Models Of Diffusion-Mediated Gas Exchange In Animal Burrows

Published: July 20, 2018

Summary

This paper seeks to identify the factors which have the greatest effect on gas diffusion rates into animal burrows, and in turn on the maximum size of burrowing animals which may rely on simple burrow designs. To do this, the author models the diffusion of metabolically important gases in and out of occupied burrows, using the to estimate the steady-state concentration gradient of these gases in the cases of a cylindrical burrow, a sealed spherical burrow, and a superposition of both cases. In this model, gases are treated as a continuous substance rather than using modeling them at the particle level. Modeling is performed using relatively simple algebraic formulae for the volumetric diffusion rate of gases; these result from simplifications of the diffusion equation (or in one section, the more specific ) in these symmetric geometries, with constant boundary conditions, and with perfect gas mixing. Some numerical estimation is done on these equations but no analysis is performed on the corresponding discrete-time system. The author also performs sensitivity analysis on this model by varying the dimensions of the burrow, the moisture content and porosity of the soil, number of occupants, and several factors related to the metabolism of the occupant.

Introduction

The exchange of gases within an animal burrow has been an extensive area of study due to the barrier that the burrowing medium (both aquatic and terrestrial) often imposes to the diffusion of gases. While smaller animals run into limited problems with meeting their necessary aerobic metabolic needs through gaseous diffusion, larger animals are often unable to survive on diffusion alone. For example, for a 1 meter deep burrow the average diffusion time of molecular oxygen approaches 27 hours, leading to a flux of $3\times 10^{{22}}$ oxygen molecules per hour, two orders of magnitude less than is needed for respiration. In addition, the buildup of hazardous substances, byproducts of respiration, have just as much difficulty escaping the burrow after being released. Studies of mammalian burrows have observed carbon dioxide concentrations exceeding 15 percent, extremely high when compared with normal atmospheric levels of only .0387 percent. Therefore, alternative mechanisms of gas exchange are needed in order for these large animals to survive within their burrows. Advective flushing is one such mechanism used to mediate gaseous exchange, a process much more rapid than gaseous diffusion. Prairie dog burrows are often cited as the gold standard as they take full advantage of advection due to their means of construction, which will be discussed later within this review.

The first part of this summary will focus on the theory of diffusion, and will provide simple calculations that show the limitations of diffusion. Within this section, the various models (both biological and physical) used to study the diffusion of gases through burrows will be discussed. The second part will summarize the results from the paper being discussed within this summary, followed by a discussion of these results. Finally, we will look at the necessity of advection and discuss how prairie dogs improve their gaseous environment by focusing on the construction of their burrow.

Modeling

Biological Model

In order to gauge the metabolic needs of organisms that live within burrows, biologists rely upon , i.e. how body size and shape effects some variable. These relationships often take the form of a power law, $y=kx^a$, where for this study, $y$听is the metabolic rate, $x$听is the mass of the organism, and $k$听and $a$听are constants derived from empirical relationships. For this study, metabolic rate will be calculated as:

$\dot{V}_{O_2}=aM^b$

Where $\dot{V}_{O_2}$听is the metabolic rate, $a$听is the magnitude of aerobic metabolism and has a constant value of $a=.064$ mL~$O_2$听per minute for mammals, $M$听is the mass of the organism (in grams), and $b$听is a constant equal to .75 for most cases. This relationship can be simplified further by assuming that the radius of the borrow~($r_b$) is 1.25 times bigger than that of the spherical animal and that the density of these mammals approach 1 g/cm^3. The resulting equation for the metabolic rate is:

$\dot{V}_{O_2}=a[\frac{4}{3}\pi(.8r_b)^3]^.75$

This equation will be used to calculate metabolic needs throughout the remainder of this summary.

Diffusion Model

The diffusion of substances on the molecular level is accomplished through the randomness of molecular collisions, also known as Brownian motion. This process can be described quantitatively through a partial differential equation of the form,

${\frac {\partial {Q}}{\partial {t}}}=AD_{m}{\frac {\partial {C}}{\partial {x}}}$

where $\frac{\partial Q}{\partial t}$ is the volumetric flow rate (cm^3/s), $D_m$ is the diffusion coefficient for the substance being diffused (cm^2/s~atm), A is the area for diffusion (cm^2) and $\frac{\partial C}{\partial x}$is the concentration gradient (atm/cm). This equation only describes transport by molecular diffusion and does not take into account other transport mechanism such as turbulent dispersion and advection.

