cortassa and Brian O’Rourke Abstract Mitochondria can behave as individual oscillators whose dynamics may obey collective, network properties. We have shown that cardiomyocytes exhibit high-amplitude, self-sustained, and synchronous oscillations of bioenergetic parameters when the mitochondrial network is stressed to a critical state. Computational studies suggested that additional low-amplitude, high-frequency oscillations were also possible. Herein, employing power spectral analysis, we show that the temporal behavior of mitochondrial membrane potential (ΔΨ) in cardiomyocytes under physiological conditions is oscillatory and characterized by a broad frequency distribution that obeys a homogeneous power law (1/) with a spectral exponent, =1.74. Additionally, relative dispersional analysis shows that mitochondrial oscillatory dynamics exhibits long-term memory, characterized by an inverse power law that scales with a fractal dimension () of 1.008, distinct from random behavior (=1.5), over at least three orders of magnitude. Analysis of a computational model of the mitochondrial oscillator suggests that the mechanistic origin of the power law behavior is based on the inverse dependence of amplitude versus frequency of oscillation related to the balance between reactive oxygen species production and scavenging. The results demonstrate that cardiac mitochondria behave as a network of coupled oscillators under both physiological and pathophysiological conditions. Abstract | Full Text | PDF (787 kb)

• A Mathematical Model of Airway and Pulmonary Arteriole Smoot...

  A Mathematical Model of Airway and Pulmonary Arteriole Smooth Muscle Biophysical Journal, Volume 94, Issue 6, 15 March 2008, Pages 2053-2064 Inga Wang, Antonio Z. Politi, Nessy Tania, Yan Bai, Michael J. Sanderson and James Sneyd Abstract Airway hyperresponsiveness is a major characteristic of asthma and is believed to result from the excessive contraction of airway smooth muscle cells (SMCs). However, the identification of the mechanisms responsible for airway hyperresponsiveness is hindered by our limited understanding of how calcium (Ca), myosin light chain kinase (MLCK), and myosin light chain phosphatase (MLCP) interact to regulate airway SMC contraction. In this work, we present a modified Hai-Murphy cross-bridge model of SMC contraction that incorporates Ca regulation of MLCK and MLCP. A comparative fit of the model simulations to experimental data predicts 1), that airway and arteriole SMC contraction is initiated by fast activation by Ca of MLCK; 2), that airway SMC, but not arteriole SMC, is inhibited by a slower activation by Ca of MLCP; and 3), that the presence of a contractile agonist inhibits MLCP to enhance the Ca sensitivity of airway and arteriole SMCs. The implication of these findings is that murine airway SMCs exploit a Ca-dependent mechanism to favor a default state of relaxation. The rate of SMC relaxation is determined principally by the rate of release of the latch-bridge state, which is predicted to be faster in airway than in arteriole. In addition, the model also predicts that oscillations in calcium concentration, commonly observed during agonist-induced smooth muscle contraction, cause a significantly greater contraction than an elevated steady calcium concentration. Abstract | Full Text | PDF (752 kb)

• …more

Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 93, Issue 11, 3753-3761, 1 December 2007

doi:10.1529/biophysj.107.110403

Biophysical Theory and Modeling

Noise-Limited Frequency Signal Transmission in Gene Circuits

Cheemeng Tan^*, Faisal Reza^*, † and Lingchong You^*, †, ,  

^* Department of Biomedical Engineering, Duke University, Durham, North Carolina
^† Institute for Genome Sciences and Policy, Duke University, Durham, North Carolina

Address reprint requests to Lingchong You, Dept. of Biomedical Engineering, Duke University, Durham, NC 27708.

To maintain their normal physiology, cells must process diverse signals such as temperature, pH, and nutrient concentrations. This process can be conceptualized as consisting of three steps: encoding, transmission, and decoding. One strategy employed to encode information is using the amplitude domain of a signal. Such amplitude signals are predominantly based on concentrations of signaling molecules.

