Translational cross talk in gene networks

Abstract

It has been shown experimentally that competition for limited translational resources by upstream mRNAs can lead to an anticorrelation between protein counts. Here, we investigate a stochastic model for this phenomenon, in which gene transcripts of different types compete for a finite pool of ribosomes. Throughout, we utilize concepts from the theory of multiclass queues to describe a qualitative shift in protein count statistics as the system transitions from being underloaded (ribosomes exceed transcripts in number) to being overloaded (transcripts exceed ribosomes in number). The exact analytical solution of a simplified stochastic model, in which the numbers of competing mRNAs and ribosomes are fixed, exhibits weak positive correlations between steady-state protein counts when total transcript count slightly exceeds ribosome count, whereas the solution can exhibit strong negative correlations when total transcript count significantly exceeds ribosome count. Extending this analysis, we find approximate but reasonably accurate solutions for a more realistic model, in which abundances of mRNAs and ribosomes are allowed to fluctuate randomly. Here, ribosomal fluctuations contribute positively and mRNA fluctuations contribute negatively to correlations, and when mRNA fluctuations dominate ribosomal fluctuations, a strong anticorrelation extremum reliably occurs near the transition from the underloaded to the overloaded regime.

Figure 1

(a) Schematic of our stochastic model for translational cross talk. A finite number of copies of two types of mRNA can either be bound or unbound to a finite pool of identical ribosomes, with at most one ribosome binding to a given mRNA molecule. Once bound to an mRNA molecule, a ribosome begins translating the transcript. A single protein of the same type as the mRNA is produced upon completion of the translation process. (b) When a ribosome completes the translation process it either rebinds to the same transcript (with probability p), or (with probability $1-p$) releases the transcript into the pool of unbound transcripts and selects an mRNA transcript at random from this pool and binds to it.

Figure 2

Plot of the steady-state correlation between protein counts for fixed numbers of mRNA (count for total mRNA of type i, $T_i$) and ribosomes (total count, R). (a) The correlation is nonnegative when rebinding is absent $\left(p=0\right)$. (b) However, a finite probability of rebinding $\left(p=0.5\right)$ can lead to negative correlations for sufficiently large values of mRNA counts. The correlation is zero in the underloaded case ($T_1+T_2\le R$, bottom left corners of plots). In both a and b, the correlation has a maximum along the line $T_1=T_2=T_s$, and this peak occurs slightly above the balance point (roughly in the range $R<2T_s<2R$). Beyond the peak, along this line, the correlation decreases monotonically toward an asymptotic value which is given in Eq. 15; this value is zero for $p=0$ and strictly negative for $p>0$. Other parameters are $R=10$, $\mu =1$, and $\gamma =0.1$.

Figure 3

Plots of the steady-state correlation between protein counts when abundances $T_i$ of the mRNAs are independently Poisson distributed with identical means, τ, and number of ribosomes, R, is constant (see the fourth section). Two different scans of parameters are shown with either (a) number of ribosomes fixed, $R=10$, and $p\text{and}\tau$ variable, or (b) $p=0$ fixed and $R\text{and}\tau$ variable. The balance point, $2\tau =R$, is plotted as a black dashed line. The correlation is very near zero for most of the underloaded regime $\left(2\tau <R\right)$ but exhibits a negative correlation resonance just after τ crosses the balance point. Furthermore, the correlation is observed to be nonpositive up to numerical error. Other parameters are $\mu =1$ and $\gamma =0.1$.

Figure 4

(a) Indirect repression of one gene’s output by another gene’s transcript. Competition for translational resources leads to an effective bottleneck for protein production (see the fourth section). This competition forms an effective repressive interaction between genes (dashed line). For illustration of our model, steady-state means and correlations are computed assuming that the abundances of mRNA copy numbers, $T_i$, are independently Poisson distributed with respective means, ${\tau }_i$, and ribosome copy number, R, is fixed. (b) Repression of the mean protein copy number, $\langle Y_2\rangle$, occurs due to increasing the mean level ${\tau }_1$ for various levels of ${\tau }_2$. (c) Correlations between the transcripts produce a substantial negative correlation resonance. There is a wider trough at the minimum when the system is unconditionally overloaded $\left({\tau }_2>R\right)$, e.g., see results for ${\tau }_2=15$. Other parameters are $R=10$, $p=0$, $\mu =1$, and $\gamma =0.1$.

Figure 5

Steady-state correlations between protein counts obtained using a quasi-steady-state approximation (solid lines). In this calculation, we use our hypergeometric function-based solutions for protein moments derived in the case of Poisson-distributed mRNA and fixed ribosome counts, and we then numerically average such moments over a given distribution for ribosome counts to derive the desired protein moments in the presence of fluctuating mRNA and ribosome counts. This quasi-steady-state calculation can thus be done by numerical evaluation of one-dimensional summations. It is assumed that the mRNA abundances, $T_i$, are independently Poisson distributed with common mean τ, and the ribosome count R has an independent discrete Gaussian distribution given by Eq. 26, with approximate mean $R_0$ and approximate variance parameter ${\sigma }_R^2$. The rebinding probability is either (a) $p=0.0$ or (b) $p=0.5$. Approximations based on a small-noise ansatz (see Eqs. 22–23) are also included (corresponding dashed lines). Other parameters are $R_0=10$, $\mu =1$, and $\gamma =0.1$.

Figure 6

Plots of the steady-state protein correlation for fluctuating mRNA counts according to Eqs. 19–21 (see the fourth section). Lower solid curves represent results obtained using the quasi-steady-state approximation in which mRNAs are independently Poisson distributed with identical means of τ. These curves are labeled with $\delta \to 0$, since this approximation becomes exact in the limit of arbitrarily slowly varying mRNA copy number. Upper solid curves represent results from stochastic simulations when mRNA copy numbers fluctuate on a timescale commensurate with that of the protein dynamics, $\delta \approx \gamma$. Simulation parameters were $\nu ={10}^5$, $\delta =0.1$, and ${\alpha }_i=\tau \delta$, with ensemble size $=12,800$. Approximate solutions derived in the Supporting Material, Sections C.3 and C.4, and discussed at the end of the fourth section of the main text, are included as nearby dashed lines. Results are plotted for (a) $p=0$, (b) $p=0.25$, (c) $p=0.5$, and (d) $p=0.75$. Other parameters are $R=10$, $\mu =1$, and $\gamma =0.1$.