Path-preference cellular-automaton model for traffic flow through transit points and its application to the transcription process in human cells

We all use path routing everyday as we take shortcuts to avoid traffic jams, or by using faster traffic means. Previous models of traffic flow of RNA polymerase II (RNAPII) during transcription, however, were restricted to one dimension along the DNA template. Here we report the modeling and application of traffic flow in transcription that allows preferential paths of different dimensions only restricted to visit some transit points, as previously introduced between the $5^{\prime }$ and $3^{\prime }$ end of the gene. According to its position, an RNAPII protein molecule prefers paths obeying two types of time-evolution rules. One is an asymmetric simple exclusion process (ASEP) along DNA, and the other is a three-dimensional jump between transit points in DNA where RNAPIIs are staying. Simulations based on our model, and comparison experimental results, reveal how RNAPII molecules are distributed at the DNA-loop-formation-related protein binding sites as well as CTCF insulator proteins (or exons). As time passes after the stimulation, the RNAPII density at these sites becomes higher. Apparent far-distance jumps in one dimension are realized by short-range three-dimensional jumps between DNA loops. We confirm the above conjecture by applying our model calculation to the SAMD4A gene by comparing the experimental results. Our probabilistic model provides possible scenarios for assembling RNAPII molecules into transcription factories, where RNAPII and related proteins cooperatively transcribe DNA.

Figure 1

The time-evolution rules for RNAPII molecules based on the ASEP and jump transfer rule. Usually RNAPII molecules move according to the ASEP. When an RNAPII molecule arrives at the end of a gene (an exon), the molecule jumps to the beginning of the next gene (or the next exon) with probability $p$, or it does not jump and obeys the ASEP with probability $1-p$.

Figure 2

The motion of RNAPII molecules in looped DNA of a gene. In the initial state, RNAPII molecules are randomly arranged and move according to the ASEP with periodical boundary condition.

Figure 3

From the intermediate state to the steady state, the DNA loop in the gene transforms its shape according to the condition (i) [i.e., whenever an RNAPII molecule passes an intergene region (or intron), the RNAPII molecule keeps away the intergene region (or intron)]. An RNAPII molecule becomes able to jump from the end of the gene (or an exon) to the beginning of the next gene (next exon) with probability $p$, except at the edge of the last gene (last exon).

Figure 4

The obtained time-evolution pattern for the parameters $N=155$ cells [the number of intergenes (introns) = 5, the number of cells per genes (exon) = 5, the ratio of the length of the intergene and gene region (intron to that of an exon) = 5], $M=15$, $h=2$ (cells/step), $t=300$ steps, $w_0=10$, $\epsilon =2$. The cell in which an RNAPII molecule exists is shown in white, while an empty cell is shown in black. The RNAPII molecules gradually assemble in genes (exons).

Figure 5

The average speed of RNAPIIs in the introns and exons according to our simulation. We used the parameters $N=12020$ (the number of introns = 100, the number of cells per exon = 20, the ratio of the length of an intron to that of an exon = 5), $M=200$, $t=10000$ steps, $w_0=10$, $\epsilon =2$, (a) $h=1$ (cell/step), (b) $h=2$ (cells/step). With time, the average speed of RNAPII increases at both (a) $h=1$ and (b) $h=2$, while that in the exon is constant at (a) $h=1$ and is slowly changing (but looks constant) at (a) $h=2$. Then, the velocity gap in the RNAPIIs at the intron and exon is increasing with time for both $h=1$ and $h=2$. At early time steps, it is 1.08 (251 steps) and 1.56 (501 steps) for $h=1$, and 1.59 (251 steps), and 6.04 (501 steps) for $h=2$, respectively.

Figure 6

The time dependence of flow as a function of the RNAPII density (fundamental diagrams) for $h=1$. The total number of cells $N=12020$ (the number of introns = 100, the number of automaton cells per exon = 20, the ratio of the length of an intron to that of an exon = 5), $h=1$ (cell/step), $w_0=10$, $\epsilon =2$. (a) $t=1$ steps, (b) $t=10$ steps, (c) $t=50$ steps, (d) $t=500$ steps, (e) $t=10000$ steps.

Figure 7

The time dependence of flow as a function of the RNAPII density (fundamental diagrams) for $h=2$. The total number of cells $N=12020$ (the number of introns = 100, the number of automaton cells per exon = 20, the ratio of the length of an intron to that of an exon = 5), $h=2$ (cells/step), $w_0=10$, $\epsilon =2$. (a) $t=1$ steps, (b) $t=10$ steps, (c) $t=50$ steps, (d) $t=500$ steps, (e) $t=10000$ steps.

Figure 8

The parameter $h$ dependence of flow as a function of the RNAPII density at $t=2000$ steps. The other parameters are the same as those in Fig. 6. (a) $h=1$ (cell/step), (b) $h=2$ (cells/step), (c) $h=3$ (cells/step), and (d) $h=4$ (cells/step).

Figure 9

Comparison RNAPII density profiles obtained from the experiment and simulation with different models. The position on the chromosome is given at the bottom. The gene length of the SAMD4A gene is 221.209 kbp from the TSS (chromosomal position 54,104,387) to the terminus (chromosomal position 54,325,595). The introns are denoted by the red bars, the exons are in yellow, and the highly enriched CTCF-cohesin binding sites are denoted by the green rings. (a) The experimental results of CTCF enrichment at $t=0$ (min) after stimulation and RNAPII density at $t=0$ and $t=30$ (min) [3]. (b) The simulation result of RNAPII density at $t=30$ (min) based on the model without taking into account the 3D diffusion at CTCF enriched positions [8]. It provides an agreement with the experimental results overall but fails to explain the experimental result of RNAPII enrichment at the positions indicated by red arrows at $P4$ and $P5$, when the leading RNAPII reaches at position $L$ (indicated by black arrow) along the DNA template. The RNAPII (the blue sphere) is the leading one (at $\mu$ in the Gaussian distribution) that is immediately activated by the stimulation, not the maximum of the distribution, because the beginning of the increase in the abundance of the pre-mRNA synthesized by the leading RNAPII is experimentally detectable [3]. (c) The simulation result of RNAPII density at $t=30$ (min) based on the model with taking into account the 3D diffusion at CTCF enriched position between $P1$, $P2$, $P3$, and $P4$. Here the jump probability from TSS to the transient points are the same: $p$ = 0.2. It is in agreement with experimental result of RNAPII enrichment at the position indicated by red arrows at $P4$ and $P5$. The blue spheres indicate the leading RNAPII in the distribution.