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Period doubling induced by thermal noise amplification in genetic circuits

View Article: PubMed Central - PubMed

ABSTRACT

Rhythms of life are dictated by oscillations, which take place in a wide rage of biological scales. In bacteria, for example, oscillations have been proven to control many fundamental processes, ranging from gene expression to cell divisions. In genetic circuits, oscillations originate from elemental block such as autorepressors and toggle switches, which produce robust and noise-free cycles with well defined frequency. In some circumstances, the oscillation period of biological functions may double, thus generating bistable behaviors whose ultimate origin is at the basis of intense investigations. Motivated by brain studies, we here study an “elemental” genetic circuit, where a simple nonlinear process interacts with a noisy environment. In the proposed system, nonlinearity naturally arises from the mechanism of cooperative stability, which regulates the concentration of a protein produced during a transcription process. In this elemental model, bistability results from the coherent amplification of environmental fluctuations due to a stochastic resonance of nonlinear origin. This suggests that the period doubling observed in many biological functions might result from the intrinsic interplay between nonlinearity and thermal noise.

No MeSH data available.


Comparison between theory and numerical simulations.Panel (a) compares the normalized subharmonic current power density  (circle markers) versus analytic theory (solid line) for different normalized frequencies Ω = ωp/2ω0. In the numerical simulations we choose V0 = 2 V, T = 10−7 W and change the input frequency in the range ωp ∈ [ω0/2, 2ω0]. For each ωp we averaged over 40 simulations to guarantee convergence of the mean value and standard deviation. Panel (b) shows the behavior of the normalized power density spectrum in the surroundings of the input frequency ωp (solid red line) and near the subharmonic peak at ωp/2 (solid green line). Spectra are calculated for a single numerical simulation with V0 = 2 V, T = 10−7 W and ωp/ω0 = 1.4.
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f6: Comparison between theory and numerical simulations.Panel (a) compares the normalized subharmonic current power density (circle markers) versus analytic theory (solid line) for different normalized frequencies Ω = ωp/2ω0. In the numerical simulations we choose V0 = 2 V, T = 10−7 W and change the input frequency in the range ωp ∈ [ω0/2, 2ω0]. For each ωp we averaged over 40 simulations to guarantee convergence of the mean value and standard deviation. Panel (b) shows the behavior of the normalized power density spectrum in the surroundings of the input frequency ωp (solid red line) and near the subharmonic peak at ωp/2 (solid green line). Spectra are calculated for a single numerical simulation with V0 = 2 V, T = 10−7 W and ωp/ω0 = 1.4.

