Controlling particle currents with evaporation and resetting from an interval

We investigate the Brownian diffusion of particles in one spatial dimension and in the presence of finite regions within which particles can either evaporate or be reset to a given location. For open boundary conditions, we highlight the appearance of a Brownian yet non-Gaussian diffusion: At long times, the particle distribution is non-Gaussian but its variance grows linearly in time. Moreover, we show that the effective diffusion coefficient of the particles in such systems is bounded from below by $1-2/\pi$ times their bare diffusion coefficient. For periodic boundary conditions, i.e., for diffusion on a ring with resetting, we demonstrate a “gauge invariance” of the spatial particle distribution for different choices of the resetting probability currents, in both stationary and nonstationary regimes. Finally, we apply our findings to a stochastic biophysical model for the motion of RNA polymerases during transcriptional pauses, deriving analytically the distribution of the length of cleaved RNA transcripts and the efficiency of RNA cleavage in backtrack recovery.

Figure 1

Diffusion with evaporation within a finite region (Brownian tunneling). (a) Sketch of the model: A Brownian particle (gray sphere) moves in one dimension from an initial position $x_0$ with diffusion coefficient $D$. The particle evaporates (light green arrows) with rate $r$ only from an interval $[-a,a]$, i.e., with the evaporation rate $r\left(x\right)$ in Eq. (2) (green line). (b) Fraction of “tunneled” particles $P_{\mathrm{tun}}\left(t\right|x_0)$ (multiplied by $D$) at time $t$ that survived evaporation by reaching the region $x\ge a$: results from numerical simulations (symbols) and analytical calculations (solid lines). Here $x_0=-3.2, r=25, D=200$ (green circles), 150 (red squares), 100 (blue diamonds), and $a=0.05\times \sqrt{D}$ respectively. Simulations were done using the Euler integration scheme with time step $\mathrm{\Delta }t=5\times {10}^{-4}$ and $N={10}^4$ runs. The inset shows the collapse of the three curves in dimensionless units.

Figure 2

Diffusion with stochastic resetting from an interval. (a) Sketch of the model: Diffusion of a Brownian particle (gray sphere) with initial position $x_0$, diffusion coefficient $D$, and resetting at rate $r$ from an interval to a resetting destination $x_r$. (b) Effective diffusion coefficient $D_{\mathrm{eff}}={lim}_{t\to \infty }{\sigma }^2\left(t\right)/2t$, with ${\sigma }^2\left(t\right)$ the variance of the position, in units of the “bare” diffusion coefficient $D$, as a function of normalized resetting point $y_r=x_r/a$. The simulations (symbols) and the analytical expression in Eq. (7) (solid lines) correspond to $\rho \simeq 1.12$ realized with $r=2.5, a=5, D=50$ (blue diamonds and solid line) and $\rho \simeq 1.89$ realized with $r=5, a=5, D=35$ (red squares and dashed line). The horizontal dotted line indicates the theoretical lower bound predicted by Eq. (8). The simulations were done using Euler's numerical integration scheme with time step $\mathrm{\Delta }t=5\times {10}^{-2}$, time duration $t={10}^3$, and $N={10}^5$ runs. The error bars are given by the standard deviation.

Figure 3

Brownian yet non-Gaussian diffusion induced by resetting. Probability density $P(x,t|x_0)$ for a Brownian particle to be at position $x$ at time $t=10$ with $D=1, r=1.0, a=2.5, x_0=x_r=-5$, for the model in Fig. 2: numerical simulations (symbols), numerical inverse Laplace transform of Eq. (B7) (solid line), and exact (Gaussian) distribution in the absence of resetting, i.e., $r=0$ (dashed line). The gray shaded area highlights the interval where resetting occurs. The numerical simulations were done using the Euler numerical integration scheme with time step $\mathrm{\Delta }t=0.05$ and $N={10}^5$ runs.

