Fig 1.
Main characteristics of the TASEP model.
(a) Phase diagram. A low density (LD) phase, with density ρ < 1/2, arises when the current is limited by the input rate α. Symmetrically, a high density (HD) phase, with ρ > 1/2 is observed when the flow is governed by the output rate β. When neither is limiting, in the ‘maximal current’ phase (MC), the current is as large as the hopping process in the bulk permits (J = k/4 for ρ = 1/2). The line separating the LD and HD phases is where a shock phase (SP) is observed, where LD and HD zones coexist along the lattice. In this case the (few) sites corresponding to the interface between these these two zones is known as the ‘domain wall’. (b)-(e) Density profiles ρi of the original TASEP for a lattice of N = 500 sites and hopping rate k = 1. (b): LD regime: α = 0.2, β = 1; (c): HD regime: α = 1, β = 0.1; (d): MC regime: α = β = 0.5; (e): SP regime: α = β = 0.3. The solid black line represents the analytical solution from the mean-field approximation, and the blue points correspond to numerical simulations using the Gillespie algorithm [41].
Fig 2.
Illustration of the ribosome drop-off model.
Ribosomes bind to the first codon of the mRNA with rate α and leave the lattice representing the mRNA strand at the stop codon, with rate β. Throughout the mRNA lattice they can either hop to the next codon, if it is free, with rate k, or drop off the lattice with rate γ.
Fig 3.
Both real branches W0(y) (bold blue line) and W−1(y) (red line) of the Lambert function.
The solid part of the blue line corresponds to the low density and maximal current phases, and the solid part of the red line corresponds to the high density phase.
Fig 4.
Density and current profiles numerically computed via stochastic simulations (blue and magenta dots, respectively) and analytically estimated (black lines) for and L = 1: (a, b) LD:
= 0.2,
= 1; (c, d) HD:
= 1,
= 0.2; (e, f) MC:
= 1,
= 1.
Hopping and drop-off events are scheduled based on the Gillespie algorithm [41]. All simulations, here and in the following, are for 107 iteration steps (each iteration corresponds to one reaction: initiation, elongation, drop-off or termination). The first 2 ⋅ 107 iterations were discarded to make sure that the system was in a steady state. Unless otherwise stated, a lattice size of 500 sites was used.
Fig 5.
Density and current profiles in the shock phase, for , L = 1 and
.
(a,b) , (c,d)
and (e,f)
. Numerical simulations are shown as blue (density) and magenta (current) points, whereas analytical predictions for the density profiles are shown as black lines. For the current profiles the analytical expressions for JLD(x) and JHD(x) are shown with green and red dashed lines, respectively.
Fig 6.
Illustration of finite size effects in the shock phase, based on the density profile for ,
,
and L = 1.
The blue dots represent the numerical results for N = 500 sites,whereas red dots are for N = 1000 sites. The solid black line corresponds to the analytical solution.
Fig 7.
Phase diagram for and L = 1 showing the average density on the lattice (heat map) numerically computed via stochastic simulations, and the borders among the phases (black lines) estimated analytically within the mean-field approximation.
Fig 8.
SP phase in the phase diagram, for and L = 1.
The colour map represents the value of xw for the position of the domain wall between LD and HD zones, as determined numerically. Where no data is shown xw was negative or larger than L, indicating respectively a LD or a HD state point. The represented points thus identify the zone corresponding to a SP, which is to be confronted to the analytical estimation of its phase boundaries (black lines).
Fig 9.
Extent of the MC-HD shock phase for L = 1, depending on (analytical results); black line:
, blue lines:
, green lines:
and the purple line:
.
The insets show the full phase diagram for these three values.
Fig 10.
Analytically computed upper and lower boundaries to the MC-HD shock phase, depending on for L = 1.
The green line represents the lower boundary and the blue line corresponds to the upper boundary
. The horizontal red dashed line serve as illustration. The intersections (at
) with
and
indicate the values of the drop-off rates at which an mRNA with termination rate
would undergo a transition from HD to SP, and from SP to MC, respectively.
Fig 11.
Resilience to drop-off for various choices of initiation and termination rates, computed analytically (solid lines) and numerically (dots) for L = 1.
The blue line represents a mRNA sequence in the low density regime, with
. The red line corresponds to a sequence starting in the high density regime with
,
. The maximal current regime is represented by the green line with
and
.
Fig 12.
Resilience to drop-off, computed analytically (solid lines) and numerically (dots) for L = 1 and .
Red: , blue:
, and green:
. For these simulations, the transient time was 109 iterations, followed by an integration time of 109 iterations.
Fig 13.
Drop-off resilience versus elongation rate analytically (solid lines) and numerically computed (dots) for L = 1 and Γ = 0.05.
Blue (LD): α = 0.1, β = 1; Red (MC): α = 1, β = 1; Black (HD): α = 1, β = 0.1. For these simulations, the transient time was 107 Gillespie iterations, followed by an integration time of 108 iterations.
Fig 14.
Density (a,c,e) and current (b,d,f) profiles for CDC7 (YDL017W), for α = 7 × 10−2 s−1, in a low density-like phase.
In the density profiles we represent the coverage density of ribosomes, i.e., the probability for a codon to be covered by a ribosome. (a),(b): no ribosome drop-off γ = 0; (c), (d): physiological value of ribosome drop-off γ = 1.4 × 10−3 s−1; (e), (f): ribosome drop-off under ethanol stress γ = 5.6 × 10−3 s−1. The slope in the current profile clearly increases with the ribosome drop-off rate, and in the density profile the slight slope towards the 3’ is also reinforced by drop-off.
Fig 15.
Density (a, c, e) and current (b, d, f) profiles for SLT2 (YHR030C), for α = 1s−1, a in high density-like phase.
Details, including the values of γ, are as in Fig 14. (a),(b): no ribosome drop-off γ = 0; (c), (d): physiological value of ribosome drop-off γ = 1.4 × 10−3 s−1; (e), (f): ribosome drop-off under ethanol stress γ = 5.6 × 10−3 s−1. The increasing slope in the 5’ to the 3’ direction characteristic of the HD phase is clearly visible in panel (c). Moreover, panel (d) illustrates the marked increase in the current at the 5’ end, compared to the case without ribosome drop-off (see panel (b)), also characteristic of the HD phase.
Fig 16.
Density (a, c, e) and current (b, d, f) profiles for CRH1 (YGR189C), for α = 1s−1, in a maximal current-like phase.
Details, including the values of γ, are as in Fig 14. (a),(b): no ribosome drop-off γ = 0; (c), (d): physiological value of ribosome drop-off γ = 1.4 × 10−3 s−1; (e), (f): ribosome drop-off under ethanol stress γ = 5.6 × 10−3 s−1. The current profiles clearly show an increase in the slope as ribosome drop-off increases, qualitatively similar to the effects predicted for sequences in MC.
Table 1.
Table summarising the results for the drop-off resilience (defined in Eq (31) for the 3 mRNA sequences CDC7 (LD), SLT2 (HD), and CRH1 (MC), and for three different values of γ.
Fig 17.
Histogram of ribosome drop-off resilience values χγ for the S. cerevisiae genome under physiological conditions.
The colours of the different regions correspond to the following χγ intervals: green: χγ < 0.7; yellow: 0.7 ≤ χγ < 0.8; orange 0.8 ≤ χγ < 0.9, and red: χγ ≥ 0.9.
Table 2.
Table summarising the results for Gene Ontology (GO) enrichment analysis for the 4 different regions of the histogram of the ribosome drop-off resilience values (Fig 17) obtained for the S. cerevisiae genome.