Cellular-automaton model of the cooperative dynamics of RNA polymerase II during transcription in human cells

RNA polymerase II (RNAPII) is the responsible motor protein for transcription. Here we report the formulation and results of a cellular automaton model of the RNAPII dynamics of gene transcription that takes account the effect of the velocity change according to the gene position, such as occurs in introns and exons. We describe RNAPII dynamics in terms of the properties in the time domain, such as elapsed time, residence time, and time intervals. We found that the RNAPII molecules move as a free-flow state, though regions of reduced velocity do exist such as exons, as far as the time interval between nearest RNAPII molecules is larger than the time required for an RNAPII passing the exclusion length in the velocity reduction region. On the other hand, if the reduction is strong enough to reach a certain threshold, at the maximally reductive velocity region, a transition occurs from the RNAPII free-flow state to the states with congested and repetitive flows. We analytically obtained the conditions for these flow states and the transition threshold. From simulations of high-density RNAPII in the SAMD4A gene with the strong blockade, we confirmed the transition from free flow to the repetitive and congested flows, suggesting that the transition may serve as a regulatory mechanism of gene expression. By fitting the experimentally observed RNAPII density profile of the SAMD4A gene during the course of transcription of the normal and altered gene (in knock-down cells) with or without roadblock, we found that the RNAPII density flow is a free state. However, even in this free state, there is a long-range correlation between RNAPII molecules, ranging from 1 to 20 min, with the corresponding distance from 3 to 80 kbp, during transcription in normal cells. This long-range correlation probably relates to the higher-order DNA loop structure.

Figure 1

The states and densities conversion lines. The RNAPII dynamics on $({\Delta }_j^k,{\Delta }_{j+1}^k)$ plane at the intron-exon boundary described in Sec. 2c1 The motion of the RNAPII at the head of many RNAPIIs, which is in the state transition phase from the congested state to the free state, has a different character as shown by the coordinate $(a,b)$.

Figure 2

Simulation results of the RNAPII density $I_j^t$ in different $\gamma$ regions with reduced velocity. Color indicates the density abundance ranging from 0 (blue) to 1 (red). (a) The simulations with $\gamma =$ 1, 2, and 5 show the enrichments of the density at the sites of velocity reduction. The density distribution of RNAPII (vertical axis) that corresponds to the enrichment of the binding sites increases with the parameter $\gamma$. From left, middle, to right, RNAPII is passing before ($t<0$), during ($t=0$), and after ($t>0$) exon. (b) The density distribution of RNAPII (vertical axis) for the gene is fixed at $\mu =1$, but the velocity gap $\gamma$ is variable at 1, 5 and 50. (top) $\mu =1$ and $\gamma =1$, (middle) $\mu =1$ and $\gamma =5$, and (bottom) $\mu =1$ and $\gamma =50$. (c) The distribution of RNAPII with the velocity gap is fixed at $\gamma =50$, but with $\mu =5$ (top), 10 (middle), and 20 (bottom).

Figure 3

RNAPII density profiles $I_j^t$ obtained from the various simulation parameters. The position on the chromosome is given at the top, and the RNAPII density profile for the SAMD4A gene with a gene length of 221.209 kbp from the TSS (chromosomal position 54 104 387) to the terminus (chromosomal position 54 325 595) is in the lower column. 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)–(d) Simulation models of increasing complexity. The color bar represents the abundance ranging from 0 (bottom: gray-white) to 1 (top: red). The corresponding abundance of RNAPII indicated by this thin colored line. In the simulations, the blue sphere on the gene indicates the leading RNAPIIs in the distribution. The pink sphere on the gene indicates the re-njected RNAPII that has once ejected from the gene. (a) $\mu =5$ and $\gamma =5$. The number of RNAPII molecules on the gene body, $M=6$. The RNAPII injection timing $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,12,12,12,12,12} min. (b) $\mu =$10, $\gamma =5$, $M=8$, and $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,8,8,8,8,8,8,8} min, (c) $\mu =5$, $\gamma =5$, $M=9$, and $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,10,10,2,2,2,10,3,3} min, (d) $\mu =7$, $\gamma =5$, $M=12$, and $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,1,2,3,4,5,10,10,6,5,4,3} min.