Figure 4

Diffusion with stochastic resetting on a ring according to different protocols (resetting “gauge invariance”). (a) Sketch of two resetting protocols: The particles are reset to a fixed destination $x_r>0$ at a rate $r$ only when located in the window $[-a,a]$. Top (model I): Resetting moves the particle along the path that does not cross the endpoints $\pm L$. Bottom (model II): Resetting occurs via the path of minimal distance between the positions before and after resetting. (b) Numerical results for the stationary spatial distribution for $y_r=x_r/a=-2$ (blue diamonds) and $y_r=0.8$ (red circles), compared with the corresponding analytical predictions (solid lines; see Appendix pp3) with $D=50, r=1, a=25, L=70$. (c) Stationary total particle current $J$ given by the right-hand side of Eq. (9) for models I (dotted lines) and II (solid lines) as a function of the normalized resetting destination $y_r$, for different resetting rates: $r=0.01$ (red), 0.1 (blue), and 10 (green) with $D=5, a=7.5, L=30$. (d) Space-dependent decomposition of the total stationary current $J\left(x\right)=J_{\mathrm{diff}}\left(x\right)+J_{\mathrm{res}}\left(x\right)$ (dotted line) as a function of the normalized position $y=x/a$ for the model II: diffusive current $J_{\mathrm{diff}}\left(x\right)$ (blue line) and resetting current $J_{\mathrm{res}}\left(x\right)$ (red line). With $\rho =15.0, y_r=-2.5$, and $l=L/a=5$. A discontinuous jump of both $J_{\mathrm{diff}}\left(x\right)$ and $J_{\mathrm{res}}\left(x\right)$ occurs at $x=x_r$. In all the panels, numerical results are obtained from $N={10}^5$ simulations with time step $\mathrm{\Delta }t=0.1$ while the gray area corresponds to the resetting region.

Figure 5

Modeling backtrack recovery of RNA polymerase. (a) Illustration of an RNA polymerase (gray circle) diffusing along a DNA template (black ladder). During a transcriptional pause, the polymerase cannot resume RNA polymerization because its active site (yellow box)—where new nucleotides are added—is blocked by backtracked RNA (red line). Polymerases “recover” from the backtracking state when the distance $x\ge 0$ (“backtrack depth,” yellow arrows) between the active site and the $3^{\prime }$ end of the RNA vanishes. Recovery from backtracking is due to Brownian diffusion of the polymerase with diffusion coefficient $D$ (black arrows) or to the action of active processes that allow cleavage (orange scissors) of backtracked RNA of length up to a finite length $a\ge 0$ and at random times with constant rate $r$ [11]. (b) Model for the evolution of the backtrack depth $x$ from an initial value $x_0$. Polymerases recover by reaching the absorbing state $x=0$ either by diffusion or by resetting, representing RNA cleavage [11, 23]. (c) Distribution of the length of cleaved RNA for Pol I, Pol II, and Pol II TFIIS with initial backtrack depth $x_0=5$ (nucleotides). (d) Cleavage (resetting) efficiency as a function of the initial backtrack depth. The numerical prediction for ${\eta }_{\mathrm{res}}\left(y_0\right)\equiv \eta (x_0=a{y}_0)$ has been already reported in Ref. [11]. In both panels (b) and (c), the symbols are obtained from numerical simulations and the solid lines are theoretical predictions given by Eqs. (10) in panel (c) and (11) in panel (d). Values of the parameters $D, r$ and $a$ can be found in Appendix pp5. Simulations were done with time step $\mathrm{\Delta }t=0.1$ for panel (c) and $\mathrm{\Delta }t=0.025$ for panel (d) with number of simulations $N={10}^5$.

Figure 6

Statistics of the Brownian tunneling: (a) Probability density $P_{\mathrm{nc}}(x,t|x_0)$ at time $t=15$ with $D=1, r=0.5, a=2.5, x_0=-5$. The solid line represents the inverse numerical Laplace transform of Eq. (B4) while the symbols correspond to the result of numerical simulations. The gray shaded area highlights the interval within which resetting occurs. (b) Probability of first reset time $P_{\mathrm{res}}\left(t\right|x_0)$ up to time $t=15$ with the same parameters as panel (a): comparison between inverse Laplace transform of Eq. (B5) and numerical simulations. (c) Probability density $P(x,t|x_0)$ with $x_r=x_0$, indicated by the vertical dashed line. The solid line represents the inverse Laplace transform of Eq. (B7) while symbols indicate the results of numerical simulations. All numerical simulations were done using the Euler numerical integration scheme with time step $\mathrm{\Delta }t=0.05$, and they were repeated $N={10}^5$ times.

