Critical time scales for advection–diffusion–reaction processes

The concept of local accumulation time (LAT) was introduced by Berezhkovskii and coworkers in 2010–2011 to give a finite measure of the time required for the transient solution of a reaction–diffusion equation to approach the steady–state solution (Biophys J. 99, L59 (2010); Phys Rev E. 83, 051906 (2011)). Such a measure is referred to as a critical time. Here, we show that LAT is, in fact, identical to the concept of mean action time (MAT) that was first introduced by McNabb in 1991 (IMA J Appl Math. 47, 193 (1991)). Although McNabb’s initial argument was motivated by considering the mean particle lifetime (MPLT) for a linear death process, he applied the ideas to study diffusion. We extend the work of these authors by deriving expressions for the MAT for a general one–dimensional linear advection–diffusion–reaction problem. Using a combination of continuum and discrete approaches, we show that MAT and MPLT are equivalent for certain uniform–to-uniform transitions; these results provide a practical interpretation for MAT, by directly linking the stochastic microscopic processes to a meaningful macroscopic timescale. We find that for more general transitions, the equivalence between MAT and MPLT does not hold. Unlike other critical time definitions, we show that it is possible to evaluate the MAT without solving the underlying partial differential equation (pde). This makes MAT a simple and attractive quantity for practical situations. Finally, our work explores the accuracy of certain approximations derived using the MAT, showing that useful approximations for nonlinear kinetic processes can be obtained, again without treating the governing pde directly.

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

In 2010 Berezhkovskii and coworkers introduced the concept of local accumulation time (LAT) 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.

Critical timescales and time intervals for coupled linear processes

In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and coworkers rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single–species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalise the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications including the analysis of models describing coupled chemical decay and cell differentiation processes, amongst others.

A simplified approach for calculating the moments of action for linear reaction-diffusion equations

The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time under-approximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.

An analytical framework for quantifying aquifer response time scales associated with transient boundary conditions

A major challenge in studying coupled groundwater and surface-water interactions arises from the considerable difference in the response time scales of groundwater and surface-water systems affected by external forcings. Although coupled models representing the interaction of groundwater and surface-water systems have been studied for over a century, most have focused on groundwater quantity or quality issues rather than response time. In this study, we present an analytical framework, based on the concept of mean action time (MAT), to estimate the time scale required for groundwater systems to respond to changes in surface-water conditions. MAT can be used to estimate the transient response time scale by analyzing the governing mathematical model. This framework does not require any form of transient solution (either numerical or analytical) to the governing equation, yet it provides a closed form mathematical relationship for the response time as a function of the aquifer geometry, boundary conditions, and flow parameters. Our analysis indicates that aquifer systems have three fundamental time scales: (i) a time scale that depends on the intrinsic properties of the aquifer; (ii) a time scale that depends on the intrinsic properties of the boundary condition, and; (iii) a time scale that depends on the properties of the entire system. We discuss two practical scenarios where MAT estimates provide useful insights and we test the MAT predictions using new laboratory-scale experimental data sets.

Posterior Vertical Bite Reconstructions of Erosively Worn Dentitions and the "Stamp Technique" - A Case Series with a Mean Observation Time of 40 Months

PURPOSE In the present case series, the authors report on seven cases of erosively worn dentitions (98 posterior teeth) which were treated with direct resin composite. MATERIALS AND METHODS In all cases, both arches were restored by using the so-called stamp technique. All patients were treated with standardized materials and protocols. Prior to treatment, a waxup was made on die-cast models to build up the loss of occlusion as well as ensure the optimal future anatomy and function of the eroded teeth to be restored. During treatment, teeth were restored by using templates of silicone (ie, two "stamps," one on the vestibular, one on the oral aspect of each tooth), which were filled with resin composite in order to transfer the planned, future restoration (ie, in the shape of the waxup) from the extra- to the intraoral situation. Baseline examinations were performed in all patients after treatment, and photographs as well as radiographs were taken. To evaluate the outcome, the modified United States Public Health Service criteria (USPHS) were used. RESULTS The patients were re-assessed after a mean observation time of 40 months (40.8 ± 7.2 months). The overall outcome of the restorations was good, and almost exclusively "Alpha" scores were given. Only the marginal integrity and the anatomical form received a "Charlie" score (10.2%) in two cases. CONCLUSION Direct resin composite restorations made with the stamp technique are a valuable treatment option for restoring erosively worn dentitions.