Mentions: During an orbit in the (x, y) plane, during the ON state we can distinguish between two different regions for the noise-free solution ξ(t). The first region is characterized by ξ + Δx > xo, which is represented by the segment B-C-A (or B-D-E) in Fig. 5a, with points B, A (or B, E) and a small interval around them excluded. In this case θ = 1 and f = x, which yields h′ = 0 and f′ = 1. Equations (16) can be trivially solved in the frequency domain ω and the solution for the current (c.c. stands for complex conjugate) is a damped harmonic oscillator, with maximum amplitude at ω = ω0 and bandwidth : being the Fourier transform of η(t). The second region of the ON state, which is the most critical, is in the vicinity of point A and is represented by , with . In this situation, the term Δx leads the dynamics to continuously oscillate between the A-B (or E-B) segment (ON state) in Fig. 5a and the rest of the cycle B-C-A (or B-D-E) (OFF state). When the system switches between these two segments, the charge Δx experiences a discontinuous dynamics originated from the term , with rapidly changing from zero to a small but finite value due to the noise fluctuations. The current Δy, conversely, evolves smoothly thanks to . In this condition, equations of motion (16) become: The term ζ(t) appearing in the RHS of the first equation oscillates with frequency ωp, and this imposes a resonance condition with the oscillation frequency of Δx(t). In the Fourier domain, in fact, the only frequency admissible is the one where the term oscillates at the same frequency with Δx and Δy. This condition imposes phase-matching ωp ± ω = ω, which is equivalent to ω = ωp/2. The only oscillation that can be observed in the evolution of Δx, Δy is therefore at ω = ωp/2, cause all the frequencies that do not satisfy phase-matching are not allowed in the second region and ruled out from the dynamics. Equations (19) allow to qualitatively assess the generality of our findings with respect to the particular shape of the promoter activity function g(p). In the most general situation, the function g(p) can be expanded in Taylor series following Eq. (2). Linear terms ∝ p0 and ∝ p give rise, in the electric circuit representation, to linear dissipative electric components RLC. High order nonlinear terms ∝ pn with n ≥ 2, conversely, originate nonlinear dissipative terms that do not qualitatively affect the existence of the parametric resonance at ωp/2. The latter, in fact, originates from the nonlinear response of the diode that, as shown in Eqs. (6) – (7), is sustained by the mechanism of cooperative stability and the linear first order terms in the expansion of g(p). In order to solve for the current , we represent δ(x) = H · rect(Hx), being rect(x) = θ(x + 1 − 2) − θ(x − 1/2). After taking the limit for H → ∞, we obtain the following solution: The spectral power of the current at ω = ωp/2 can be then obtained by combining in time the two expressions for appearing in Eqs. (20) and (18) for ω = ωp/2 with weights 1 − α and α respectively. The latter indicates the time spent in the different region of the cycle and critically depends on the dynamics of the noise free solution (ξ, ζ). When we combine in time different spectra with different time duration, the bandwidth and the Q–factor of the single spectra change as well, as we are convolving the spectrum arising from an infinite signal with the Fourier transform of a box function of a finite length. We can therefore define an effective Q–factor , which here must be consider as a fitting parameter as it depends on the time spent by ξ and η in the different regions of the cycle. After some straightforward algebra, we obtain the following expression for the current power density: being Ω = ωp/2ω0, /ηr/R/2 the current noise spectral power at the peak frequency ωp/2. Equation (21) has a nonlinear stochastic resonance at Ω ≈ 0.5, which predicts a strong noise amplification of the subharmonic ωp/2 when ωp ≈ ω0. In order to verify our theory, we calculated the behavior of Jy versus Ω by a series of simulations with a sinusoidal input with a varying frequency ωp. For each ωp, we averaged over 40 realization to calculate mean value and standard deviation of Jy. We then compared numerical simulations versus Eq. (21), with parameters α = 0.51 and given by a nonlinear least square fit (Fig. 6a). We observed an excellent agreement between Eq. (21) and the results from numerical simulations, confirming the strongly resonant nature of the process. Equation (20) allows also to interpret the quantitative differences between the power density spectra in Fig. 2i and Fig. 4c. According to the phase matching condition imposed by Eq. (20), in fact, the bandwidth of the amplified noise is expect to match the bandwidth of coherent part the input source near ωp, where each component resonate with its subharmonics and get amplified through parametric resonance. This process is highlighted in Fig.6b, where we superimpose different part of the current power density spectrum (Fig. 6b) obtained from a numerical simulation with V0 = 1.6 V, T = 10−6 W and ωp/ω0 = 1.4. As seen in the figure, the power density of the amplified noise near ωp/2 (Fig. 6b solid green line) matches very well the spectrum of the input signal near ωp (Fig. 6b solid red line). In the case of short pulses, parametric resonance is triggered by a larger spectrum near ωp, due to the larger harmonic content of a short pulse with respect to a purely sinusoidal source, and therefore results into a larger amplified band near ωp/2.


Period doubling induced by thermal noise amplification in genetic circuits
Comparison between theory and numerical simulations.Panel (a) compares the normalized subharmonic current power density  (circle markers) versus analytic theory (solid line) for different normalized frequencies Ω = ωp/2ω0. In the numerical simulations we choose V0 = 2 V, T = 10−7 W and change the input frequency in the range ωp ∈ [ω0/2, 2ω0]. For each ωp we averaged over 40 simulations to guarantee convergence of the mean value and standard deviation. Panel (b) shows the behavior of the normalized power density spectrum in the surroundings of the input frequency ωp (solid red line) and near the subharmonic peak at ωp/2 (solid green line). Spectra are calculated for a single numerical simulation with V0 = 2 V, T = 10−7 W and ωp/ω0 = 1.4.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5382689&req=5