Figure 7

Statistics of the Brownian tunneling: (a) Probability density $P_{\mathrm{nc}}(x,t|x_0)$ at time $t=15$ with $D=1, r=0.5, a=2.5, x_0=1$. The solid line represents the inverse numerical Laplace transform of Eq. (B8) while the symbols correspond to the result of numerical simulations. The gray shaded area highlights the interval within which resetting occurs. (b) Probability of first reset time $P_{\mathrm{res}}\left(t\right|x_0)$ up to time $t=15$ with the same parameters as panel (a): comparison between inverse Laplace transform of Eq. (B9) and numerical simulations. (c) Probability density $P(x,t|x_0)$ with $x_r=x_0$, indicated by the vertical dashed line. The solid line represents the inverse Laplace transform of Eq. (B10) while symbols indicate the results of numerical simulations. All numerical simulations were done using the Euler numerical integration scheme with time step $\mathrm{\Delta }t=0.05$, and they were repeated $N={10}^5$ times.

Figure 8

Brownian tunneling with resetting: Comparison between numerical simulation (solid line) and theory (symbols) of the time evolution of the average position $\langle x\left(t\right)\rangle$ of the particle with parameters $D=50, a=5, r=2.5$, and $x_0=x_r=-7.5$. Simulations are performed with a time step $\mathrm{\Delta }t=0.05$ and are repeated $N={10}^5$ times.

Figure 9

Brownian tunneling with resetting: Comparison between numerical simulation (solid line) and theory (symbols) of the time evolution of the variance $\langle {\sigma }^2\left(t\right)\rangle$ of the particle with parameters $D=50, a=5, r=2.5$, and $x_0=x_r=-7.5$. Simulations are performed with a time step $\mathrm{\Delta }t=0.05$ and are repeated $N={10}^5$ times.

Figure 10

Statistics of resetting with periodic boundary conditions: (a) Probability density $P_{\mathrm{nc}}(x,t|x_0)$ at time $t=15$ with $D=5, r=1, a=5, x_0=-7.5$. The solid line represents the inverse numerical Laplace transform of Eq. (C2) while symbols correspond to simulations. The gray shaded area indicates the region within which resetting may occur. (b) Probability of first reset time $P_{\mathrm{res}}\left(t\right|x_0)$ as a function of $t$ with same parameters: comparison between inverse Laplace transform of Eq. (C5) and simulations. (c) Probability density $P_{\mathrm{nc}}(x,t|x_0)$ as a function of position $x$ with $x_r=x_0$, indicated by the vertical dashed line. The solid line represents the inverse Laplace transform of Eq. (C6) while diamonds represent simulations. Simulations were done using the Euler numerical integration scheme with time step $\mathrm{\Delta }t=0.05$, and they were repeated $N={10}^5$ times.

Figure 11

Statistics of resetting with periodic boundary conditions: (a) Probability density $P_{\mathrm{nc}}(x,t|x_0)$ at time $t=15$ with $D=5, r=1, a=5, x_0=1.5$. The solid line represents the inverse numerical Laplace transform of Eq. (C8) while symbols correspond to simulations. The grey shaded area indicates the region within which resetting may occur. (b) Probability of first reset time $P_{\mathrm{res}}\left(t\right|x_0)$ as a function of $t$ with same parameters: Comparison between inverse Laplace transform of Eq. (C9) and simulations. (c) Probability density $P(x,t|x_0)$ as a function of position $x$ with $x_r=x_0$, indicated by the vertical dashed line. The solid line represents the inverse Laplace transform of Eq. (C10) while diamonds simulations. Simulations were done using the Euler numerical integration scheme with time step $\mathrm{\Delta }t=0.05$, and they were repeated $N={10}^5$ times.