Alternatively, information may be encoded in the frequency domain of an oscillatory signal. Oscillatory signals have been observed in diverse cellular processes, such as circadian clocks 1, segmentation clocks 2, Ca²⁺ signaling pathways 3,4, p53 DNA repair pathways 5, NF-κB pathways 6, and cell cycles 7,8,9. These signals can directly control the spatiotemporal dynamics of downstream cellular processes, and are likely to be the prevalent mechanism for regulating cellular processes where oscillations are involved. For example, the frequency of Notch protein oscillations in the posterior presomitic mesoderm (PSM) of mice controls cyclic expression of downstream genes such as HES110 and Lfng11, which in turn modulates periodic patterning in somite development 12,13,14. Loss of these oscillations, through perturbation of either the Notch protein or the downstream genes, will cause chaotic segmental organization 15,16,17,18. In mammalian circadian clocks, the mCLK/BMAL1 protein oscillations control cyclic expression of an albumin-D-binding protein containing a basic leucine zipper. The albumin-D-binding protein regulates critical cellular processes, such as circadian sleep consolidation and rhythmic EEG activity 19. Disruption of the oscillation frequency may be detrimental to the circadian rhythm and ultimately to human physiology 20.

In other cases, frequencies of oscillatory signals are correlated with activities of specific cellular processes that further suggest frequency encoding. During inflammatory response of T-lymphocytes, [Ca²⁺] oscillation frequencies of ∼0.01/s were found to maximize expression of interleukin-2 and interleukin-8 cytokines 21. During growth and differentiation of HeLa cells, [Ca²⁺] oscillation frequencies of ∼0.008/s significantly increased the activity of Ras proteins 22. Similarly, the frequency of NF-κB oscillations was proposed to regulate the activity of downstream genes involved in cell division, apoptosis, and inflammation response 6. In addition, chemical networks with specific architectures may function as frequency decoders 23. Table S1 lists many biological systems where frequency signal encoding is potentially adopted.

Regardless of encoding strategies, cellular signals are processed in the presence of noise, which arises from reactions between small number of molecules and perturbations inside a cell or from the environment 24,25,26,27,28,29,30,31,32. The presence of noise presents an important challenge for cellular signal processing. To understand the effects of noise on biological systems, considerable research has been performed to study noise generation and propagation 33,34, noise frequency modulation by negative feedback 35,36, and noise filtering in bacterial chemotaxis pathway 37. However, little is understood about how cellular noise impacts transmission of frequency signals. There is evidence for small amplitude oscillations in genome wide transcription of yeast respiratory cycles 38,39, which could be affected by cellular noise. Therefore, if cells do use frequency encoding, what are the basic characteristics of frequency-signal processing, in terms of processing speed and fidelity? How do these characteristics depend on the biochemical parameters and the architecture of underlying cellular networks? More generally, have cells evolved to take advantage of different strategies to properly control the transmission?

We address these questions by analyzing the impact of cellular noise on frequency-signal transmission in simple gene circuits. By using both analytical and numerical methods, we define a metric—critical frequency—that quantifies the speed limit of frequency-signal transmission. We argue that the critical frequency is an intrinsic property of a cellular network. Strategies to vary the critical frequency will introduce a tradeoff between speed and metabolic cost of signal transmission: an increase in transmission speed will come with an increase in metabolic cost. Our results further indicate that the critical frequency is dependent on network architectures. Our findings suggest a classification scheme for gene regulatory motifs, such as feedback regulations, based on their performance in transmitting frequency signals. Furthermore, insights into fundamental fidelity and speed limits may guide gene circuit designs for cellular computing in the long term 40. Finally, this work presents a general framework for analysis of frequency signal transmission in other types of cellular networks, such as signaling networks and metabolic networks.

To elucidate the questions raised above, we analyzed transmission of frequency signals in simple gene circuits by mathematical modeling. To start, we consider a one-stage gene circuit where an output protein (P) is driven by an oscillatory input signal (A) (Figure 1A). In the cellular context, the input oscillations may be directly derived from environmental conditions (e.g., day-night cycles) or endogenous cellular oscillators (e.g., circadian clocks). Without loss of generality, we assume that the oscillation in Figure 1A is characterized by a sinusoid function (Figure 1B):

(1)

Display large version of this figure

Figure 1

Analysis of frequency signals with noise. (A) A one-stage gene circuit where the output protein P is controlled by a transcription activator, A. (B) An oscillatory input signal can generate an output signal with oscillations compounded with noise. The mean and standard deviation of the output signal of the linearized model can be analytically computed. Here, we define the mean value as the oscillatory component and the standard deviation as the noise component. Alternatively, the stochastic simulations of the output signal for the nonlinear system can be analyzed by the FFT method to obtain its dominant frequency (see Methods for more details).

