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Interactions among oscillatory pathways in NF-kappa B signaling.

Wang Y, Paszek P, Horton CA, Kell DB, White MR, Broomhead DS, Muldoon MR - BMC Syst Biol (2011)

Bottom Line: Power spectra of the ratio of nuclear-to-cytoplasmic concentration of NF-kappa B suggest that the cells' responses are entrained by the pulsing frequency.Our results show that, at least in simulation, frequencies other than those of the forcing and the main NF-kappa B oscillator can be excited via sub- and superharmonic resonance, producing quasiperiodic and even chaotic dynamics.Further simulations suggest that the nonlinearities of the NF-kappa B feedback oscillator mean that even sinusoidally modulated forcing can induce a rich variety of nonlinear interactions.

View Article: PubMed Central - HTML - PubMed

Affiliation: Mathematical Biosciences Institute, The Ohio State University, Jennings Hall, Columbus, Ohio 43210, USA. ywang@mbi.osu.edu

ABSTRACT

Background: Sustained stimulation with tumour necrosis factor alpha (TNF-alpha) induces substantial oscillations--observed at both the single cell and population levels--in the nuclear factor kappa B (NF-kappa B) system. Although the mechanism has not yet been elucidated fully, a core system has been identified consisting of a negative feedback loop involving NF-kappa B (RelA:p50 hetero-dimer) and its inhibitor I-kappa B-alpha. Many authors have suggested that this core oscillator should couple to other oscillatory pathways.

Results: First we analyse single-cell data from experiments in which the NF-kappa B system is forced by short trains of strong pulses of TNF-alpha. Power spectra of the ratio of nuclear-to-cytoplasmic concentration of NF-kappa B suggest that the cells' responses are entrained by the pulsing frequency. Using a recent model of the NF-kappa B system due to Caroline Horton, we carried out extensive numerical simulations to analyze the response frequencies induced by trains of pulses of TNF-alpha stimulation having a wide range of frequencies and amplitudes. These studies suggest that for sufficiently weak stimulation, various nonlinear resonances should be observable. To explore further the possibility of probing alternative feedback mechanisms, we also coupled the model to sinusoidal signals with a wide range of strengths and frequencies. Our results show that, at least in simulation, frequencies other than those of the forcing and the main NF-kappa B oscillator can be excited via sub- and superharmonic resonance, producing quasiperiodic and even chaotic dynamics.

Conclusions: Our numerical results suggest that the entrainment phenomena observed in pulse-stimulated experiments is a consequence of the high intensity of the stimulation. Computational studies based on current models suggest that resonant interactions between periodic pulsatile forcing and the system's natural frequencies may become evident for sufficiently weak stimulation. Further simulations suggest that the nonlinearities of the NF-kappa B feedback oscillator mean that even sinusoidally modulated forcing can induce a rich variety of nonlinear interactions.

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Response to periodic trains of weak pulses. A heat map summarising the power spectral density of NF-κB localisation when the model is subjected to pulsed forcing with TR = 0.1. For forcing frequencies in the range 1.5-3.0 × 10-4. Hz. the lowest-lying line segment, which has slope ≈ 1/2, provides evidence of subharmonic resonance.
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Figure 5: Response to periodic trains of weak pulses. A heat map summarising the power spectral density of NF-κB localisation when the model is subjected to pulsed forcing with TR = 0.1. For forcing frequencies in the range 1.5-3.0 × 10-4. Hz. the lowest-lying line segment, which has slope ≈ 1/2, provides evidence of subharmonic resonance.

Mentions: However, when the stimulating pulses are weaker the nonlinearity of the system leads to more complex patterns of resonance. In particular, both subharmonic resonance—when periodic forcing excites a response at a rationally-related lower frequency—and superharmonic resonance—in which a pure sinusoid excites responses containing higher harmonics—are possible. Note that these terms are defined with reference to the forcing frequency, a convention used in, for example, [26]. Superharmonic resonance is harder to identify when, as with the rectangular pulses used here, the periodic forcing already has power at higher harmonics, but subharmonic resonance occurs when 0.01 <TR < 0.2, as is evident in the power spectral densities summarized in Figure 5. In addition to the bright lines with integer slopes, lines with slopes and also appear. The first of these provides evidence that subharmonic resonance occurs over a wide range of pulsing frequencies.