Figure 12

Pictorial representation of the contribution for the resetting current $J_{\mathrm{res}}\left(x\right)$ in the case of $x_r<-a$. The shaded area corresponds to the resetting region in which the resetting current $J_{\mathrm{res}}$ originates. Horizontal arrows depict the particles moving towards the resetting point $x_r$ in the two possible directions (left-moving and right-moving): Thick arrows represent particle currents that contribute to $J_{\mathrm{res}}$ while thin ones to the particles that reach $x_r$ without crossing (“stop” sign) the region in which we compute the current.

Figure 13

Stationary current $J-J_{\mathrm{res}}$ in Eq. (D12) as a function of the normalized position of the resetting point $x_r/a$ and for various values of $r=0.1$ (red), 1.0 (blue), and 20 (green), with $D=5, a=7.5$, and $L=20$. The gray area corresponds to the resetting region.

Figure 14

Schematic representation of the contribution to the current $J_{\mathrm{res}}(-L)$ from particles resetting in the region $(x_r,L)$ according to the minimal path protocol, the forbidden complementary region $(-L,x_r)$ is denoted by oblique lines. In both cases, $x_r\in (-L,-a)$, panel (a), and $x_r\in (-a,a)$, panel (b), the protocol identifies two distinct regions: The yellow shaded region $(x_r+L,a)$ refers to the particles resetting following $l_2$ (right moving) and the white region $(-a,x_r+L)$ following $l_1$ (left moving). The resetting particles, horizontal arrows, that contribute to the current $J_{\mathrm{res}}(-L)$ are only those following $l_2$.

Figure 15

Schematic representation of the contribution to the current $J_{\mathrm{res}}(-L)$ from particles resetting in the region $(-L,x_r)$ according to the minimal path protocol, the forbidden complementary region $(x_r,L)$ is denoted by oblique lines. In both cases $x_r\in (-a,a)$, panel (a), and $x_r\in (a,L)$, panel (b), the protocol identifies two distinct regions: The yellow shaded region $(-a,x_r-L)$ refers to the particles resetting following $l_2$ (left moving) and the white region $(x_r-L,a)$ following $l_1$ (right moving). The resetting particles, horizontal arrows, that contribute to the current $J_{\mathrm{res}}(-L)$ are only those following $l_2$.

Figure 16

(a) Total stationary current $J=J_{\mathrm{diff}}+J_{\mathrm{res}}$ as a function of $x_r/a$ and for various of $r=0.5$ (red), 2.5 (blue), and 75 (green), with $D=5, a=5$, and $L=20$ according the minimal path protocol. (b) Comparison between the analytic prediction for $J$ [red curve in (a)] and the numerical simulations (red symbols with error bars). The gray shaded area corresponds to the resetting region. Simulations were done using the Euler numerical integration scheme with time step $\mathrm{\Delta }t=0.5$, and they were repeated $N={10}^5$ times. The error bars are given by the standard deviation.

Figure 17

(a) Total stationary current $J=J_{\mathrm{diff}}+J_{\mathrm{res}}$ as a function of $x_r/a$ and different values of $r=0.5$ (red), 1.5 (blue), and 2.5 (green), with $D=5, a=5, L=20$, and $\lambda =0.75$ fixed. (b) Comparison between the analytic prediction for $J$ [red curve in (a)] and the numerical simulations (red symbols with error bars). The gray shaded area corresponds to the resetting region. Simulations were done using the Euler numerical integration scheme with time step $\mathrm{\Delta }t=0.2$, and they were repeated $N={10}^5$ times. The error bars are given by the standard deviation.

Figure 18

Survival probability $S\left(t\right)$ with parameters $r=1, D=10, a=10$, and $x_0=5.$ Solid line indicates the result of numerical simulations and indicates the result of numerical inverse Laplace transform of Eq. (E4). Simulations were done using the Euler numerical integration scheme with time step $\mathrm{\Delta }t=0.01$, and they were repeated $N={10}^5$ times.