Supporting Information S1
نویسندگان
چکیده
Various quantities in our work are based upon the model by Chevalier et al. [2]. Here we briefly discuss the key components of that approach and we provide details of how we have used their mathematical development to obtain mean first passage time (MFPT) expressions, as well as how we have tested the accuracy of such expressions. The needle complexes are represented by small circular (2d) and spherical (3d) targets with radius ε and center positions on the boundary of the confining regions, respectively, a circle in 2d and a sphere in 3d. An effector is represented as a random walker that moves either diffusively or subdiffusively, and gets absorbed when it hits one of the available targets. If the particle hits any other part of the boundary, it gets reflected inside. In presence of just one target one needs to compute the first passage probability to reach the target, from which one can deduce the mean first passage time to the target. Here, however, the situation is more complicated due to the presence of multiple targets. In this case one needs to compute splitting probabilities P1(r)...PN (r) for each of the N targets. For illustrative purpose in Fig. 1a we show a sample trajectory of an effector that starts at the origin and hits one of the N = 20 available targets on the boundary of a disk. In Fig. 1b we display the splitting probability associated with the specific target at coordinates (x = 1,y = 0) as function of the initial effector location. In Fig. 1c we plot the MFPT with N = 20 needle complexes as function of the initial effector position. In general terms given a confined domain the splitting probability Pi(r) represents the chance that, for a particle starting its movement at position r, a target with label i will eventually be hit without visits to any other targets beforehand. When the movement of the effectors is diffusive, i.e. when the random walkers are Brownian, the splitting probabilities for both the unit disk and the unit sphere, P(r) = [P1(r)...PN (r)]T , satisfy the following linear algebraic system [2]: MP(r) = b(r), (1.1)
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