Figure 4

The density profile of RNAPII and the corresponding CTCF and RAD21 enrichment. The color bar represents the abundance ranging from 0 (bottom: gray-white) to 1 (top: red). (a) The experimental density profile of RNAPII for genes at $t=30$ and 60 min after the stimulation, as observed in the ChIP-chip experiments, and also CTCF/cohesin (RAD21) enrichment at $t=0$ min [14]. (b) The simulation results of density $I_j^t$ at $t=30$ and 60 min with the optimized parameters: $\mu =7$ and $\gamma =5$. The number of RNAPII molecules on the gene body, $M=5$, and the RNAPII injection timing $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,4,4,6,6} min in the gene at $t=3$0 min will be changed to $M=7$, and $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,4,4,6,6,12,20} min at $t=60$ min. The sphere on the gene indicates the leading RNAPIIs in the distributions.

Figure 5

The obtained time-evolution pattern in SAMD4A for the parameters: $\gamma =10$, $0⩽t⩽300$ min, and $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,4,4,6,6,12,20,20,20,$\dots$} min. The intensity of $I_j^t$ at $G_j$, in which an RNAPII molecule exists, is ranging from gray-white to red while an empty position is shown in white on time-position plane. Higher (red) intensities at sites in the gene suggest that the RNAPII molecules gradually assemble in exons and at CTCF/cohesin (RAD21) binding sites. This generates the same homogeneous flow as before, after passing through the $\gamma$ region, since $\left\{\delta t_{\text{in}}\left(k\right)\right\}$ satisfies the condition in Eq. (25).

Figure 6

The RNAPII density profile in the knock-down cells at $t=60$ min and the putatively corresponding simulation results. (a) Experimental results for the density profile in RAD21 knock-down cells at $t=60$ min. (b) The simulation results for the RNAPII density profile $I_j^t$ at $t=120$ min, where the transcription status of the SAMD4A gene is stable and RNAPII is deliberately reinjected, in addition to the transient injections. The parameters are the same as in Fig. 4. The number of RNAPII molecules on the gene body $M=10$, and the RNAPII injection timing is $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,4,4,6,6,12,20,20,20,20} min. The color bar represents the abundance ranging from 0 (bottom: gray-white) to 1 (top: red). (c) The simulation result of the RNAPII density profile $I_j^t$ at $t=60$ min for the knock-down cells. Here the simulation parameters are $\mu =7$, $\gamma =1$, $v_i=v_e=7.0$ kbp/min, $M=13$, and $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,4,4,6,6,12,20,20,20,20,20,20,20} min.

Figure 7

The obtained time-evolution pattern in SAMD4A for the parameters $\gamma =100$, $0⩽t⩽300$ min, $M=18$, and $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1} min. (a) The intensity of $I_j^t$ at $G_j$, in which an RNAPII molecule exists, is ranging from gray-white to red while an empty position is shown in white on time-position plane. If the injection delay $\delta t_{\text{in}}\left(k\right)$ is shorter than the critical conversion point in Eq. (27), the RNAPII molecules were congested around the exon region, and after that, they traveled with the constant delay ${\gamma }_{j+1}\eta /v_i$, which resulted in a repetitive pattern of flow over a certain range. (b)–(g) The RNAPII density profile $I_j^t$ at (b) $t=30$, (c) $t=60$, (d) $t=90$, (e) $t=120$, (f) $t=150$, and (g) $t=300$ min. In the simulations, a blue sphere on the gene indicates the preceding RNAPIIs with a distribution for the cell population. Pink spheres on the gene indicate the re-injected RNAPII that were once ejected from the gene. The color bar represents the abundance ranging from 0 (bottom: gray-white) to 1 (top: red).

Figure 8

The obtained time-evolution pattern in SAMD4A for the parameters: $\gamma =200$, $0⩽t⩽300$ min, $M=100$, and the same time delays with $\left\{\delta t_{\text{in}}\left(k\right)\right\}=$ {0,2,2,2,2,2,2, …} min. The intensity of $I_j^t$ at $G_j$, in which an RNAPII molecule exists, is ranging from gray-white to red while an empty position is shown in white on time-position plane. One can see that the congestion (red) stretches out from the first encountered exon of the large $\gamma$ to the $5^{\prime }$ side intron region. However, the $3^{\prime }$ side intron region remains a homogeneous flow state over a certain range.