Tunability enhancement of gene regulatory motifs through competition for regulatory protein resources

Gene regulatory networks (GRNs) orchestrate the spatiotemporal levels of gene expression, thereby regulating various cellular functions ranging from embryonic development to tissue homeostasis. Some patterns called “motifs” recurrently appear in the GRNs. Owing to the prevalence of these motifs they have been subjected to much investigation, both in the context of understanding cellular decision making and engineering synthetic circuits. Mounting experimental evidence suggests that (1) the copy number of genes associated with these motifs varies, and (2) proteins produced from these genes bind to decoy binding sites on the genome as well as promoters driving the expression of other genes. Together, these two processes engender competition for protein resources within a cell. To unravel how competition for protein resources affects the dynamical properties of regulatory motifs, we propose a simple kinetic model that explicitly incorporates copy number variation (CNV) of genes and decoy binding of proteins. Using quasi-steady-state approximations, we theoretically investigate the transient and steady-state properties of three of the commonly found motifs: Autoregulation, toggle switch, and repressilator. While protein resource competition alters the timescales to reach the steady state for all these motifs, the dynamical properties of the toggle switch and repressilator are affected in multiple ways. For toggle switch, the basins of attraction of the known attractors are dramatically altered if one set of proteins binds to decoys more frequently than the other, an effect which gets suppressed as the copy number of the toggle switch is enhanced. For repressilators, protein sharing leads to an emergence of oscillation in regions of parameter space that were previously nonoscillatory. Intriguingly, both the amplitude and frequency of oscillation are altered in a nonlinear manner through the interplay of CNV and decoy binding. Overall, competition for protein resources within a cell provides an additional layer of regulation of gene regulatory motifs.

Figure 1

(a) Schematic representation of various processes for a positive autoregulatory motif in the presence of decoy sites. The promoter switches between a protein bound and empty states with rates $k_{-}$ and $k_{+}$, respectively. Basal and regulated mRNA production rates are $\alpha$ and $\beta$. The mRNA translation rate is $\sigma$, which leads to monomeric protein production. Monomers reversibly bind to form a dimer with rate ${\kappa }_{+}$, which subsequently can dissociate into monomers at rate ${\kappa }_{-}$. mRNA and monomeric proteins degrade at rates ${\gamma }_m$ and ${\gamma }_p$, respectively. We assume that the dimeric protein can bind or unbind to the promoter or decoys with the same rates. (b) Governing equations characterizing the time evolution of the relevant variables in the system. Variables refer to concentrations of protein monomers $x$, protein dimers $y$, unoccupied and occupied promoters $d_u$ and $d_o$, unoccupied and occupied decoys $\overline{d_u}$ and $\overline{d_o}$, and mRNA $m$. (c) Separation of timescales, list of fast and slow processes we consider (mRNA and protein degradation rates are not shown).

Figure 2

(a) Attainment of steady state of total monomer concentration. The presence of decoys affects the transient dynamics; the time it takes to reach the steady state is enhanced as the number of decoys in increased. (b) Half-times as functions of decoy copy numbers are shown for different copy numbers of the motif. Here half-times is defined as the time it takes to reach half of the steady-state monomer concentration.

Figure 3

(a) Cartoon of the toggle switch. Two genes repress each other. The protein products pertaining to the genes can bind to decoys. (b) Phase space of the toggle switch in the presence of homogeneous decoys. The dashed line indicates a separatrix which defines the two stable states of the toggle switch (high, low) and (low, high). The gray lines are nullclines for the toggle switch system which intersect at two stable and one unstable fixed points (see Appendix pp2 for details). (c) An example of the dynamics in the basin of attraction for (low, high) state with and without decoys. Note that the presence of decoys increases the time to reach the steady state.

Figure 4

(a) Phase space of the toggle switch in the presence of heterogeneous decoys, i.e., when the numbers of decoys corresponding to two genes are different (number of decoys for protein 2 is higher than the number for protein 1). The separatrix begins to exhibit a curvature that changes with the degree of dissimilarity in decoy numbers. The basin of attraction associated with (high, low) state thus expands at the expense of the one related to (low, high) state. The gray lines are nullclines for the toggle switch system, which intersect at two stable and one unstable fixed points (see the Appendixes for details). (b) The example of Fig. 2 is replotted in the presence of dissimilar decoys. Initial conditions that evolved to (low, high) state earlier evolved to (high, ow) state. (c) Phase space of the toggle switch for $N_p=3$ copies of the switch in the presence of heterogeneous decoy binding. Increase in gene copy number rescues the effect of heterogeneous decoy binding; the effect of decoys is suppressed. It must be noted that the average expression level of both genes is higher when the copy number of the switch is higher. (d) Such a rescue effect is further evident by the transient behavior of the system; time to reach the steady state is reduced.

Figure 5

(a) Cartoon of the repressilator. Three genes repress each other in a cyclic structure. The proteins produced from these genes bind to decoy sites. (b) Stability diagram in $\mathrm{\Gamma }-\kappa$ parameter space, where $\mathrm{\Gamma }={\gamma }_p/{\gamma }_m$ and $\kappa =\alpha \sigma /\left({\gamma }_m{\gamma }_p\right)$ [see Fig. 1 and Appendixes for details of the parameters]. The blue curve is for the case without decoys, while the red curve indicates the case when decoys are added. (c) We show stability diagrams for $N_p=1,3$ with decoy number $N_d=1000$. A part of the steady-state regime becomes oscillatory in the presence of decoys and vice versa. (d) An example of dynamics from the region exhibiting oscillatory behavior without decoys becomes steady in the presence of decoys. (e) A complementary example from the region for the opposite case is shown.

Figure 6

(a) Frequency ratio of oscillation with and without decoy binding is shown. Ratios for three different gene copy numbers are shown. The dashed line in black indicates ${\omega }_r=1$ for reference. (b) Amplitude ratios with and without decoys for different copy numbers of genes are shown. It shows a nonlinear behavior as a function of decoy number; initially it does not vary much but then shows a hump followed by rapid decline as the decoy number is increased. The dashed line in black indicates $A_r=1$ for reference.

Figure 7

(a) Comparison of the prefactor method (dashed curve) with the full unreduced system (solid curves). (b) Transient dynamics due to increasing decoy sites is sustained for longer times in the full system. The blue curve is for the case with decoy sites, and the red curve shows the case with decoy sites. This behavior is again similar to what we have observed with the prefactor method.

Figure 8

(a) The curves show a linear increase in half-times to reach the steady state with increase in gene copy number $N_p$. For computational reliability, these half-times are computed by averaging over an ensemble of 100 random initial conditions chosen in the neighborhood of steady state. (b) Separatrixes have been plotted for $N_p=1$ for different protein binding affinities of decoys. The curve's change is significant and therefore, basins of attraction of the two stables states are considerably altered. (c) Amplitude ratio is highly sensitive to variation in protein binding affinities of decoys for $N_p=1$. The effect appears to be suppressed for high binding affinities.

Figure 9

List of reactions used for the toggle switch and repressilator systems. For the toggle switch, $i$ takes values 1 and 2, corresponding to two genes of the motif. For the repressilator, $j$ takes values 1, 2, and 3, corresponding to three genes of the motif.

Figure 10

Examples to show variation in amplitude of monomer concentration with time for different numbers of decoy sites at $N=1$. The blue curve indicates the variation in the absence of decoy sites, while the red curve shows that in the presence of decoy sites. (a) The case for the absence decoys, i.e., $N_d=0.0$. (b) For $N_d=1000$, amplitude of oscillation increases. (c) Amplitude decreases at $N_d=40000$.