Interactions among oscillatory pathways in NF-kappa B signaling.

Wang Y, Paszek P, Horton CA, Kell DB, White MR, Broomhead DS, Muldoon MR - BMC Syst Biol (2011)

Response to periodic trains of weak pulses. A heat map summarising the power spectral density of NF-κB localisation when the model is subjected to pulsed forcing with TR = 0.1. For forcing frequencies in the range 1.5-3.0 × 10-4. Hz. the lowest-lying line segment, which has slope ≈ 1/2, provides evidence of subharmonic resonance.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3050740&req=5

Figure 5: Response to periodic trains of weak pulses. A heat map summarising the power spectral density of NF-κB localisation when the model is subjected to pulsed forcing with TR = 0.1. For forcing frequencies in the range 1.5-3.0 × 10-4. Hz. the lowest-lying line segment, which has slope ≈ 1/2, provides evidence of subharmonic resonance.
Mentions: However, when the stimulating pulses are weaker the nonlinearity of the system leads to more complex patterns of resonance. In particular, both subharmonic resonance—when periodic forcing excites a response at a rationally-related lower frequency—and superharmonic resonance—in which a pure sinusoid excites responses containing higher harmonics—are possible. Note that these terms are defined with reference to the forcing frequency, a convention used in, for example, [26]. Superharmonic resonance is harder to identify when, as with the rectangular pulses used here, the periodic forcing already has power at higher harmonics, but subharmonic resonance occurs when 0.01 <TR < 0.2, as is evident in the power spectral densities summarized in Figure 5. In addition to the bright lines with integer slopes, lines with slopes and also appear. The first of these provides evidence that subharmonic resonance occurs over a wide range of pulsing frequencies.

Bottom Line: Power spectra of the ratio of nuclear-to-cytoplasmic concentration of NF-kappa B suggest that the cells' responses are entrained by the pulsing frequency.Our results show that, at least in simulation, frequencies other than those of the forcing and the main NF-kappa B oscillator can be excited via sub- and superharmonic resonance, producing quasiperiodic and even chaotic dynamics.Further simulations suggest that the nonlinearities of the NF-kappa B feedback oscillator mean that even sinusoidally modulated forcing can induce a rich variety of nonlinear interactions.

View Article: PubMed Central - HTML - PubMed

Affiliation: Mathematical Biosciences Institute, The Ohio State University, Jennings Hall, Columbus, Ohio 43210, USA. ywang@mbi.osu.edu

ABSTRACT

Background: Sustained stimulation with tumour necrosis factor alpha (TNF-alpha) induces substantial oscillations--observed at both the single cell and population levels--in the nuclear factor kappa B (NF-kappa B) system. Although the mechanism has not yet been elucidated fully, a core system has been identified consisting of a negative feedback loop involving NF-kappa B (RelA:p50 hetero-dimer) and its inhibitor I-kappa B-alpha. Many authors have suggested that this core oscillator should couple to other oscillatory pathways.

Results: First we analyse single-cell data from experiments in which the NF-kappa B system is forced by short trains of strong pulses of TNF-alpha. Power spectra of the ratio of nuclear-to-cytoplasmic concentration of NF-kappa B suggest that the cells' responses are entrained by the pulsing frequency. Using a recent model of the NF-kappa B system due to Caroline Horton, we carried out extensive numerical simulations to analyze the response frequencies induced by trains of pulses of TNF-alpha stimulation having a wide range of frequencies and amplitudes. These studies suggest that for sufficiently weak stimulation, various nonlinear resonances should be observable. To explore further the possibility of probing alternative feedback mechanisms, we also coupled the model to sinusoidal signals with a wide range of strengths and frequencies. Our results show that, at least in simulation, frequencies other than those of the forcing and the main NF-kappa B oscillator can be excited via sub- and superharmonic resonance, producing quasiperiodic and even chaotic dynamics.

Conclusions: Our numerical results suggest that the entrainment phenomena observed in pulse-stimulated experiments is a consequence of the high intensity of the stimulation. Computational studies based on current models suggest that resonant interactions between periodic pulsatile forcing and the system's natural frequencies may become evident for sufficiently weak stimulation. Further simulations suggest that the nonlinearities of the NF-kappa B feedback oscillator mean that even sinusoidally modulated forcing can induce a rich variety of nonlinear interactions.

Show MeSH
Related in: MedlinePlus