Searching for a partially absorbing target by a run-and-tumble particle in a confined space

To obtain a comprehensive picture of the underlying physics, we express the searching time $T$ for a partially absorbing target in the following form:

where $T_{first}$ is first-passage time, i.e., the time it takes for a searcher starting from a given initial position to arrive at the target for the first time, $T_{return}^{(i)}$ is the first-return time to the target starting from the target position after $i$-th visit to the target, and $N_{pass}$ is the number of times that the searcher passes the target until it is finally absorbed. Since they are all independent random numbers, the average of searching time can be written as

where $\langle T_{return}\rangle=\langle T_{return}^{(i)}\rangle$. The boundary condition at $x=0$, Eq. (5), determines the probability that RTP finds the target when visiting the target site, $p_{find}=1-e^{-\tilde{k}/\tilde{v}}$. The probability distribution of $N_{pass}$ is thus given as

which yields

$\langle T_{first}\rangle$ and $\langle T_{return}\rangle$ are the mean first-passage times for different initial distributions of searcher; $\langle T_{first}\rangle$ is the mean first-passage time when the initial searcher distribution is uniform, and $\langle T_{return}\rangle$ is the mean first-passage time when the searcher starts from the target site. In Appendix A and B, the mean and the variance of $T_{first}$ and $T_{return}$ are analytically evaluated to give

By substituting these into Eq.(30), the expression of the mean searching time, Eq.(28), is reproduced.

In the following subsections, we investigate the asymptotic behaviors of $\langle T\rangle$ in different limits and find the optimal velocity that minimizes the searching time.

Let us begin with two limiting cases: $\tilde{v}\rightarrow 0$ limit and $\tilde{v}\rightarrow\infty$ limit. The rescaled velocity $\tilde{v}$ is, by definition, the ratio between the persistence length $\xi=v/\lambda$ and the system size $L$. Depending on whether $\tilde{v}$ is small or large, the motion of searcher is considered to be diffusive or ballistic.

In the limit of $\tilde{v}\rightarrow 0$, i.e., the persistence length $\xi$ is much smaller than the system size $L$, the trajectory of a run-and-tumble particle resembles that of a diffusing particle with an effective diffusion coefficient $\tilde{D}_{eff}=\tilde{v}^{2}/2$. Then, $\langle T_{first}\rangle$ can be identified with the mean first-passage time of a diffusive searcher,

As $\tilde{v}$ decreases, the effective diffusion coefficient decreases, so the mean first-passage time increases. In the same limit, $\langle N_{pass}\rangle\langle T_{return}\rangle$ is vanishingly small: Although the mean returning time [Eq.(34)] increases as the velocity $\tilde{v}$ decreases, the average number of passing becomes exponentially smaller, $\langle N_{pass}\rangle\simeq e^{-\tilde{k}/\tilde{v}}$ for large $\tilde{k}/\tilde{v}$. The net returning time $\langle N_{pass}\rangle\langle T_{return}\rangle$ is given as

which goes to zero as $\tilde{v}\rightarrow 0$. In the limit of $\tilde{v}\rightarrow 0$, the mean searching time is dominated by $\langle T_{first}\rangle$,

which is a decreasing function of $\tilde{v}$.

In the opposite limit of $\tilde{v}\rightarrow\infty$, the searcher undergoes the ballistic motion to the confining boundaries. When the searcher reaches the hard-wall boundary, it exerts pressure against the wall and stays stuck until the next tumbling occurs to reverse its orientation. As the direction of the searcher flips the searcher leaves this boundary, passes the target almost instantaneously, and finds the target with a probability of $1-e^{-\tilde{k}/\tilde{v}}$.

In this limit, the particle spends most of its time stuck at the hard wall boundary, waiting for the flipping of the direction. As normalized by the inverse of the flipping rate, the mean returning time is given as

since a single flipping is needed for returning, which takes one on average, and the mean first-passage time is given as

because the number of flips required for a particle to first reach its target is either 0 or 1 with the same probability, depending on its initial orientation. Thus the total mean searching time is determined by the number of passing $\langle N_{pass}\rangle$, which diverges as $\tilde{v}$ increases. In the limit of $\tilde{v}\rightarrow\infty$, the total returning time $\langle N_{pass}\rangle\langle T_{return}\rangle$ dominates the first-passage time $\langle T_{first}\rangle$, and the mean searching time is well-approximated as

which is an increasing function of $\tilde{v}$.

