Figure 1.
Three models for stochastic gene expression.
(A) Burst model in which transcription of the DNA is always active. (B) Two-state model in which the DNA switches with constant rates between active and repressed states. (C) Inducible genetic switch in which an inducer both controls the rate of switching between active and inactive transcription states and is also positively regulated by the protein product – a positive feedback loop (PFB). The gray dotted connection indicates a weak effect of the inducer in promoting the unbinding of repressor at high inducer concentrations.
Figure 2.
Overview of the lac genetic circuit in E. coli.
(A) In the absence of inducer, the lac repressor (LacI) binds to the lac operator preventing transcription of genes in the lac operon. (B) Following an increase in the extracellular inducer concentration, inducer enters the cell via both diffusion across the membrane and active transport by lactose permease (LacY). Once inside, inducer binds free LacI molecules preventing them from binding to the operator. (C) After the intracellular inducer concentration reaches a threshold, any bound repressor is “knocked-off” the operator leading to expression of the lac genes. (D) At high intracellular inducer concentrations the genes for lactose metabolism are fully induced. (E) After inducer is removed, repressor rebinds to the operator preventing further expression of the lac operon and the enzymes for lactose metabolism are either degraded or diluted through cellular division.
Table 1.
Reactions and rate constants used in the stochastic model of the lac circuit.
Figure 3.
Fits of rate constants for IPTG binding to the lac repressor.
(A) Pseudo first order rate constants observed during stochastic simulations of IPTG binding to (blue) repressor and (red) repressor-operator complex. At each inducer concentration 1000 simulations starting with 2 free (or operator-complexed) repressor dimers in a volume of L were performed. The mean fraction of free repressor monomers as a function of time was fit to a single exponential to obtain the observed rate constant for binding at the inducer concentration. x and o are data from Dunaway et al. [75]. (B) Equilibrium binding of IPTG to (blue) repressor and (red) repressor-operator complexes. In a stochastic simulation at each inducer concentration, 20 free (or operator-complexed) repressor dimers in
L were first equilibrated with inducer to reach the steady state. Following, 5 minutes of data were collected from which the equilibrium fraction of inducer bound repressor monomers was calculated. x and o are data from O'Gorman et al. [74].
Table 2.
Obstacle abundance in in vivo spatial models.
Figure 4.
Markov diagram for transcriptional bursting in the lac circuit.
Under low-to-moderate inducer concentrations, a burst begins when the operator enters the state and ends when it transitions to a repressor bound state.
.
Figure 5.
Parameter space of repressor binding parameter .
(A) Mean burst size as a function of inducer concentration for various values of , where
. Parameters used were
=
M,
=
M,
=
, and
=
. (B) The rate of change in the burst size with respective to the inducer concentration.
Figure 6.
Linear fit of burst size to inducer concentration.
x are data from Choi et al. [22].
Figure 7.
Burst analysis of stochastic simulations of a simple two-state process.
The two-state process was described by: . Rate constants were chosen such that on average
bursts of Z with a constant burst size
were produced during Z's mean lifetime with the mean duration of each burst lasting for the indicated fraction of the lifetime. At each point, 250 stochastic simulations were run until the probability density was stationary and then the distributions of Z were fit to gamma distributions to obtain the
and
parameters. The ratios of (A)
/
and (B)
/
as a function of the burst duration show the range of burst durations for which a gamma distribution fit can reliably recover the original parameters. In this example
and
.
Figure 8.
Parameter fitting for inducer–repressor–operator interactions.
(A) Fraction of operator regions bound by a repressor as a function of time following an increase of IPTG to the indicated concentration. In these simulations, . (B) Number of bursts over the mean protein lifetime as a function of inducer concentration for a variety of values of the
parameter. x are data from Choi et al. [22].
Figure 9.
Steady state LacY distributions from the well-stirred NPF model.
Distributions at inducer concentrations of (A) 0, (B) 100, and (C) 200 TMG. Shown are (gray bars) histograms from 10,000 Gillespie trajectories and (red dash) gamma distributions from Choi et al. [22]. (D) Mean LacY as a function of inducer concentration along with 95% ranges. (E) The noise in the LacY distributions as quantified by the Fano factor (variance over the mean). (F) The fraction of time spent in the transcriptionally active state.
Figure 10.
Response of an uninduced PFB population to the addition of external inducer.
(A) Probability density (arbitrary units, darker = higher) of the number of LacY in a cell over the course of 24 hours. Shown are representative responses for populations in the uninduced range (0–10
; left), the bimodal range (10–25
; center), and the concerted induction range (>25
; right). Lines show the mean value of the (green) uninduced and (red) induced subpopulations. (B) Fraction of the cells in each of the subpopulations. (C) The (solid) mean and (dotted) variance of LacY in the uninduced subpopulation.
Figure 11.
Effect of positive feedback on GRF noise.
(A) Mapping of the mean internal inducer concentration for a given external concentration for the (green) uninduced and (red) induced subpopulations. (black dotted) The values for the lac circuit without positive feedback are shown for reference. (B) The mean number of LacY in the subpopulations as a function of internal inducer concentration. (C) The noise in the LacY distribution.
Figure 12.
The effect of in vivo crowding on repressor rebinding.
Each line represents the mean of 5000 trajectories. (A) The observed diffusion coefficient, , as a function of time scale for a repressor diffusing in a volume with the indicated fraction occupied by in vivo obstacles. (B)
–exponent arising from fitting
to a model of anomalous diffusion,
. (C) The probability for a repressor to rebind with the operator before diffusing into the bulk (64 nm from operator) following unbinding, as a function of the in vivo packing. (D) The distribution of escape times for repressors that diffuse to bulk rather than rebind, at three packing values.
Figure 13.
LacY PFB+IV in vivo distributions.
(A) The distribution of LacY in (orange bars) 100 modeled E. coli cells at 13 TMG concentration compared with (green dotted) the PFB well-stirred distribution. (B) Mean number of LacY proteins in the (circles) PFB+IV and (green dotted) PFB models. (C) The noise in the distributions.
Figure 14.
Analysis of cryoelectron tomography based cell model.
(A) Slow growth E. coli cell model based in part on data from a tomographic reconstruction. Shown are (orange) ribosomes, (light gray) membrane, (dark grey) condensed nucleoid, and (red) lac operator. (B+C) Distribution of repressor–operator complex lifetimes for the fast and slow growth models, respectively. Curves show fits to an exponential distribution with the given mean. (D) Position of mRNA–membrane contact after diffusion of mRNA produced at the lac operon in (blue x) fast growth and (red o) slow growth models. Dotted lines show the length of the respective cells.
Figure 15.
Maximum-likelihood fitting of two models for gene expression to stochastic simulations of an inducible genetic circuit.
(A and B) Parameter fits from the burst model. (C–F) Parameter fits from the two-state model. Shown are fits for (black dotted) NPF simulations, (green dotted) PFB simulations, and (orange circles) PFB+IV simulations. Also shown are (blue solid) actual parameter values calculated from the simulation data. Shaded areas indicate the 95% confidence intervals for ML fits using distributions from 50 and 200 NPF cells.
Figure 16.
Probability landscape of protein–mRNA abundances in the inducible lac switch model.
(A) Steady-state probability landscape (arbitrary units, darker = higher) for the NPF model at 500 TMG. The dotted line shows the trajectory of a representative cell during a ∼3 hour interval starting at the open circle and ending at the closed circle. (B) Probability landscape of the PFB circuit over a period of 24 hours following the addition of external TMG to 16
. The line follows a single cell switching from the uninduced to the induced state over the course of ∼13 hours.