Fig 1.
The simple telegraph model and four relatively complex gene expression models.
A: In the telegraph model (TM), the gene switches between an inactive (off) and an active (on) state with rates and
. The gene product (mRNA or protein, denoted by P) is synthesized with rate
when the gene is active, and is degraded with rate d. B: The telegraph model can generate three different shapes of steady-state distributions: a unimodal distribution with a zero peak (left panel), a unimodal distribution with a nonzero peak (middle panel), and a bimodal distribution with both a zero and a nonzero peak (right panel). C: In the three-state model (TSM), the gene exhibits a “refractory” behavior: after leaving the active state with rate γ, the gene has to progress through two sequential inactive states with rates λ1 and λ2 before becoming active again. D: In the cross-talk pathway model (CPM), the gene can be activated via two signalling pathways with rates λ1 and λ2. The competition between the two pathways is modelled by equipping them with two selection probabilities q1 and q2 = 1 − q1. E: In the positive feedback model (PFM), the protein produced from the gene activates its own expression with feedback strength μ. F: In the negative feedback model (NFM), the protein produced from the gene inhibits its own expression with feedback strength ν.
Fig 2.
Fitting the steady-state distributions of complex models to the simple telegraph model.
For each complex model, synthetic data of gene product numbers are generated using the SSA under 625 parameter sets. A: In steady state, all the simulated distributions (blue bars) are well captured by the predictions of the effective telegraph model (red curve). For each complex model, the left panel shows a typical gene product distribution and the right panel shows the distribution with worse telegraph model approximation, i.e. maximum HD value. B: For each complex model, the HD between the simulated distribution and its telegraph model approximation is shown as a function of the mean expression level for the 625 parameter sets. The HD is less than 0.08 for all complex models. C: In steady state, the telegraph model not only captures the total gene product distribution of a complex model, but also captures the conditional distribution in the active gene state. In contrast, for all complex models except the three-state model, the conditional distribution in the inactive gene state in general fails to be captured by the telegraph model. For each complex model, the left (right) panel shows the conditional distribution when the gene is on (off) with worse telegraph model approximation, i.e. maximum HD value. D: For each complex model, the HD is shown as a function of the mean expression level for the 625 parameter sets. The blue circles (grey diamonds) show the HD between the conditional distribution when the gene is on (off) and its telegraph model approximation. The maximum HD for blue circles is only 0.08 for all complex models, while the maximum HD for grey diamonds can be as large as 0.78.
Fig 3.
Linking effective parameters to their real values in complex models.
A: For each complex model, the absolute values of relative errors of the three effective parameters ,
, and
are computed under 625 parameter sets, along with their sample means, sample variances, and the sample frequencies of relative errors being greater than 0.2. The effective parameter
is closed to the synthesis rate ρ for the three-state, cross-talk pathway, and positive feedback models, while the effective parameter
is closed to the gene activation rate λ for the negative feedback model. B: Accuracy of the three effective parameters
,
, and
for each complex model. For the three-state model, all effective parameters are over-estimated; for the cross-talk pathway and positive feedback models, all effective parameters are under-estimated; for the negative feedback model,
is over-estimated, while
and
are under-estimated. C: For each complex model, 150 parameter sets are randomly generated such that 1/〈Toff〉 and 1/〈Ton〉 are between 0 and 2.5d (grey diamonds). For the three-state model, the scatter plot of
escapes from the potential bimodal region of
; for the cross-talk pathway and positive feedback models, the scatter plot of
moves towards the potential bimodal region; for the negative feedback model, the scatter plot of
neither escapes from nor moves towards the potential bimodal region. The yellow (orange) bar shows the proportion of parameter sets that give rise to a unimodal (bimodal) distribution.
Fig 4.
Determining the most competitive model to describe single-cell data at multiple time points.