Since the searching time diverges for both $\tilde{v}\rightarrow 0$ and $\tilde{v}\rightarrow\infty$, there exists an optimal velocity $\tilde{v}^{*}$ that minimizes the searching time, as we observe in Fig.1. The optimal velocity is determined by the interplay between the first passage time $\langle T_{first}\rangle$ and the total returning time $\langle N_{pass}\rangle\langle T_{return}\rangle$. Here we separate two cases, large $\tilde{k}$ and small $\tilde{k}$, and investigate how the optimal velocity $\tilde{v}^{*}$ depends on $\tilde{k}$ for each case.

Let us first suppose that $\tilde{k}\ll 1$. In this case, we can show that the optimal velocity $\tilde{v}^{*}$ is in the range of $\tilde{k}\ll\tilde{v}^{*}\ll 1$: the optimal velocity must be within the diffusive regime $\tilde{v}^{*}\ll 1$. Otherwise, the total returning time $\langle N_{pass}\rangle\langle T_{return}\rangle\sim O(\tilde{v}/\tilde{k})$ dominates the mean first-passage time $\langle T_{first}\rangle$ which is at most $\sim O(1)$. In the diffusive regime ($v\ll 1$), $\langle T_{first}\rangle\sim O(1/\tilde{v}^{2})$ [Eq.(35)] is much greater than $\langle T_{return}\rangle\sim O(1/\tilde{v})$ [Eq.(34)], and for $\langle T_{first}\rangle$ and $\langle N_{pass}\rangle\langle T_{return}\rangle$ to be comparable with each other, $\langle N_{pass}\rangle$ must be much greater than 1, which leads $\tilde{v}^{*}\gg\tilde{k}$.

When $\tilde{k}\ll\tilde{v}\ll 1$, the average number of passing can be approximated as

and by multiplying with the mean returning time given by Eq.(34), we have

Among the terms in Eq.(42), only the first and the second terms are relevant since other terms are always negligible compared to $\langle T_{first}\rangle\simeq 2/3\tilde{v}^{2}$. Therefore, we can write $\langle T\rangle$ in this range of $\tilde{v}$ as

and the optimal value $\tilde{v}^{*}$ is given as

The minimum value of $\langle T\rangle$ is then evaluated to be

and the minimum value of mean searching time can be expressed in terms of the optimal value $\tilde{v}^{*}$,

as presented in Fig.1.

Now, let us consider the case where $\tilde{k}\gg 1$. In this regime, the optimal velocity must be much greater than 1 because, otherwise, $\langle N_{pass}\rangle\langle T_{return}\rangle$, which is at most $\sim O(e^{-\tilde{k}/\tilde{v}}/\tilde{v})$, would always be negligible compared to $\langle T_{first}\rangle$, which is a decreasing polynomial function of $\tilde{v}$. For $\tilde{v}\gg 1$, the mean first-passage time is given as

and the leading order of $\langle N_{pass}\rangle$ and $\langle T_{return}\rangle$ are given as

Collecting the relevant terms in $\langle T\rangle$, we obtain

from which the optimal value of the rescaled speed $\tilde{v}$ is calculated as

The minimum value of the mean searching time is given as,

which is a half of the mean flipping time and correctly captures the behavior of $\langle T\rangle$ for $\tilde{k}\gg 1$ in Fig. 1.

Here, we consider the first-passage problem of a diffusive RTP, i.e., a RTP subject also to diffusion, with a small perfect target in a three-dimensional elongated domain. Despite the relevance of the translational diffusion in case of a small target in high dimensions, an analytic solution of this problem is unavailable to date. As a possible application of the current result, we show that the first-passage time of a 3D diffusive RTP can be qualitatively understood in terms of a 1D RTP with a reactive target.

For this purpose, we consider a three-dimensional rectangular box, with hard wall boundary condition at $x=\pm L$, periodic boundary condition along the lateral directions, i.e., at $y$ (and $z$)$=\pm d/2$, and a spherical perfect target of a radius $a$ at the center ($\mathbf{r}=0$) which absorbs the searcher perfectly on the surface. We assume the high aspect ratio of the space and the small target size, $a\ll d\ll L$. The motion of a diffusive RTP is described by the overdamped Langevin equation,

