Reaction–diffusion constraints in living tissue: Effectiveness factors in skeletal muscle design

Abstract

A mathematical model was developed to analyze the effects of intracellular diffusion of O(2) and high-energy phosphate metabolites on aerobic energy metabolism in skeletal muscle. We tested the hypotheses that in a range of muscle fibers from different species (1) aerobic metabolism was not diffusion limited and (2) that fibers had a combination of rate and fiber size that placed them at the brink of substantial diffusion limitation. A simplified chemical reaction rate law for mitochondrial oxidative phosphorylation was developed utilizing a published detailed model of isolated mitochondrial function. This rate law was then used as a boundary condition in a reaction-diffusion model that was further simplified using the volume averaging method and solved to determine the rates of oxidative phosphorylation as functions of the volume fraction of mitochondria, the size of the muscle cell, and the amount of oxygen delivered by the capillaries. The effectiveness factor, which is the ratio of reaction rate in the system with finite rates of diffusion to those in the absence of any diffusion limitations, defined the regions where intracellular diffusion of metabolites and O(2) may limit aerobic metabolism in both very small, highly oxidative fibers as well as in larger fibers with lower aerobic capacity. Comparison of model analysis with experimental data revealed that none of the fibers was strongly limited by diffusion, as expected. However, while some fibers were near substantial diffusion limitation, most were well within the domain of reaction control of aerobic metabolic rate. This may constitute a safety factor in muscle that provides a level of protection from diffusion constraints under conditions such as hypoxia.

The formation and functional consequences of heterogeneous mitochondrial distributions in skeletal muscle

“…In support of this view, shifts in mitochondrial distribution toward the fiber periphery during hypertrophic fiber growth in a variety of comparative models appear to be independent of opposing shifts in nuclear distribution toward the fiber center (Nyack et al, 2007;Hardy et al, 2009;Hardy et al, 2010;Priester et al, 2011), and experiments with isolated cells indicate that mitochondrial DNA replication is independent of proximity to nuclei (Magnusson et al, 2003). Moreover, reactiondiffusion models indicate that spatial gradients in high-energy phosphate molecules like AMP are very slight in muscle (Jimenez et al, 2008;Dasika et al, 2011;Pathi et al, 2011), and we know of no biogenesis signals that vary spatially across the radius of the fiber. In contrast, as mentioned above, mitochondrial degradation is a regionally variable, selective process, whereby damaged mitochondria are specifically targeted and removed.…”

“…The full model of Beard (Beard, 2005) is too computationally intensive to apply to our population of mitochondria. Therefore, Dasika et al (Dasika et al, 2011) developed a rate law that closely approximates the full model of Beard (Beard, 2005) that expresses ATP production in the mitochondria as a function of the concentration of oxygen, ADP and inorganic phosphate (P i ). This rate law was used to describe reactions in the mitochondria.…”

“…The intracellular partial pressure of O 2 (P O2 ) in the model was set as a fixed parameter as in our previous studies (Jimenez et al, 2008;Dasika et al, 2011;Pathi et al, 2011) based on literature indicating that this is a reasonable estimation of the concentration of oxygen available at the capillaries (2666.45Pa, i.e. 35.1mmoll -1 ) in fishes (Egginton et al, 2002) and mammals (Groebe, 1995).…”

“…fiber with complete limitation of reaction rate by diffusion). The  for the different cases was evaluated as previously reported (Dasika et al, 2011;Pathi et al, 2011).…”

Critical Role of Coaptive Strain in Aortic Valve Leaflet Homeostasis: Use of a Novel Flow Culture Bioreactor to Explore Heart Valve Mechanobiology

“…It has been shown that leaflets exist on the verge of core hypoxia because of their lack of vascularization,10 and that leaflet motion including coaptation helps to convectively deliver nutrients to the deeper regions of valves 11, 12. Indeed, human valves are in the 300‐ to 700‐μm thickness range,13 beyond the theoretical passive oxygen diffusion range of 100 μm14 when unloaded, yet remain fully viable. Of note, AV area increases at least 20% to 30% at coaptation 15.…”