Figure 1

Schematic diagram of the stochastic directed intermittent search model.Reuse & Permissions

Figure 2

(Color online) Labeling scheme for a general dendritic tree. (a) An example of a tree $\Gamma$ with three generations (Gen) of branch nodes, a primary closed terminal node (corresponding to the soma), and a set of open terminal nodes. (b) The branch node $\alpha \left(k\right)$ is shown in relation to the neighboring branch node ${\alpha }^{\prime }\left(k\right)$ closest to the primary node. The branch segments extending out from $\alpha \left(k\right)$ in the positive direction together comprise the set ${\overline{\mathcal{I}}}_{\alpha \left(k\right)}$.Reuse & Permissions

Figure 3

The effect of a single branch on directed intermittent search. [(a), (c)] The hitting probability $\Pi$ and MFPT $T$ for a finite interval of length $2L$ (no branching) as a function of target location $X$ and for various values of the drift velocity $V$. [(b), (d)] The corresponding hitting probability $\Pi$ and MFPT $T$ in the case of a single branch point at $X=L$. Nondimensional parameter values used are $L=10$, $\lambda =0.2$, and $D=0.1$.Reuse & Permissions

Figure 4

(Color online) Numerical results for the hitting probability $\Pi$ and MFPT $T$ on a semi-infinite Cayley tree with coordination number $z=3$ and branch length $L=10\mu \text{m}$. Comparison of Monte Carlo simulations of the full three-state Markov model with analytical results obtained for the reduced scalar FP equation. The solid gray line represents the analytical approximation of the hitting probability and MFPT obtained from the generating function (3.34). The latter is calculated by solving the integral Eq. (3.32) using the Green’s function for the Cayley tree. The corresponding results from Monte Carlo simulations of the three-state model are denoted by “x.” (a) The hitting probability $\Pi$ as a function of the target location $X$. (b) The mean first passage time $T$ as a function of the target location. (The gray bars represent the standard deviation of the first passage time evaluated using Monte Carlo simulations.) The diffusivity $D$, drift velocity $V$, and absorption rate $\lambda$ of the scalar FP equation are calculated using the parameter values of the three-state model: $k=0.2{\text{s}}^{-1}$, $v_{\pm }=0.2\mu \text{m}/\text{s}$, $l=0.5\mu \text{m}$, $\alpha =1{\text{s}}^{-1}$, ${\beta }_{+}=1{\text{s}}^{-1}$, and ${\beta }_{-}=2{\text{s}}^{-1}$. These values are extracted from experimental values of mRNA transport in dendrites [13, 14, 15, 16].Reuse & Permissions

Figure 5

MFPT on a finite interval of length $L=20\mu \text{m}$ with a hidden target at $X=10\mu \text{m}$. (a) Plot of MFPT $T$ as a function of $\alpha$ for fixed $\Pi =0.8$ and various ${\beta }_{-}$. Other parameters are $v_{\pm }=0.1\mu \text{m}{\text{s}}^{-1}$ and $k=0.05{\text{s}}^{-1}$. Unidirectional case is shown as a dotted curve. (b) Corresponding plot of minimum search time $T_{\text{min}}$ (with respect to variation in $\alpha$) as a function of the hitting probability $\Pi$. Unidirectional case (dotted curve) is plotted for ${\alpha }_{\text{max}}=0.5{\text{s}}^{-1}$.Reuse & Permissions

Figure 6

(Color online) MFPT on a semi-infinite Cayley tree with coordination number $z=3$ and branch length $L=10\mu \text{m}$. Target is located at a distance $X=5\mu \text{m}$ along a branch adjacent to the primary branch $\left(ň=1\right)$. (a) Plot of MFPT $T$ as a function of $\alpha$ for fixed hitting probability $\Pi =0.4$ and various values of ${\beta }_{-}$. Other parameter values are $k=0.1{\text{s}}^{-1}$ and $v_{\pm }=0.1\mu \text{m}/\text{s}$. (b) Corresponding plots of $T$ for $\Pi =0.5$. The minimum MFPT for each fixed value of ${\beta }_{-}$ is indicated by the black “o” mark, and the corresponding value of the MFPT obtained from Monte Carlo simulations is shown by the “x.”Reuse & Permissions

Figure 7

(Color online) Two phases of behavior for directed intermittent search on a semi- infinite Cayley tree with coordination number $z=3$ and branch length $L=10\mu \text{m}$. Target is located at a distance $X=5\mu \text{m}$ along a branch that is one generation away from the primary branch $\left(\check{n}=1\right)$. Each thin solid curve is a parameterized plot of $\left[\Pi \left({\beta }_{+}\right),T\left({\beta }_{+}\right)\right]$ for fixed ${\beta }_{-}$ which terminates at the point $\Pi ={\Pi }_c$ where the search is unbiased. [Since the accuracy of the quasi-steady-state reduction breaks down in the unbiased limit, we determine the termination points using Monte Carlo simulations (closed circles). Other points obtained from simulations are indicated by crosses.] The thick black curve represents the limiting envelope of unidirectional search. Other parameter values are $\alpha =2{\text{s}}^{-1}$, $k=1{\text{s}}^{-1}$, and $v_{\pm }=0.1\mu \text{m}/\text{s}$.Reuse & Permissions