where $D$ is the diffusion constant, $\eta(t)$ a Gaussian white noise satisfying $\langle\eta_{\mu}(t)\eta_{\nu}(t^{\prime})\rangle=\delta_{\mu\nu}\delta(t-t^{\prime})$, and the tumbling occurs in the Poisson process with a rate $\lambda$, at which the self-propulsion direction $\mathbf{e}_{\Omega}$ is updated to a randomly chosen solid angle $\Omega$. By performing the Langevin dynamics simulations of Eq. (53), we numerically estimate the mean first-passage times for various parameters, which are shown in Fig. 2. For numerical evaluations, we rescale all lengths by $L$ according to $\tilde{\mathbf{r}}=\mathbf{r}/L$ and introduce dimensionless velocity $\tilde{v}=v/L\lambda$ and diffusion constant $\tilde{D}=D/L^{2}\lambda$. The first-passage time, measured in units of $\lambda^{-1}$ as $T=\lambda t$, is averaged over 100 realizations. The Fokker-Planck equation (FPE) for the probability distribution function of a diffusive RTP reads as

where $p_{\Omega}$ is the probability distribution function of the searcher to be at position $\mathbf{r}$ with self-propulsion direction $\Omega$. $\ell_{c}=D/v$ is the length scale separating a diffusion-dominant regime from a drift-dominant regime; on a length scale of $\ell\ll\ell_{c}$, the diffusion term dominates in Eq. (54), while on a length scale of $\ell\gg\ell_{c}$, the drift term does.

When $L\ll D/v$ (or equivalently, $\tilde{v}\ll\tilde{D}$), the diffusion is always dominant, i.e., even on the largest length scale of the system, $L$. Trajectories of the particle then strongly resemble those of a diffusing particle and the first-passage dynamics can be well-approximated by that of a purely diffusive system. For a 1D diffusive particle with a reactive target represented by a reaction rate $k_{eff}$, the mean first-passage time is given by

or when rescaled using $\lambda^{-1}$,

where the 1D diffusion constant $D_{x}$ is related to the 3D diffusion constant as $3D_{x}=D$. The 1D reaction rate, $k_{eff}$, can be estimated following the scheme described below [see Eq. (60)]. Our prediction, Eq. (56), is clearly supported by the numerical results of Langevin dynamics simulations, shown in Fig 2(a): For the regime of $\tilde{v}\ll\tilde{D}$, the mean first-passage time $\langle T\rangle$ does not depend on the propulsion speed $v$ and agrees well with Eq. (56).

Now we consider the opposite limit of $L\gg D/v$ (or equivalently, $\tilde{v}\gg\tilde{D}$). In this case, depending on the length scales under consideration, either diffusion or drift prevails. On the length scale of $\ell\sim L\gg\ell_{c}$, the diffusion term is negligible in Eq. (54), which leads to

This suggests that the self-propulsion velocity manifests itself in the first-passage dynamics, occurring over a length scale of $L$, and that the first-passage problem reduces to that of a RTP. What remains in order to apply our results derived in the previous sections is to determine the 1D reaction rate, $k_{eff}$. The reaction (absorption) rate is identified with the number of particles arriving at the target boundary per unit time, which is determined by the local concentration profile of the particle close to the target. When we are interested in the vicinity of the target, i.e., on a length scale of $\ell\sim a\ll D/v$, the diffusion dominates so that the particle concentration profile near the target obeys the diffusion equation,

where ${p}=\int d\Omega\,p_{\Omega}$. The 3D reaction rate $\kappa$ is then found from a solution of the diffusion equation by integrating the particle current over the target surface,

where $p_{st}$ is the steady state solution of the Laplace’s equation $\nabla^{2}p=0$. Considering the boundary conditions, $p(\mathbf{r})|_{r=a}=0$ and $p(\mathbf{r})|_{r\rightarrow\infty}=p_{\infty}$, one recovers the well-known result of the reaction rate, $\kappa=4\pi Da$, obtained for a spherical perfect target of radius $a$ in 3D with a spherical symmetry. Now we relate this 3D reaction rate $\kappa$ to the 1D reaction rate $k_{eff}$. For this, we adopt the scheme proposed in [38] in which a 2D reactive boundary layer at $x=0$, containing the target, was considered. The main idea is to replace a small perfectly absorbing target in 3D with a 2D reactive layer which absorbs the particles through the cross-sectional area $S$ with an effective reaction rate $k_{eff}$ [38]. At a steady state, the total number of particles absorbed through this reactive layer surface is equal to the number of particles arriving at the target surface, which yields

where $S\simeq d^{2}$ for the 2D planar layer considered here. Assuming that except the small region around the target, the concentration profile of particles is almost constant and can be considered homogeneous along the lateral directions, the projection is carried out onto the $x$-axis, and the 2D reactive layer is reduced to the 1D reactive boundary with the reaction rate $k_{eff}$.

