Stochastic dynamics of genetic broadcasting networks

The complex genetic programs of eukaryotic cells are often regulated by key transcription factors occupying or clearing out of a large number of genomic locations. Orchestrating the residence times of these factors is therefore important for the well organized functioning of a large network. The classic models of genetic switches sidestep this timing issue by assuming the binding of transcription factors to be governed entirely by thermodynamic protein-DNA affinities. Here we show that relying on passive thermodynamics and random release times can lead to a “time-scale crisis” for master genes that broadcast their signals to a large number of binding sites. We demonstrate that this time-scale crisis for clearance in a large broadcasting network can be resolved by actively regulating residence times through molecular stripping. We illustrate these ideas by studying a model of the stochastic dynamics of the genetic network of the central eukaryotic master regulator $\text{NF}\kappa \text{B}$ which broadcasts its signals to many downstream genes that regulate immune response, apoptosis, etc.

Figure 1

(a) Schematic picture of the broadcasting network of master regulator $\text{NF}\kappa \text{B}$. Under external stimulation $\text{NF}\kappa \text{B}-\text{I}\kappa \text{B}$ is marked for degradation of $\text{I}\kappa \text{B}$ with the rate $\alpha$. Once freed, $\text{NF}\kappa \text{B}$ goes inside the nucleus and binds to a myriad of DNA sites, including functional targets for downstream signaling, nonfunctional decoys, and the promoter for $\text{I}\kappa \text{B}$ itself. Binding to the promoter site initiates the synthesis of $\text{I}\kappa \text{B}$. The negative feedback loop of $\text{I}\kappa \text{B}$ is shown with the dashed line with its inhibition of $\text{NF}\kappa \text{B}$ taking place either via direct binding to dissociated $\text{NF}\kappa \text{B}$ (solid black arrow) or via molecular stripping. (b) Shown are the mechanisms of (1) the broadcasting network with no stripping, where the DNA unbinding rate takes place via passive dissociation, and (2) the switch with stripping, where switching is being controlled kinetically via broadcasting signals of $\text{I}\kappa \text{B}$ stripping $\text{NF}\kappa \text{B}$ off of DNA sites directly in addition to a passive dissociation step.

Figure 2

Shown are 50 representative trajectories of $\text{NF}\kappa \text{B}\text{−}\text{DNA}$ complexes in broadcasting networks. We have examined cases of networks with passive dissociation (no stripping), $k_s=0\mu {\text{M}}^{-1}{\text{min}}^{-1}$ [first row, (a, b)], and with stripping, $k_s=10\mu {\text{M}}^{-1}{\text{min}}^{-1}$ [second row, (a, b)] under conditions of either terminated ($\alpha =0$) or continuous ($\alpha >0$) stimulation. (a) The slow on-off limit where rebinding of $\text{NF}\kappa \text{B}$ to DNA sites is negligible ($k_{\text{doff}}=10{\text{M}}^{-1}{\text{min}}^{-1}$) and the unbinding rates for all the bound decoys are $k_{\text{doff}}=0.1{\text{min}}^{-1}$. (b) The fast on-off rate limit where clearance times are dominated by frequent rebinding events ($k_{\text{don}}={10}^3\text{–}{10}^4{\text{M}}^{-1}{\text{min}}^{-1}$) of $\text{NF}\kappa \text{B}$ to $\text{DNA}$ sites and the unbinding rates for all the bound decoys are set to be $k_{\text{doff}}=10{\text{min}}^{-1}$.

Figure 3

(a) The dependence of the mean clearance time of occupied decoys $\text{NF}\kappa \text{B}\text{−}\text{DNA}$ on the unbinding rate for cases of no stripping and stripping with rates $k_s$. (b) The dependence of the standard deviation of clearance times for cases of no stripping and stripping with rates $k_s$. (c) The probability distributions corresponding to different unbinding rates in the case with no stripping, $k_s=0$.

Figure 4

(a) Mean and (b) coefficient of variation for the broadcasting network with no stripping ($k_s=0\mu {\text{M}}^{-1}{\text{min}}^{-1}$) as a function of numbers of decoys.

Figure 5

The survival probabilities of single bound decoy sites as a function of time for the networks (a) with no stripping and (b) with stripping. Rebinding probabilities are computed as a function of time for the cases (c) with no stripping and (d) with stripping. Different curves correspond to different dissociation rates of the $\text{NF}\kappa \text{B}$ bound decoys.

Figure 6

(a) Dependence of the mean fraction of cleared times on the decoy unbinding rate computed for networks' different stripping rates $k_s$. (b) Temporal autocorrelation function of DNA bound $\text{NF}\kappa \text{B}$ molecules for the unbinding rate of $k_{\text{doff}}=0.1{\text{min}}^{-1}$.

Figure 7

The values of the (a) mean clearance time and (b) coefficient of variation of the network with terminated stimuli as a function of dissociating $\mathrm{rates}{k}_{\text{doff}}$ with fixed thermodynamic affinities $K_d=k_{\text{doff}}/k_{\text{don}}=\text{const}$. (c) Mass action ratio of decoy-$\text{NF}\kappa \text{B}$ quasiequilibrium, with the corresponding macroscopic dissociation as a function of dissociation rates $k_{\text{doff}}$. (d) Probability of the unoccupied $\text{I}\kappa \text{B}$ promoter.