A: The three-state and positive feedback models have smaller time-dependent mean curve compared to the effective telegraph model, while the cross-talk pathway and negative feedback models have larger time-dependent mean curve. B: Box plots of d(τe − τc) for each complex model, where the τc is the response time for a complex model and τe is the response time for the effective telegraph model. Here the response time is defined as the time for the mean curve to reach half of its steady-state value [44]. C: In E. coli cells, the mRNA of interest, under the control of an inducible promoter Plac/ara, was consisted of the coding region for a red fluorescent protein mRFP1, followed by a tandem array of 96 MS2 binding sites (left panel) [1]. The GFP, independently produced from the promoter PLtetO, tagged the target transcript by binding to the MS2 binding sites. The number of target transcripts in a single cell was computed using fluorescence intensities of GFP at nine time points from 0—120 min. The steady-state mRNA distribution (at 120 min) was fitted to the telegraph model with measured decay rate d = 0.014 minȒ1 [1] (upper-right panel) and the three effective parameters are estimated to be . The real time-dependent mean expression levels (blue triangles in the lower-right panel) are much larger than the mean expression levels predicted by the effective telegraph model (red curve), suggesting that the cross-talk pathway model is a potential candidate to describe the data. D: Point estimates (red points) and confidence intervals (blue lines) for the six parameters q1, q2, λ1, λ2, γ, and ρ when fitting the data to the cross-talk pathway model. Here the confidence intervals are computed using the profile likelihood method. E: The cross-talk pathway model (blue curves) provides a much better fit of the time-dependent mRNA distributions than the telegraph model (red dashed curves). The parameters for the cross-talk pathway model are estimated to be q1 = 0.13, q2 = 1 − q1, and (λ1, λ2, γ, ρ) = (0.0013, 0.055, 0.0042, 0.21) min−1.
Fig 5.
Variation patterns of effective parameters under different induction conditions in all complex models.
A: Tuning a single parameter of a complex model can generate a series of steady-state gene product distributions, along with different mean expression levels. Fitting these distributions to the telegraph model leads to a series of effective parameters ,
, and
. Plotting
,
, and
as functions of the corresponding mean expression level reveals how the effective parameters vary when a single parameter of a complex model is tuned. B: Effective parameters changed when modulating a single parameter of a complex model. For example, for the positive feedback model, the effective parameter
changes when tuning the parameter λ, while all effective parameters
,
, and
change when tuning the parameter ρ.
Fig 6.
Unravelling the regulation mechanism in a synthetic gene network integrated in human kidney cells [82].
A: In the network, a bidirectional promoter transcribes the zsGreen-LacI and dsRed transcripts. The gene network includes two architectures: a negative-feedback network and a network with no feedback. The zsGreen-LacI transcripts are inhibited by LacI, forming a network with negative autoregulation. The dsRed transcripts are not regulated, forming a network with no feedback. The activity of the promoter can be activated in the presence of Dox, and the negative feedback strength can be tuned by induction of IPTG. B: Under both high and low Dox levels, fitting the distributions of zsGreen levels under different IPTG concentrations to the telegraph model leads to increasing , decreasing
, and almost invariant
against the mean expression level. Such variation pattern of the three effective parameters coincides with that in the negative feedback model when the feedback strength ν is tuned. C: Under both high and low Dox levels, fitting the distributions of dsRed levels under different IPTG concentrations to the telegraph model leads to almost invariant values of
and
against the mean expression level. The error bars in B and C show the standard deviation of three repeated experiments [82].
Fig 7.
Robustness of results with respect to cooperative regulation and extrinsic noise.
A: Positive and negative feedback models with cooperative regulation. Feedback is mediated by cooperative binding of two protein copies to the gene. B: For each cooperative feedback model, the HD between the simulated distribution and its telegraph model approximation is shown as a function of the mean expression level for 625 parameter sets. The simulated distribution is well captured by the telegraph model, manifested by HD <0.065. C: Under cooperative regulation and fast gene switching, the deterministic rate equation for the positive feedback model is given by . It may have two stable fixed points (and an unstable fixed point) and thus gives rise to deterministic bistability. The intersections of y = ρ(λ + μx2)/(λ + γ + μx2) (blue curve) and y = dx (red dashed curve) give the locations of the three fixed points (green circles). For a positive feedback loop with deterministic bistability, the effective telegraph model still accurately captures the resulting bimodal distribution. The parameters of the positive feedback model are chosen as ρ = 50, d = 1, λ = 2, γ = 160, μ = 0.5. The effective parameters are estimated to be
. D: For each cooperative feedback model, the (absolute values of) relative errors of the three effective parameters
,
, and
are computed under 625 parameter sets, along with their sample means, sample variances, and the sample frequencies of relative errors being greater than 0.2. E: Under cooperative regulation, the positive feedback model still has smaller time-dependent mean curve compared to the effective telegraph model, while the negative feedback model still has larger time-dependent mean curve. F: For each complex model and each parameter set, the simulated distributions are fitted to the telegraph model under four noise levels (0%, 5%, 10%, and 50%). The relative errors of the three effective parameters are computed for all parameter sets, along with the three statistics of relative errors (same as in D). The three statistics of
are shown for the negative feedback model, and the three statistics of
are shown for the other three complex models.