Fig 1.
Mean first passage time in a self regulated gene.
(A) Schematic of a self-regulating gene. (B) Schematic showing the reactions of TF binding/unbinding to competing decoy sites. (C) Reactions of a target gene regulated by the auto-regulatory TF gene. (D) Number of genes activated (blue bars) or repressed (red bars) by some known autoregulated TFs in E. coli. Data is taken from RegulonDB [54]. (E) First passage time is defined as the time for the gene expression to hit the threshold for the first time as shown here. For an auto-regulated gene, the mean first passage time (MFPT) and the distribution of FPT strongly depend on the strength of auto-regulation (α) and the TF-binding affinity.
Fig 2.
Effect of TF-binding affinity on MFPT for auto-regulated genes without any competing binding sites.
(A, D) MFPT as a function of off-rate (binding affinity) of auto-activated gene and auto-repressed gene. Each curve (blue, red, yellow) is generated by keeping the transcription rate (r0) and translation rate (b) fixed and varying only the binding affinity through koff. Dashed curves correspond to doubling the translation rate (2b). Black dashed lines correspond to MFPT of a constitutive gene of transcription rate r0. (B, E) MFPT as a function of TF-promoter occupancy. Solid and dashed curves are for translation rates b and 2b, respectively. (C,F) Scaled steady state TF expression (SSE) and inverse of production rate (1/R) as a function of off-rate. The ratio of SSE and R (black curve) qualitatively predicts the behavior of the MFPT as a function of off-rate. For auto-activation we use r0 = 0.0025 s−1, b = 0.025 s−1mRNA−1; for auto-repression r0 = 0.05 s−1, b = 0.1 s−1mRNA−1. The protein and mRNA degradation rates are γ = 0.0003 s−1 and γm = 0.011 s−1 per molecule, all additional parameter values listed in the Materials and methods.
Fig 3.
Power-law behavior in auto-activated gene.
(A) Peak value of MFPT as a function of auto-regulation strength, α, shows a power-law behavior with approximate exponent of 0.5 irrespective of basal transcription (r0) or translation rates (b). (B) Peak MFPT for different expression thresholds have different exponents. (C)Power-law exponent of Peak MFPT versus auto-regulation strengths as a function of expression threshold. (D) The value of koff/(r0b) at the peak MFPT also follows a power law behavior with exponent slightly larger than 1. (E) The exponent again depends on threshold. (F) The value of the exponent as a function of threshold.
Fig 4.
MFPT of an auto-regulated gene in a network of competing binding sites.
The MFPT of an auto-activated gene as a function of (A) number of competitive binding sites and (B) steady-state occupancy of TF at the promoter when decoy number is varied. The red curve is for weaker TF binding while the blue curve is for stronger TF binding. Inset: MFPT as function of free TF in steady state. Black dashed lines correspond to MFPT of a constitutive gene of transcription rate r0. (C) Heatmap showing phase space of MFPT in number of competing binding sites and affinity space for auto-activating gene (α = 10). (D,E) The MFPT of an auto-repressed gene as a function of competing binding sites and steady-state occupancy. Dashed blue line correspond to complete repression (α = 0). Inset: MFPT as function free TF in steady state. (F) Heatmap showing phase space of MFPT in number of competing binding sites and affinity space for auto-repressing gene (α = 0.1). We use α = 10, b = 0.05 s−1mRNA−1, r0 = 0.0025 s−1 (activation) and α = 0.1, b = 0.05 s−1mRNA−1, r0 = 0.05 s−1 (repression). γ = 0.0003 s−1 is used for protein degradation corresponding to τ = 38.5 min, all additional parameter values listed in the Materials and methods.
Fig 5.
Distribution of first passage time (FPT) of an auto-activated TF gene.
(A-B) FPT distribution for several values of koff for (A) auto-activation and (B) auto-repression. Dark blue curves represent distribution corresponding to koff where the MFPT peaks or have a minimum. (C) The CV of first passage time as a function of MFPT when koff is changed to vary MFPT for auto-activation (solid lines) and auto-repression (dashed lines) of differing regulatory strength. Parts (D-F) show the same information but for varying decoy instead.
Fig 6.
Expression timing of target gene is dictated by the nature of TF regulation.
(A, B) MFPT of a target gene as a function of binding affinity of TF to the target gene (koff,t). Filled symbols represent the TF being auto-activated (α > 1), open symbols represent when the TF gene is auto-repressed (α < 1), and black filled circles are when the TF gene is constitutive (α = 1). Circles represent target gene activation whereas squares represent target gene being repressed by the TF gene. (C) MFPT of target gene as a function of MFPT of TF gene. α is varied from from 0 to 50 while keeping koff = 0.002 s−1 and koff,t = 0.2 s−1 constant. The MFPTs of TF and target genes are normalized by their respective MFPTs when α is one or equivalently when TF gene is constitutive. Translation rate of the TF gene is adjusted to achieve constant level of TF number (∼50) for varying binding affinities and α in (A-C). The target gene is purely activated or repressed, i.e., when a TF is bound to target gene it is expressed in the case of activation and is completely repressed in the case of repression. Parameter used for target gene: r0,t = 0, rt = 0.05 s−1, b = 0.1 s−1mRNA−1 (target activation), r0,t = 0.05 s−1, rt = 0, b = 0.1 s−1mRNA−1 (target repression). (D) Plot showing asymmetry in the expression timing of TF and target gene when they have identical binding affinity, transcription and translation rates. Target gene repressed by an auto-repressed TF gene always has higher MFPT compared to the TF gene. TF binding affinity is tuned to generate the curves. Target gene activated by an auto-activated TF gene, on contrary, do not show any significant differences (target gene being moderately faster than the TF gene). Parameters: r0 = r0,t = 0.025 s−1, b = 0.1 s−1mRNA−1. Binding affinity is varied to generate the curve.