Fig 1.
Experimental evidence for amplitude and frequency modulation.
(A and B) Example data showing amplitude modulation from [10]. (A) Single-cell nuclear localization of Msn2 transcription factor in response to H2O2 stress as a function of time. The stimulus profile (input) is a step change applied at t = 0 (inset) which applies to all figure panels. (B) Average time trace for different concentrations of H2O2 stress. (C and D) Example data showing frequency modulation from [15]. (C) Single-cell nuclear localization of Crz1 in response to calcium stress as a function of time, showing bursts of Crz1. (D) The average frequency of bursts against calcium concentration, showing an increased frequency with increased concentration. (Inset) Burst duration distribution for low (blue bars) and high (red bars) concentration. Both histograms are well described by the Gamma distribution , with τb = 70s (black solid line), demonstrating that pulse duration is independent of calcium concentration. Experimental data in arbitrary units (AU) of fluorescence.
Fig 2.
Advantages and disadvantages of amplitude and frequency modulation.
AM may be less noisy than FM (A,B), but FM may allow coordinated expression of many genes (C,D) [15, 19]. (A) In AM, low/high stimuli result in low/high levels of transcription factor (TF) inside the nucleus. (B) In AM, different nuclear TF concentrations (blue and red curves) lead to gene expression of proteins A and B (see orange and green promoter functions respectively) with variable ratios (order of dot and square changes). (C) In FM, the stimulus strength only affects the frequency of bursts, not their amplitude. (Inset) Schematic of TF (purple dots) binding promoter PA of gene A (orange) and promoter PB of gene B (green) with different binding strengths. (D) In FM, the nuclear TF concentration is always the same during a burst, only the frequency of occurrence changes. As a consequence, the protein ratio stays constant.
Fig 3.
Schematic view of signaling and gene regulation.
(A) Cartoon of S. cerevisiae in presence of extracellular calcium, considered a paradigm of bursty frequency modulation. Calcium enters through plasma-membrane ion channels and can be stored (released) in (from) vacuoles. Intracellular calcium activates calcineurin, which dephosphorylates Crz1p. Once dephosphorylated, Crz1 binds inporting Nmd5p and enters the nucleus. Exportin Msn5p subsequently removes Crz1 from the nucleus. Cytoplasmic calcium pulses may correspond to Crz1 bursts in the nucleus [15]. Red arrows indicate movement while blue arrows stand for chemical signaling. (B) Single receptor/ion channel activity, r(t) (blue line), depends on the concentration of extra-cellular stimulus c. The signaling rate u differs between continuous (CM) and bursty modulation (BM). In CM, u is constant rate α during bound intervals, with pb the probability of being bound. In BM, ζ molecules are realized at the time of binding with τbursts the duration between consecutive bursts (binding events). (C) Different regulatory networks. Linear pathway used for concentration sensing. Incoherent feedforward loop and integral feedback control allow chemical ramps to be sensed.
Fig 4.
The two regimes in the linear pathway model based on the master equation.
(A-B) fast (k+ c0 = 20s−1, k− = 100s−1, γ = 0.1s−1, α = 100s−1, ζ = 1) and (C-D) slow (k+ c0 = 0.01s−1, k− = 0.05s−1, γ = 1s−1, α = 25s−1, ζ = 500) switching. (A,C) Protein number as a function of time from Gillespie simulations for CM (blue lines) and BM (red lines). (B) The probability distribution for n target proteins is unimodal for both AM (blue) and FM (red). (D) The probability distribution is bimodal for AM (blue) and remains unimodal for BM (red) but with a long tail in the slow switching regime.
Fig 5.
First three moments of the protein distribution in concentration sensing from the master equation.
Averages (A,B), variance (C,D), and skewness (E,F) as a function of the frequency of binding events, f = k+ c0/(1+k+ c0/k−). (Insets) Magnification of small-noise approximation region (fast switching). Analytical results for CM (blue) and numerical results for BM (red) as function of the frequency of binding events (logarithmic scale). Two regimes are shown: k− = 10 k+ c0 (α = 100s−1, γ = 1s−1, ζ from 1000 to 1) (left column) and k− = 0.1 k+ c0 (α = 10s−1, γ = 1s−1, ζ from 1000 to 1) (right column). Averages from CM and BM are constrained to be equal, i.e. . Variances of CM and BM exhibit two different regimes for fast switching: for k+ c0 < k− BM is more accurate than CM (inset in C), while for k+ c0 > k− CM is generally more accurate (inset in D), except for ζ = 1. Third moments show that, for large noise, the probability distributions become asymmetric.
Fig 6.
Two regimes in incoherent feedforward loop based on the small-noise approximation.
Output noise, i.e. relative variance of x (top) and y (bottom), as function of the non-dimensional ramp time u1 t/u0 for k+ c0 < k− i.e. ⟨τb⟩ < ⟨τu⟩ (left) and k+ c0 > k− i.e. ⟨τb⟩ > ⟨τu⟩ (right). CM and BM are shown by blue and red lines respectively. (A,B) BM is more accurate than AM for k+ c0 = 107 s−1 and k− = 6.7 × 107 s−1. (C,D) CM is more accurate then BM for k+ c0 = 107 s−1 and k− = 6.7 × 106 s−1. Remaining parameters: k+ c1 = 105 s−2, kx = 5s−1 and ky = 10s−1.