Diffusion-mediated absorption by partially-reactive targets: Brownian functionals and generalized propagators
Paul C. Bressloff
Abstract
Abstract Many processes in cell biology involve diffusion in a domain Ω that contains a target <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">U</mml:mi> </mml:math> whose boundary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>∂</mml:mi> <mml:mi mathvariant="script">U</mml:mi> </mml:math> is a chemically reactive surface. Such a target could represent a single reactive molecule, an intracellular compartment or a whole cell. Recently, a probabilistic framework for studying diffusion-mediated surface reactions has been developed that considers the joint probability density or generalized propagator for particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on a totally reflecting boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. In this paper we extend the theory of diffusion-mediated absorption to cases where the whole interior target domain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">U</mml:mi> </mml:math> acts as a partial absorber rather than the target boundary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>∂</mml:mi> <mml:mi mathvariant="script">U</mml:mi> </mml:math> . Now the particle can freely enter and exit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">U</mml:mi> </mml:math> , and is only able to react (be absorbed) within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">U</mml:mi> </mml:math> . The appropriate Brownian functional is then the occupation time (accumulated time that the particle spends within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">U</mml:mi> </mml:math> ) rather than the boundary local time. We show that both cases can be considered within a unified framework, which consists of a boundary value problem (BVP) for the propagator of the corresponding Brownian functional and an associated stopping condition. We illustrate the theory by calculating the mean first passage time (MFPT) for a spherical target <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">U</mml:mi> </mml:math> located at the center of a spherical domain Ω. This is achieved by solving the propagator BVP directly, rather than using spectral methods. We find that if the first moment of the stopping time density is infinite, then the MFPT is also infinite, that is, the spherical target is not sufficiently absorbing.