Theoretical analysis of the distribution of isolated particles in totally asymmetric exclusion processes: Application to mRNA translation rate estimation

The Totally Asymmetric Exclusion Process (TASEP) is a classical stochastic model for describing the transport of interacting particles, such as ribosomes moving along the messenger ribonucleic acid (mRNA) during translation. Although this model has been widely studied in the past, the extent of collision between particles and the average distance between a particle to its nearest neighbor have not been quantified explicitly. We provide here a theoretical analysis of such quantities via the distribution of isolated particles. In the classical form of the model in which each particle occupies only a single site, we obtain an exact analytic solution using the matrix ansatz. We then employ a refined mean-field approach to extend the analysis to a generalized TASEP with particles of an arbitrary size. Our theoretical study has direct applications in mRNA translation and the interpretation of experimental ribosome profiling data. In particular, our analysis of data from Saccharomyces cerevisiae suggests a potential bias against the detection of nearby ribosomes with a gap distance of less than approximately three codons, which leads to some ambiguity in estimating the initiation rate and protein production flux for a substantial fraction of genes. Despite such ambiguity, however, we demonstrate theoretically that the interference rate associated with collisions can be robustly estimated and show that approximately 1% of the translating ribosomes get obstructed.

Figure 1

Illustration of the TASEP with open boundaries. (a) A schematic representation of the TASEP model. Particles are introduced at the start of the lattice with exponential rate $\alpha$ and move along with exponential rate 1, provided that there is no particle occupying the next site. At the end of the lattice, they exit with exponential rate $\beta$. (b) Phase diagram of the average particle density along the lattice. The profile of average density of particles along the lattice can be classified according to a phase diagram in $(\alpha ,\beta )$ space, separating different regions: the maximal current regime (MC), the low density regime (LD), and the high density regime (HD). The LD and HD regions can also be decomposed into two separate ones: LD I/II, and HD I/II, respectively.

Figure 2

The density of isolated particles in different regions of the TASEP phase diagram. For the different regimes of the TASEP (see also Fig. 1), the asymptotic formulas from Table 1 (red points) are compared with the exact densities (black points) of isolated particles given by (4, 5, 6, 7, 8).

Figure 3

The density of particles in the $\ell$-TASEP model. We simulated and plot (in black) the density of particles of the $\ell$-TASEP ($\ell =10$) in the different regimes LD, HD, and MC. In red, we plot the estimates of the density in the bulk from Lakatos and Chou [2].

Figure 4

Comparison of the results from the refined mean-field approach with Monte Carlo simulations. (a) We simulated the TASEP with extended particles (size $\ell =10$, sample size $={10}^9$) and plotted (in red) the densities of isolate particles in the three different regimes of the phase diagram. We compared these simulation results with the asymptotics obtain from (13), (16), and (19) (in black). (b) For fixed values of $\beta$, these plots show how the total density, the density of isolated particles, and their ratio vary as a function of $\alpha$. The results obtained using Monte Carlo simulations (open circles) of the TASEP with extended particles (size of particles $\ell =10$, sample size of isolated particles ${10}^4$, lattice size $=400$) are compared with the results obtained from the refined mean-field approach (solid lines). Note that there are discontinuities when transitioning from LD to HD regime (first and second rows).

Figure 5

A schematic representation of ribosome profiling. (a) Positions of ribosomes along the mRNA are obtained by nuclease digestion and allowed to count the number of ribosomes found at a specific position. However, it is possible that the nuclease cannot cleave stacked ribosomes [6, 14, 17, 18, 19]. (b) As a result, the profile of ribosome count along the mRNA recorded from isolated ribosomes (plotted in red) might be different from the true profile (plotted in black).

Figure 6

Estimation of undetected ribosomes from ribosome profiling experiment. (a) This plot shows experimental ribosome profiling data of S. cerevisiae from Weinberg et al. [29] against the total ribosome density obtained from polysome profiling by Arava et al. [30] (482 genes). Also shown are plots of $y=ax{\left(\frac{1-10x}{1-9x}\right)}^{2d}$, obtained from computing the density of detected particles of size $\ell =10$ as a function of the total density in the $\ell$-TASEP [see (26)] with various isolation range $d=0,...,10$. We set $a=0.82$, obtained by linear fit to genes with total density less than 1 ribosome per 100 codons. (b) For values of $d\in \{0,...,10\}$, we plot the root-mean-square error obtained from comparing experimental data to the theoretical plots in (a). (c) For $d\in \{0,...,10\}$, we plot the corresponding fraction of genes with anomalous detected densities, where a detected density said to be anomalous if it is larger than the theoretical maximum value implied by (26), used in (a).

Figure 7

Analysis of initiation and traffic obstruction. (a) For different values of isolation range $d$, we plot the density of isolated particles [see (23)] as a function of the initiation rate $\alpha$ in the LD regime. (b) For different ranges $d$ of isolation, we studied the identifiability of the initiation rate. Black line: We estimated the fraction of genes for which there exists a corresponding value for the initiation rate $\alpha$, such that the associated density of isolated particles is equal to the detected density. This happens when the detected density is less than ${\langle {\tau }^{\left(d\right)}\rangle }_{\text{max}}\left(\beta \right)$ [see (27)], where $\beta$ is the inferred termination rate Red line: We estimated the fraction of genes for which the initiation rate can be inferred without ambiguity from the plotted curves in (a), which happens when the detected density is less than ${\langle {\tau }^{\left(d\right)}\rangle }_{\text{id}}\left(\beta \right)$ [see (28)]. (c) From our data set of 3712 genes, we used (23) to estimate the fraction of detected ribosomes for different values of the detection gap threshold $d\in \left\{1,...,6\right\}$. To compute these fractions when there is an ambiguity in identifying the initiation rate $\alpha$ [see (b)], we considered two possible estimates: a lower estimate and an upper one [see also Fig. 8]. The left plot represents the average fraction of detected ribosomes, with error bars indicating the standard deviation, using lower estimates (in blue) and upper estimates (in red) of $\alpha$. (d) The same as in (c) for obstruction rates, using (25) [see also Fig. 8].

Figure 8

The fraction of isolated particles and obstruction rate as a function of $\langle {\tau }^{\left(d\right)}\rangle$. (a) For different isolation ranges $d\in \left\{1,...,6\right\},$ we plot the fraction of isolated particles as a function of the average density of isolated particles $\langle {\tau }^{\left(d\right)}\rangle$, according to (23). Note that for given $d$, some values of $\langle {\tau }^{\left(d\right)}\rangle$ can lead to two possible fractions of isolated particles. (b) As in (a), we plot the isolation rate as a function of the average density of isolated particles $\langle {\tau }^{\left(d\right)}\rangle$, according to (25). Note that for $\langle {\tau }^{\left(d\right)}\rangle \le 0.02$ and all $d$, the initiation rates associated with the lower branch are very close.