Figure 1B illustrates the two approaches we took to analyze transmission of the frequency signal. The first approach was to decompose the output P time course into its mean and standard deviation, which is an application of a linear genetic network method 41,42. The output signal P would oscillate when the gene circuit was driven by the oscillatory input signal A (Eq. (1)). We define the mean as the oscillatory component and the standard deviation as the noise component, which tends to obscure or mask the oscillatory component (see Supplementary Materials ). We anticipate that the frequency signal is transmitted accurately if the oscillation amplitude (α) exceeds the noise level (σ). For simplicity in terminology, we call α and σ the amplitude and noise level, respectively, of the output signal.

To complement the analytical method, we analyzed the P time course for its dominant frequency using numerical methods. If the signal transmission was accurate, this dominant frequency would be the same (within numerical errors) as the input frequency. The dominant frequency of the P time course was calculated by using the fast Fourier transform (FFT) method. Methods of numerical simulation and data analysis are detailed in Supplementary Materials. Briefly, the steady-state portion of the P time course for each simulation was analyzed using the FFT method (see Supplementary Materials, Fig. S1 ). Results from the FFT analysis were then used to extract the dominant output frequency. This output frequency would correspond to the signal frequency “perceived” by downstream processes.

By linearizing the mathematical model of the gene circuit and then decomposing the output using established methods 41,42,43 (also see Supplementary Materials ), we could obtain the average output level (b):

(2)

When changing a circuit parameter, we maintained the average output level at a constant (500 molecules) by adjusting the k_p. For instance, if g_m is increased 10-fold, b can be kept constant by increasing k_p 10-fold. By doing so, we ensured that different circuit configurations or parameter settings would on average elicit the same average level of downstream gene expression (whether or not the input frequency was maintained through transmission).

The amplitude of the output oscillations (α) follows:

(3)

(4)

Eq. (3) defines a decreasing function of α with increasing input frequency (f_in). This dependency reflects the low-pass filter characteristic of linear gene circuits 44. In contrast, σ is independent of f_in (Eq. (4)). Although the noise level oscillated in response to an oscillatory input (see Supplementary Material ), we only used σ for analysis because the amplitude of noise oscillations was negligible (Fig. S2 ). Therefore, α would decrease below σ for sufficiently high f_in (Figure 2A). In this region, frequency signals will be masked by the noise. The intersection between the σ curve and the α curve thus defines a critical frequency (f_c), beyond which the circuit will fail to transmit the input signals. For the given circuit configuration, f_c was ∼0.02/min.

Display large version of this figure

Figure 2

Critical frequency for the one-stage gene circuit. (A) The amplitude of output oscillations decreased with f_in. f_c was calculated as the intersection between the “average noise level” curve and the “oscillation amplitude” curve. (B) Calculations of f_out for varying f_in using stochastic simulations. (C) Fraction of stochastic simulations that generated correct f_out (i.e., where f_out=f_in).

The results of the decomposition method were consistent with those from stochastic simulations. Specifically, we varied f_in from 0.002/min to 0.033/min. For each f_in, we carried out 200 stochastic simulations using the Gillespie algorithm 45. We then determined the dominant frequency for each output time course (f_out) using FFT (Fig. 1). Figure 2B shows a parity plot between f_in (x axis) and corresponding f_out (y axis). The estimated f_c (0.02/min) using the decomposition method corresponds to a transition region in the parity plot. In most simulations, when f_in was <0.02/min, f_out was equal to the corresponding f_in. We consider these signals to be accurately transmitted despite cellular noise. Beyond 0.02/min, however, the average f_out started to deviate from the corresponding f_in and the deviation increased drastically with further increase of f_in (Figure 2B, shaded region).