Global Ocean Biomes: Mean and time-varying maps (NetCDF 7.8 MB)

Large-scale studies of ocean biogeochemistry and carbon cycling have often partitioned the ocean into regions along lines of latitude and longitude despite the fact that spatially more complex boundaries would be closer to the true biogeography of the ocean. Herein, we define 17 open-ocean biomes classified from four observational data sets: sea surface temperature (SST), spring/summer chlorophyll a concentrations (Chl a), ice fraction, and maximum mixed layer depth (maxMLD) on a 1° × 1° grid. By considering interannual variability for each input, we create dynamic ocean biome boundaries that shift annually between 1998 and 2010. Additionally we create a core biome map, which includes only the grid cells that do not change biome assignment across the 13 years of the time-varying biomes. These biomes can be used in future studies to distinguish large-scale ocean regions based on biogeochemical function.

Democratic planning in action; time for timing.

Mode of access: Internet.

Minimum mean running time function generation using read only memory /

Vita.

Comment on "Local accumulation times for source, diffusion, and degradation models in two and three dimensions" [J. Chem. Phys. 138, 104121 (2013)]

In a recent paper, Gordon, Muratov, and Shvartsman studied a partial differential equation (PDE) model describing radially symmetric diffusion and degradation in two and three dimensions. They paid particular attention to the local accumulation time (LAT), also known in the literature as the mean action time, which is a spatially dependent timescale that can be used to provide an estimate of the time required for the transient solution to effectively reach steady state. They presented exact results for three-dimensional applications and gave approximate results for the two-dimensional analogue. Here we make two generalizations of Gordon, Muratov, and Shvartsman’s work: (i) we present an exact expression for the LAT in any dimension and (ii) we present an exact expression for the variance of the distribution. The variance provides useful information regarding the spread about the mean that is not captured by the LAT. We conclude by describing further extensions of the model that were not considered by Gordon,Muratov, and Shvartsman. We have found that exact expressions for the LAT can also be derived for these important extensions...

How long does it take for aquifer recharge or aquifer discharge processes to reach steady state?

Groundwater flow models are usually characterized as being either transient flow models or steady state flow models. Given that steady state groundwater flow conditions arise as a long time asymptotic limit of a particular transient response, it is natural for us to seek a finite estimate of the amount of time required for a particular transient flow problem to effectively reach steady state. Here, we introduce the concept of mean action time (MAT) to address a fundamental question: How long does it take for a groundwater recharge process or discharge processes to effectively reach steady state? This concept relies on identifying a cumulative distribution function, $F(t;x)$, which varies from $F(0;x)=0$ to $F(t;x) \to \infty$ as $t\to \infty$, thereby providing us with a measurement of the progress of the system towards steady state. The MAT corresponds to the mean of the associated probability density function $f(t;x) = \dfrac{dF}{dt}$, and we demonstrate that this framework provides useful analytical insight by explicitly showing how the MAT depends on the parameters in the model and the geometry of the problem. Additional theoretical results relating to the variance of $f(t;x)$, known as the variance of action time (VAT), are also presented. To test our theoretical predictions we include measurements from a laboratory–scale experiment describing flow through a homogeneous porous medium. The laboratory data confirms that the theoretical MAT predictions are in good agreement with measurements from the physical model.

Desiccation tolerance in seeds of Annona emarginata (Schldtl.) H. Rainer and action of plant growth regulators on germination

Annona emarginata (Schldtl.) H. Rainer (araticum-de-terra-fria) is used as a rootstock for several species of Annonaceae. It is suggested that these seeds should be sown immediately after extraction and, therefore, they could be intolerant to desiccation. There are several mechanisms involved with desiccation tolerance. Soluble sugars, for example, can accumulate and act as osmoprotectants for the membrane system during desiccation. The aim of this study is to assess desiccation tolerance in A. emarginata seeds. In addition, we examined the soluble sugars involved in desiccation tolerance. Finally, we determined the effect of gibberellic acid (GA4+7) and N-(phenylmethyl)-aminopurine in promoting the germination of seeds with different water contents. The experiment consisted of a randomized 4×5 factorial design (desiccation levels × concentration of growth regulators). After drying, seeds containing 31 (control), 19, 12 and 5% water were incubated in different concentrations of GA4+7 N-(phenylmethyl)-aminopurine (0, 250, 500, 750 and 1000 mg L-1) for 60 hours. The experiment was conducted in a germination chamber with alternating temperature and photoperiod of 20oC for 18 hours of darkness and 30oC for 6 hours of light. We analyzed electrical conductivity, germination rate, mean germination time, germination speed, frequency and uniformity of germination, percentage of dormant seeds and soluble sugar profile in intact seeds through high-performance liquid chromatography (HPLC). The data were subjected to analysis of variance, and the means were compared using Tukey's test at a threshold of p<0.05. The results showed that seeds of A. emarginata appears to be desiccation tolerant and, also, that sucrose increases when seed water content is reduced to values as low as 12%, exogenous GA4+7+N-(phenylmethyl)-aminopurine improves its germinability.