Moments of action provide insight into critical times for advection-diffusion-reaction processes

Abstract

Berezhkovskii and co-workers introduced the concept of local accumulation time as a finite measure of the time required for the transient solution of a reaction-diffusion equation to effectively reach steady state [Biophys J. 99, L59 (2010); Phys. Rev. E 83, 051906 (2011)]. Berezhkovskii's approach is a particular application of the concept of mean action time (MAT) that was introduced previously by McNabb [IMA J. Appl. Math. 47, 193 (1991)]. Here, we generalize these previous results by presenting a framework to calculate the MAT, as well as the higher moments, which we call the moments of action. The second moment is the variance of action time, the third moment is related to the skew of action time, and so on. We consider a general transition from some initial condition to an associated steady state for a one-dimensional linear advection-diffusion-reaction partial differential equation (PDE). Our results indicate that it is possible to solve for the moments of action exactly without requiring the transient solution of the PDE. We present specific examples that highlight potential weaknesses of previous studies that have considered the MAT alone without considering higher moments. Finally, we also provide a meaningful interpretation of the moments of action by presenting simulation results from a discrete random-walk model together with some analysis of the particle lifetime distribution. This work shows that the moments of action are identical to the moments of the particle lifetime distribution for certain transitions.

Simplified approach for calculating moments of action for linear reaction-diffusion equations

“…To determine whether it is appropriate to work with such steady state solutions, we must decide whether a sufficient amount of time has passed so that the transient solution has effectively reached steady state. Several definitions of critical time have been proposed for this purpose [1][2][3][4][5][6][7]. One such definition, the mean action time (MAT) [3][4][5][6][7][8], also known as the local accumulation time [9][10][11][12], is the mean of a probability density function (PDF) associated with the linear reaction-diffusion problem of interest.…”

“…The MAT is an objective definition of the critical time which can be determined without solving for the transient solution of the reactiondiffusion equation. For high-variance PDFs, the MAT underapproximates the critical time since it neglects to account for the spread about the mean [5,6,13]. To address this limitation we can calculate the higher moments of the PDF, also known as the moments of action [6].…”

“…For high-variance PDFs, the MAT underapproximates the critical time since it neglects to account for the spread about the mean [5,6,13]. To address this limitation we can calculate the higher moments of the PDF, also known as the moments of action [6]. We anticipate that an improved estimate of the critical time for high-variance PDFs would be the MAT plus one standard deviation of the PDF [6,13,14].…”

Kinetics of receptor occupancy during morphogen gradient formation

“…In a number of applications, one is interested in characteristic time scales of concentration dynamics at a given location. [18][19][20][21][22][23][24][25][26] Here, we provide analytical expressions for such time scales, with the emphasis on receptor-bound component of the model. This reflects the fact that morphogens control cellular processes through cell surface receptors.…”

Accurate and efficient calculation of response times for groundwater flow

“…In this work, we use a different approach based on the concept of mean action time [14,15,26]. The benefit of working with this framework is that it avoids the need for solving the underlying parabolic partial differential equation model of transient flow, and it provides explicit information about how the response time varies with position [22,32].…”