Using this $k_{eff}$, we conjecture that the first-passage problem of a 3D diffusive RTP can be captured by means of a RTP in 1D with a reactive target. To illustrate this point, the mean first-passage time $\langle T\rangle$ is plotted in Fig. 2 as a function of self-propulsion velocity for various parameters. When $\tilde{v}\gg\tilde{D}$, $\langle T\rangle$ exhibit a non-monotonic behavior in $\tilde{v}$, similarly to Fig.1. For $\tilde{v}\gg 1$, the mean first-passage times with different parameters merge with one another if the estimated $\tilde{k}_{eff}=4\pi\tilde{D}\tilde{a}/\tilde{d}^{2}$ are the same [Fig. 2(a)]. To emphasize this, we plot $\langle T\rangle$ as a function of $\tilde{v}/\tilde{k}_{eff}$ in Fig. 2(b) where all the simulation results collapse on a single curve, which clearly indicates $\langle T\rangle\simeq\tilde{v}/\tilde{k}_{eff}$ for large $\tilde{v}$, as consistent with Eq. (40). This supports our conjecture that the first-passage dynamics of active particles in elongated high dimensional systems can be qualitatively understood by using the result presented in this work, i.e., the 1D RTP in the presence of a reactive target with an effective reaction rate $k_{eff}$. This mapping is no longer valid when $v$ increases further to $a\sim D/v$ because in that case, the particle concentration around the target does not satisfy the diffusion equation and thus, $k_{eff}$ cannot be estimated as described above.

The existence of the optimal speed ${v}^{*}$ implies that we can in principle extract RTPs having speed $v^{*}$. Consider a mixture of non-interacting RTPs with different self-propulsion speeds spread out in a one-dimensional confined space with a reactive target at the center. It is then expected that the particles with velocities close to the optimal speed $v^{*}$ are more likely to be absorbed faster.

In practice, this one-time attempt may be insufficient to achieve a meaningful resolution in filtering. More efficient procedure can be conceptually designed as following. Many copies of the identical experimental set-up are prepared as a column, and a mixture of RTPs to be sorted is placed in the first set-up (see Fig.3). As soon as particles are extracted from one experiment, they are put into the next. Over time, the particles of the first experimental set-up propagate into next set-ups, and the propagation speed depends on the mean searching time of each active particle. Then we can finally obtain a chromatographic spectrum of active particles along the column, and the particle with optimal speed $\tilde{v}^{*}$ will be placed at the edge of the spectrum.

To be more specific, we can define the propagation speed along the column as the number of passed floors (set-ups) $N$ divided by the time it takes to migrate $T(N)$. If $N$ is large, we can estimate $T(N)$ as

where $\sigma_{T}$ is the standard deviation of the mean first-passage time, which is evaluated in Appendix C [see, e.g., Eq. (113)], and $\eta$ is a standard normal random number. Then, the propagation speed $V$ along the column is written as

and the standard deviation of the propagation speed is given as

On the other hand, if the velocity $\tilde{v}$ of an active particle differs from $\tilde{v}^{*}$ by $\delta\tilde{v}$, the mean value of the propagation velocity differs from the optimal one by

Here we define the resolution of the filtering as the ratio between $\tilde{v}$ and the minimum difference $\delta\tilde{v}_{min}$ that causes the difference in $V$ that can be distinguished from the standard deviation $\sigma_{V}$. Since the minimum distinguishable difference $\delta\tilde{v}_{min}$ is given as

the resolution of the filtering is given as

If $\tilde{k}\ll 1$, we can prove $\partial_{\tilde{v}}^{2}\langle T\rangle=(9/2)^{2/3}\tilde{k}^{-4/3}$ and $\sigma_{T}\simeq 2\tilde{k}^{-1}$, and the resolution is given as

and the height of the column must be at least $N\simeq 1.92v^{-2}R^{-4}$ to achieve the resolution $R$. If $\tilde{k}\gg 1$, we have $\partial_{\tilde{v}}^{2}\langle T\rangle\simeq\frac{3}{2}\tilde{k}^{-3}[\log\tilde{k}]^{4}$ and $\sigma_{T}=\sqrt{3/4}$, which gives

which means that if we want to extract particles of self-propulsion speed $\tilde{v}\gg 1$ with a required resolution $R$, the height of the column, i.e., the minimum number of experimental setups must be at least $N\simeq 0.33\tilde{v}^{2}(\log\tilde{v})^{4}R^{-4}$.