The drastic deviation was due to the fact that most output time courses gave incorrect f_out. To gain better insight, we analyzed the percentage of the outputs that oscillated at the correct f_out for each f_in. This analysis can be considered as quantifying the fraction of a cell population that could correctly transmit the frequency signal, where behavior of each cell was represented by one stochastic simulation. It provided a quantitative measure of signal transmission fidelity for each f_in (Figure 2C). Again, the estimated f_c defines a transition point that corresponds to a drastic reduction of cells that generated the correct f_out. When f_in was <0.02/min, nearly 100% of the cells produced the correct f_out, indicating high fidelity in signal transmission. However, for f_in>f_c, the percentage decreased drastically, indicating that the majority of cells failed to transmit the frequency signal accurately.

We also calculated signal/noise ratio (SNR) of each cell to assess the fidelity of frequency-signal transmission (Fig. S3 ). The SNR was calculated by dividing the power spectrum density (PSD) at f_in by the maximum PSD of output signals. We assumed that a SNR of 1.0 would ensure accurate transmission of a signal. Again, the f_c calculated from the analytical method corresponds to a sharp transition point where SNR dropped drastically due to the decreasing PSD at f_in and the increasing PSD at noise frequencies. In this region (Fig. S3 , shaded region), downstream processes may have a lower probability of reading the frequency signals due to the dominant PSD of the noise frequencies.

Therefore, both the analytical method and the “brute-force” method by stochastic simulation revealed an intrinsic property of frequency-signal transmission in the simple gene circuit: it is “all or none”, with the transmission fidelity limited by f_c (Figure 2AC). The analytical method also suggests how the f_c emerges as the interplay between the amplitude and the noise level of each output oscillation. For subsequent analyses, we present results from the analytical method, unless noted otherwise.

Frequency multiplexing is a mechanism where multiple frequencies are encoded in one signal. For example, frequency of [Ca²⁺] oscillations regulate several cellular processes, such as exocytosis 46, gene expression 21,47, cell growth, and differentiation 22, suggesting the possibility that multiple frequencies are multiplexed in [Ca²⁺] signals 48. To analyze the impact of noise on frequency multiplexing, we modeled transmission of a composite signal carrying three distinct f_in that are <f_c (Figure 3A), which generated a corresponding multiplexed output (Figure 3B). Frequency-domain analysis indicated that this composite signal was transmitted with absolute fidelity, where the three input frequencies (Figure 3C) were reproduced by the output (Figure 3D). The PSD of the three frequencies were at least 10-fold higher than the PSD of noise frequencies. These findings suggest an advantage of frequency encoding whereby multiple frequencies can be encoded in a composite signal and transmitted to downstream target genes or proteins with high fidelity. In addition, frequency signals may also be more cost-effective than amplitude signals. It has been shown that an oscillatory [Ca²⁺] signal is more effective than a constant [Ca²⁺] signal in inducing translocation of the nuclear factor NF-AT 49. In light of these observations, our results suggest that encoding multiple frequencies in cellular signals may be an efficient yet accurate signaling strategy.

Display large version of this figure

Figure 3

Transmission of a multiplexed signal. (A) A multiplexed input signal. (B) The corresponding output signal computed by stochastic simulation. (C) Power spectra of the input signal. (D) Power spectra of the output signal. Power spectra of the output signal indicate that all three frequencies were transmitted with complete fidelity. Even though power spectra decreased when the input frequency increased, they were still at least 10-fold higher than the power spectra of background noise. Three frequencies (0.005/min, 0.0067/min, and 0.01/min) were multiplexed in a composite signal with an amplitude of five molecules for each input frequency.

Nevertheless, processing of a multiplex signal requires an efficient frequency decoder that can respond to a specific range of frequencies. Although frequency decoders have been suggested both theoretically 23,50 and experimentally 21, the underlying mechanisms of frequency decoding in nature have not been well established experimentally. In this study, we have assumed that a SNR <1.0 (Fig. S3 ) would likely impact processing of the corresponding frequency signal by a downstream frequency decoder.