Schematic of Diffusion Models used in the Withers Study

In order to apply this equation to specific subterranean conditions, the proper boundary conditions of the burrow need to be specified. Withers considered three different scenarios for this study; (1) subterranean tunnels through an impermeable media, (2) small animals that live in a chamber within a porous media without constructing tunnels, and (3) large animals that construct a central chamber with tunnels connecting to the surface. A schematic of these scenarios is shown in Figure 1. The solutions to the diffusion equation for each of these scenarios will be discussed below in the results section.

Results

Impermeable Tunnel

For gases diffusing along a cylindrical burrow that is impermeable, the diffusion equation can be modified to

${\dot {V_{c}}}={\frac {\pi r_{{b}}^{{2}}D_{m}\delta C}{l_{b}}}={\dot {V}}_{{O_{2}}}$

where $\dot{V}_c$听is the volumetric flow rate, $\delta C$is the difference in concentration between the ambient environment and the burrow, and $l_b$ is the length of the burrow. Using the metabolic equation discussed earlier to calculate $\dot{V}_{O_2}$, the concentration difference can be computed as a function of the radius of the burrow. Withers assumed both sub aquatic and sub-terrestrial diffusion and plotted these concentration differences versus the radius of the burrow. Fitting a line to this data then allowed him to calculate an empirical expression for concentration differences as a function of body mass, $\delta C=aM^{{0.37}}$ .

For sub-aquatic organisms, he found that animals in these burrows were completely unable to survive on passive diffusion alone, due to the low diffusivity of substances in water. Oxygen differences between the ambient environment and the burrow would be far too excessive to be maintained by average ambient conditions alone. In addition, the buildup of carbon dioxide would prove fatal. This calculation supports the earlier conclusion that diffusion alone is insufficient to transport oxygen to an animal that is larger than 1~mm in diameter. Most aquatic species, therefore, require an advective mass flow in order to survive in sub-aquatic conditions.

For sub-terrestrial organisms, however, it is indeed possible for a solitary animal to survive on passive diffusion alone due to the higher diffusivity in air. For colonies of animals, however, passive diffusion would most likely expose the burrow dwelling creatures to hypoxia and hypercapnia.

Flow Through Porous Media

For subterranean animals, the is an important variable that can facilitate or hinder gaseous diffusion, as the concentrations of $O_2$ in the interstitial soil spaces are comparable to those concentrations at the surface. Therefore, the diffusion coefficient will be altered depending upon the porosity of the soil, with a relationship $D_s=D_mf^{3/2}$, where $D_s$听is the diffusion coefficient for porous media, $D_m$听is the diffusion coefficient discussed earlier, and f is a measure of the porosity of the soil. The expression for soil porosity is, $f=\frac{V_a+V_w}{V_a+V_w+V_s}$,听where $V_x$ represents the volume of air, water, and solid within a soil sample.

For a chamber with no attached tunnels, the porosity of the soil is a vital variable in determining whether an animal will survive in the burrow. Withers derived an expression for a spherical burrow of radius $r_n=2r_b$ buried at a depth, $d$, below the surface:

${\dot {V}}_{{sphere}}={\frac {4\pi r_{n}D_{s}\delta Cd}{2d+r_{b}}}={\dot {V}}_{{O_{2}}}$

The results from this expression were then compared to a scenario where the chamber was attached to numerous burrows that connected to the surface. In this scenario, the diffusion of oxygen into the nest site is simply the superposition of the diffusion through each of the individual burrows in addition to the remaining surface area of the burrow through which oxygen can diffuse. Because of the added complication of the porous media within the burrow and nest, the concentration gradient along the length of the burrow is no longer uniform. Withers overcame this complexity by relying upon an analogous expression for the heat loss from a wire. The mathematical expressions, in this case for radial and longitudinal diffusion, are given below.

${\begin{matrix}{\dot {V}}_{{b,r}}&=\pi r_{{b}}^{2}D_{m}\delta C\mu \tanh(.5\mu l_{b})\\{\dot {V}}_{{b,l}}&={\frac {\pi r_{{b}}^{2}D_{m}\delta C\mu }{\sinh(\mu l_{b})}}\end{matrix}}$

where $\mu ={\frac {2D_{s}}{r_{{b}}^{2}D_{m}\cosh ^{{-1}}({\frac {d}{r_{b}}})}}$ . The results of this analysis agreed with the findings for the more simple spherical nest with no tunnel scenario. As expected, the nest with multiple burrows required less of a concentration gradient to meet the metabolic needs of the organism.