f6: Comparison between theory and numerical simulations.Panel (a) compares the normalized subharmonic current power density (circle markers) versus analytic theory (solid line) for different normalized frequencies Ω = ωp/2ω0. In the numerical simulations we choose V0 = 2 V, T = 10−7 W and change the input frequency in the range ωp ∈ [ω0/2, 2ω0]. For each ωp we averaged over 40 simulations to guarantee convergence of the mean value and standard deviation. Panel (b) shows the behavior of the normalized power density spectrum in the surroundings of the input frequency ωp (solid red line) and near the subharmonic peak at ωp/2 (solid green line). Spectra are calculated for a single numerical simulation with V0 = 2 V, T = 10−7 W and ωp/ω0 = 1.4.
Mentions: During an orbit in the (x, y) plane, during the ON state we can distinguish between two different regions for the noise-free solution ξ(t). The first region is characterized by ξ + Δx > xo, which is represented by the segment B-C-A (or B-D-E) in Fig. 5a, with points B, A (or B, E) and a small interval around them excluded. In this case θ = 1 and f = x, which yields h′ = 0 and f′ = 1. Equations (16) can be trivially solved in the frequency domain ω and the solution for the current (c.c. stands for complex conjugate) is a damped harmonic oscillator, with maximum amplitude at ω = ω0 and bandwidth : being the Fourier transform of η(t). The second region of the ON state, which is the most critical, is in the vicinity of point A and is represented by , with . In this situation, the term Δx leads the dynamics to continuously oscillate between the A-B (or E-B) segment (ON state) in Fig. 5a and the rest of the cycle B-C-A (or B-D-E) (OFF state). When the system switches between these two segments, the charge Δx experiences a discontinuous dynamics originated from the term , with rapidly changing from zero to a small but finite value due to the noise fluctuations. The current Δy, conversely, evolves smoothly thanks to . In this condition, equations of motion (16) become: The term ζ(t) appearing in the RHS of the first equation oscillates with frequency ωp, and this imposes a resonance condition with the oscillation frequency of Δx(t). In the Fourier domain, in fact, the only frequency admissible is the one where the term oscillates at the same frequency with Δx and Δy. This condition imposes phase-matching ωp ± ω = ω, which is equivalent to ω = ωp/2. The only oscillation that can be observed in the evolution of Δx, Δy is therefore at ω = ωp/2, cause all the frequencies that do not satisfy phase-matching are not allowed in the second region and ruled out from the dynamics. Equations (19) allow to qualitatively assess the generality of our findings with respect to the particular shape of the promoter activity function g(p). In the most general situation, the function g(p) can be expanded in Taylor series following Eq. (2). Linear terms ∝ p0 and ∝ p give rise, in the electric circuit representation, to linear dissipative electric components RLC. High order nonlinear terms ∝ pn with n ≥ 2, conversely, originate nonlinear dissipative terms that do not qualitatively affect the existence of the parametric resonance at ωp/2. The latter, in fact, originates from the nonlinear response of the diode that, as shown in Eqs. (6) – (7), is sustained by the mechanism of cooperative stability and the linear first order terms in the expansion of g(p). In order to solve for the current , we represent δ(x) = H · rect(Hx), being rect(x) = θ(x + 1 − 2) − θ(x − 1/2). After taking the limit for H → ∞, we obtain the following solution: The spectral power of the current at ω = ωp/2 can be then obtained by combining in time the two expressions for appearing in Eqs. (20) and (18) for ω = ωp/2 with weights 1 − α and α respectively. The latter indicates the time spent in the different region of the cycle and critically depends on the dynamics of the noise free solution (ξ, ζ). When we combine in time different spectra with different time duration, the bandwidth and the Q–factor of the single spectra change as well, as we are convolving the spectrum arising from an infinite signal with the Fourier transform of a box function of a finite length. We can therefore define an effective Q–factor , which here must be consider as a fitting parameter as it depends on the time spent by ξ and η in the different regions of the cycle. After some straightforward algebra, we obtain the following expression for the current power density: being Ω = ωp/2ω0, /ηr/R/2 the current noise spectral power at the peak frequency ωp/2. Equation (21) has a nonlinear stochastic resonance at Ω ≈ 0.5, which predicts a strong noise amplification of the subharmonic ωp/2 when ωp ≈ ω0. In order to verify our theory, we calculated the behavior of Jy versus Ω by a series of simulations with a sinusoidal input with a varying frequency ωp. For each ωp, we averaged over 40 realization to calculate mean value and standard deviation of Jy. We then compared numerical simulations versus Eq. (21), with parameters α = 0.51 and given by a nonlinear least square fit (Fig. 6a). We observed an excellent agreement between Eq. (21) and the results from numerical simulations, confirming the strongly resonant nature of the process. Equation (20) allows also to interpret the quantitative differences between the power density spectra in Fig. 2i and Fig. 4c. According to the phase matching condition imposed by Eq. (20), in fact, the bandwidth of the amplified noise is expect to match the bandwidth of coherent part the input source near ωp, where each component resonate with its subharmonics and get amplified through parametric resonance. This process is highlighted in Fig.6b, where we superimpose different part of the current power density spectrum (Fig. 6b) obtained from a numerical simulation with V0 = 1.6 V, T = 10−6 W and ωp/ω0 = 1.4. As seen in the figure, the power density of the amplified noise near ωp/2 (Fig. 6b solid green line) matches very well the spectrum of the input signal near ωp (Fig. 6b solid red line). In the case of short pulses, parametric resonance is triggered by a larger spectrum near ωp, due to the larger harmonic content of a short pulse with respect to a purely sinusoidal source, and therefore results into a larger amplified band near ωp/2.

View Article: PubMed Central - PubMed

ABSTRACT

Rhythms of life are dictated by oscillations, which take place in a wide rage of biological scales. In bacteria, for example, oscillations have been proven to control many fundamental processes, ranging from gene expression to cell divisions. In genetic circuits, oscillations originate from elemental block such as autorepressors and toggle switches, which produce robust and noise-free cycles with well defined frequency. In some circumstances, the oscillation period of biological functions may double, thus generating bistable behaviors whose ultimate origin is at the basis of intense investigations. Motivated by brain studies, we here study an “elemental” genetic circuit, where a simple nonlinear process interacts with a noisy environment. In the proposed system, nonlinearity naturally arises from the mechanism of cooperative stability, which regulates the concentration of a protein produced during a transcription process. In this elemental model, bistability results from the coherent amplification of environmental fluctuations due to a stochastic resonance of nonlinear origin. This suggests that the period doubling observed in many biological functions might result from the intrinsic interplay between nonlinearity and thermal noise.

No MeSH data available.