Circuit parameters such as transcription (k_m) and translation (k_p) rate constants, as well as mRNA (g_m) and protein (g_p) decay rate constants, can affect the dynamics of gene circuits. Increasing decay rate constants can speed up enzymatic kinetics 51 and increase noise frequencies in simple gene circuits 36. Based on these studies, we hypothesized that increasing mRNA and protein decay rate constants can increase f_c of a gene circuit. Indeed, Fig. 4 shows that increasing g_m, g_p, or both, led to a significant increase in f_c, thus permitting faster signal transmission. Fig. S4, A and B , highlights the interplay between the oscillation amplitude and the noise level that gave rise to the f_c curve. In all cases, increasing g_m or g_p, as well as k_p, increased the oscillation amplitude more significantly than the noise level, which led to an increase in f_c.

Display large version of this figure

Figure 4

Effects of mRNA or protein dynamics on f_c demonstrate the dependence of f_c on the speed of mRNA dynamics and the speed of protein dynamics. The speed of mRNA or protein dynamics was modulated by proportionally changing the translation rate constant, k_p, and the mRNA or protein decay rate constant (g_m or g_p), keeping other parameters constant. It was normalized with respect to the base case (g_m=0.2/min, g_p=0.02/min). The color bar represents log₁₀(f_c).

Regulatory mechanisms, such as negative feedback, positive feedback, and feedforward, are prevalent in cellular networks. They have been shown to impact generation, propagation, and control of cellular noise 24,26. For instance, negative feedback can reduce the noise in gene expression and improve response speed to a steady-state input 43,52,53,54. Here, we examined how regulatory mechanisms might impact frequency-signal transmission by simultaneous modulation of the oscillatory and noise components of the signal. Negative feedback was established by a protein that represses its own transcription; positive feedback was established by a protein that enhances its own transcription. We first analyzed feedback with an OR gate in regulating gene expression. In these models, the protein and the activator were assumed to bind to separate, independent binding sites (Fig. S5, A and B ). These models were linearized (Eqs. S2–S4 and S6–S8 ) to allow decomposition of output signals. To study the effects of network architectures, we perturbed the dissociation constant (K_d) of the binding reaction between the output protein and its operator site. The strength of feedback regulation increases with an increase in 1/K_d. While varying feedback strength, we also changed the translation rate constant accordingly to maintain the same average protein output level. Qualitative aspects of our conclusions remained true for low to intermediate feedback strength (1/K_d<10⁻³nM⁻¹) if other parameters (e.g., decay rate constants of the protein or mRNA) were changed to balance the average output level.

Figure 5A shows that negative feedback increased f_c. This is because the negative feedback increased the oscillation amplitude but modulated the noise level in a biphasic manner (Fig. S6 A ). The noise level decreased with increasing feedback strength (1/K_d) when the latter was small. However, when the feedback strength was sufficiently large (1/K_d>10⁻³nM⁻¹), its further increase would increase the noise. The noise increased with strong negative feedback because of the constraint to maintain the same average output level. In particular, very strong negative feedback would lead to a very small number of mRNA molecules. To maintain the average output level, cells would have to amplify protein production from the small number of mRNA molecules. Yet, fast translation coupled with slow transcription has been shown to amplify noise in the protein level 55,56. If we introduced negative feedback without “balancing” it by increasing the translation rate, negative feedback would reduce noise, as reported previously 43. Although increased f_c can be accounted for by the interplay between modulations of the noise amplitude and the oscillation amplitude (Fig. S6 A ), a frequency shift of noise due to negative feedback 36 might also have contributed to this f_c increment. Previous study using an oscillatory input signal has also shown that noise fluctuations resonate at a specific frequency due to negative feedback 41. This resonance, however, would not affect our conclusions here, because it occurred at a frequency much higher than f_c (results not shown).

Display large version of this figure

Figure 5

Effects of circuit architectures on f_c. (A) Dependence of f_c on negative feedback and positive feedback. A high 1/K_d value corresponds to stronger feedback regulation. Cooperativity of feedback regulation (n) was fixed at 2. (B) Dependence of f_c on feedforward rate constant (normalized with respect to the base value of 0.1/min). See Supplementary Material for description of models.

In contrast, positive feedback reduced f_c (Figure 5A). In particular, it reduced the oscillation amplitude while modulating the noise level in a biphasic manner (Fig. S6 B ). The noise level increased with increasing positive feedback strength when the latter was weak (small 1/K_d); it would decrease with the latter if it was sufficiently large (1/K_d >10⁻³nM⁻¹). This noise reduction at high positive feedback strength was due to the balancing reduction in the translation rate constant (results not shown) to maintain the same average output protein level. However, increasing positive feedback strength (balanced by reducing translation rate constant) also led to a decrease in the oscillation amplitude and this decrease was always greater than the noise reduction. As a consequence, overall positive feedback would always result in a decrease in f_c. Further increase of the positive feedback strength led to a decrease in the noise level and a similar decrease in the oscillation amplitude. This would result in a plateau for f_c.

To verify results from the linearized models of feedback regulations, we simulated the corresponding full nonlinear models by using the Gillespie algorithm (see Supplementary Materials ). Similar to the case of the unregulated circuit, we calculated f_out for a range of f_in (5-min intervals) by using FFT to determine a critical point where the mean f_out deviated significantly from the corresponding f_in (Figure 2B). This “brute-force” method generated results qualitatively consistent with those of the linearized models (Figure 5AAB). Generally, negative feedback increased f_c but positive feedback reduced it. Hence, the linear models were accurate and they were useful to decipher the underlying mechanisms that limit the speed of signal transmission.

In addition to feedback regulation with OR gates, we analyzed feedback regulation with AND gates. In a negative-feedback loop with an AND gate (see Supplementary Materials, Eq. S5 ), the protein competes with the activator for the same binding site, hence represses its own transcription activation. In a positive-feedback loop with an AND gate (Eq. S9 ), the protein binds to the activator and enhances its own transcription. In either case, the model could not be solved analytically; thus, we resorted to numerical simulations. Interestingly, we found that feedback regulation with AND gates showed results qualitatively different from those of feedback regulation with OR gates. In particular, negative feedback with an AND gate had biphasic effects on f_c: it only increased f_c if the feedback strength was small (1/K_d<0.0025nM⁻¹); further increase in 1/K_d, however, would lead to a decrease in f_c. Similar to the positive feedback with an OR gate, the positive feedback with an AND gate also reduced f_c. However, when 1/K_d increased to 0.01nM⁻¹, f_c was increased by the positive feedback. In the feedback regulation with AND gates, the noise curves changed in a pattern similar to that of their counterparts with OR gates. For negative feedback, noise first decreased and then increased with increasing feedback strength (Fig. S7 C ). For positive feedback, noise first increased and then decreased with increasing feedback strength (Fig. S7 D ). In either case, however, amplitude of output oscillations did not change significantly with feedback strength, due to competition between the protein and the activator (for negative feedback) or their nonlinear interaction (for positive feedback). As a consequence, f_c depended solely on the inverse of the noise curve: f_c increased when noise decreased, and vice versa.

We further investigated signal transmission in a two-stage circuit (Fig. S5 C ). Without regulation, the two-stage circuit f_c (0.007/min) was lower than that of the one-stage circuit (0.02/min). In general, f_c decreased progressively with increasing cascade length (results not shown). This finding is consistent with low-pass property of long-cascade circuits 33,57. However, the f_c of the two-stage cascades can be increased by incorporating a feedforward regulation. To illustrate this point, we here considered a coherent Type 1 feedforward regulation with an OR gate 58. In this circuit, either the activator molecule or the protein generated from the first stage can activate the second stage (Fig. S5 C ). Our results indicated that increasing feedforward rate constants increased f_c (Figure 5B). At small feedforward rate constants (<0.1/min), f_c did not change significantly due to negligible changes in the oscillation amplitude and noise level (Fig. S8 ). In this region, the contribution of feedforward regulation toward the expression level and the noise level of the second stage were masked by the signal and noise from the first stage. At higher feedforward rate constants, f_cincreased drastically because the oscillation amplitude increased significantly faster with respect to the noise level. Essentially, strong feedforward regulation (with an OR gate) creates a “short cut” between the input signal and the output. However, more detailed analysis is required to elucidate the effects of other types of feedforward motifs 58 on frequency signal transmission.

Our work builds upon previous studies on characteristics of noise generation, propagation, and modulation 33,34. In addition to analysis in the amplitude domain, recent work has shown that noise frequency structures are affected by autoregulation in gene circuits driven by a constant input 35,36. Along another line, it has been suggested that noise characteristics of a circuit in response to an oscillatory input may help infer the underlying network properties 41. Here, we have extended and applied these concepts to analyze the impact of cellular noise on the transmission of frequency signals, introducing the concept of critical frequency (f_c).

The critical frequency defines the fundamental speed limit of signal transmission in a gene circuit: a higher f_c will allow faster response. Only signals with frequencies below f_c can be transmitted with high fidelity (Figure 2BC). We further note that signals with different frequencies can be multiplexed as long as all these frequencies are <f_c. Thus, f_c also defines the capacity or the bandwidth of a circuit in processing frequency signals. This fundamental speed limit for signal transmission may be a general, intrinsic property of diverse cellular networks, including signaling cascades and metabolic pathways. Simulations indicate that a critical frequency also exists for a MAPK signaling pathway (Fig. S9 ) or in a gene circuit consisting of repressors (results not shown). However, aspects of our predictions may differ when considering some nonlinear gene circuits. For example, negative feedback will introduce instability and amplify noise if the time delay is considerably long 59. Stochastic resonance can occur in nonlinear circuits driven by noise 60.

The speed of reliable signal transmission can be constrained by the associated metabolic cost. In the unregulated gene circuit, for instance, increasing f_c always requires an increase in the oscillation amplitude (Fig. S4 ). This increase will require faster protein or mRNA production and turnover and, as such, will incur faster consumption of energy and resources. In this scenario, cells may need to properly balance the speed of signal transmission and the corresponding metabolic cost to maximize their fitness. It has been suggested that energy consumption can constrain evolution of proteins 61 and organization of viral genomes 62,63. It will be interesting to explore whether evolution of the cellular networks involved in frequency-signal transmission has been constrained by the available energy and resources.

In nature, the speed of signal transmission will be further influenced by extrinsic noise, which arises from other cellular processes or from the extracellular environment 25,64. The extrinsic noise will increase the total noise level and further decrease the f_c (Fig. S10 ). In general, signals with low frequencies can be transmitted in noisier environments. For example, transmission of a signal with f_in of 0.001/min in the linear gene circuit (Figure 1A) will not be affected even if the noise level increases appreciably. However, the full picture may be more complex: the frequency of extrinsic noise may also affect the fidelity of frequency signals. It has been suggested that extrinsic noise contains low-frequency signals 31,65. Perhaps these slow fluctuations and the intrinsic noise together define an optimal frequency bandwidth for frequency-signal transmission. Further analysis will be needed to elucidate these questions.

We have illustrated the high fidelity in the transmission of frequency signals if their frequencies are below the critical frequency of a given gene circuit. In natural systems, applicability of this signal-transmission strategy depends on the complexity and adaptability of the available cellular infrastructure. We expect frequency signals to be more likely adopted by higher organisms that can provide sufficiently complex infrastructure, including encoders, decoders, and metabolic capacities to transmit frequency signals. Consistent with this notion, we have found many examples where frequency signals may play an important role in regulating physiological functions in higher organisms, including immune response, metabolism, and sleep cycle (Table S1 ). In contrast, frequency-signal transmission will likely be less common in prokaryotes, as they lack long regulatory gene cascades to provide the adequate infrastructure and energy needed to transmit diverse frequency signals 66. Yet, for critical processes, frequency signals appear to be adopted even in bacteria. In cell-division regulation, for example, the oscillatory dynamics of MinD and MinE proteins at particular frequencies determine the formation of septum in the middle of cells 67,68. Consistent with our findings, it has been suggested that this complex process is highly noise-resistant 69. Nevertheless, this signal transmission strategy will require higher metabolic costs due to its complex cellular architecture. After this study, it will be interesting to examine the tradeoff between energy requirements and signaling fidelity of frequency decoders in biological systems.

Finally, we note that our modeling predictions can be tested experimentally by implementing and measuring the responses of synthetic gene circuits in model organisms such as bacteria and yeast. To obtain f_c, the gene circuits can be modulated by using oscillatory concentrations of small inducer molecules, such as isopropyl-ß-D-thiogalactopyranoside. In the long term, the high fidelity of frequency signals and the ability to multiplex frequency signals suggests a promising means of programming reliable cellular computation using synthetic gene circuits 40.