The results of these three models are shown in Figure 2. From these results, it is clear that the more complex organisms require a much greater concentration gradient in order to sustain their metabolism. Single celled organisms (u in figure) require very little oxygen intake, approximately 4 orders of magnitude less than the endotherms (e in figure). Therefore, the smaller the organism and the less complex their metabolism, the more likely they are to survive on passive diffusion alone. The differences between the terrestrial and aquatic environments can also be clearly seen from these figures. The top set of lines are for sub-aquatic organisms while the bottom set of lines are for the sub-terranean animals. The concentration gradient required for aquatic organisms to survive is approximately 6 orders of magnitude greater than for terrestrial, a significant hindrance for those that must survive in the water environment.

The influence that soil porosity has on passive diffusion can also be seen from this figure. Figure 2b plots the results for Model 2 (solid line) and Model 3 (dashed line). Comparing these results to the impermeable rock case, it is clear that the required concentration gradient is much less when gases are allowed to diffuse through the inner pores of the soil medium. This allows small animals to have a much greater chance of survival than relying upon burrow diffusion only. From this plot, it is clear that a nest with numerous burrows built within a porous media is the ideal construction scenario if the organism has the energy availability to construct this complicated network.

Sensitivity Analysis

For terrestrial organisms right on the edge of meeting their necessary metabolic needs through passive diffusion, the sensitivity of diffusion to various environmental parameters is of the utmost importance. Withers utilized Model 3 to test the sensitivity of various parameters that influence the concentration gradient, such as the porosity of the medium, burrow radius and length, depth of the nest, and metabolic rate. By varying these parameters and calculating the corresponding concentration gradients, Withers found that soil porosity and soil moisture content had the greatest sensitivity, as a moist soil was found to significantly hinder diffusion. This, of course, is not encouraging news for those organisms that rely upon passive diffusion for survival, as quickly changing environmental conditions could render their nesting site obsolete due to hypoxic conditions. The metabolic rate was shown to be the next most sensitive parameter in this study. Therefore, increasing ones metabolic rate, or simply adding additional organisms to the nest can significantly increase the oxygen content needed for respiration. This is the reason for why most organisms that rely on passive diffusion tend to be solitary creatures. Surprisingly, the radius and length of the burrow were the least sensitive variables. No explanation was given for why this was found to be the case.

Conclusion

From this study, it was determined that passive diffusion alone is an inadequate means of transporting life sustaining gases to burrowing creatures. While subterranean animals are capable of surviving in a solitary nest, assuming the porosity of the soil is adequate, a colony of animals will be unable to cope with the high carbon dioxide build-up that will inevitably result. In addition, it was found that sub-aquatic organisms cannot rely on passive diffusion, since the molecular diffusion of substances in water is so small.

To overcome the issues associated with passive diffusion, organisms have evolved to utilize additional resources in order to transport nutrients to their underground chambers. Prairie dogs, for instance, construct their burrows in such a manner that the burrow is continuously flushed via advection, making molecular diffusion insignificant. By constructing the burrow so that it has two entrances to the surface, and building one entrance mound higher than the other, the prairie dogs ensure that the chamber will have sufficient ventilation. In building one mound higher than the other, the pressure over the taller mound must be lower than that over the shorter mound in order to satisfy Bernoulli's Equation, shown in the equation below. This establishes a pressure gradient within the burrow, which leads to an advective flux of oxygen into the nest site.

${\frac {P_{{tall}}}{\gamma }}+{\frac {V_{{tall}}^{2}}{2g}}+z_{{tall}}={\frac {P_{{short}}}{\gamma }}+{\frac {V_{{short}}^{2}}{2g}}+z_{{short}}$

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Recent Citations

Holtze S, Braude S, Lemma A, et al. . Afr J Ecol. 2017;00:1鈥�11

Prompted in part by Withers's finding that larger or more gregarious species should be unable to maintain sufficient oxygen levels by diffusion alone, Holtze et al performed measurements of temperature and gas concentrations in several colonies of naked mole-rats. As noted on page 8 the paper, Withers's prediction of large mass flow in the burrows appears to have failed, as no substantial air currents have been detected in the tunnels. Instead, their measurements lead the authors to suggest that exchange could be driven by temperature gradients between the burrow and the outside air, as discussed in some of their citations. Also noted is that naked mole-rat colonies exhibit behavior which may be consistent with Withers's model when the colony is exposed to moisture; at high moisture levels, colony activity decreases, possibly due to a decrease in available oxygen.

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Inbal Brickner-Braun, Daniel Zucker-Milwerger, Avi Braun, J. Scott Turner, Berry Pinshow, Pedro Berliner. Journal of Experimental Biology 2014 217: 4141-4148

Prompted by a lack of research into the relative contributions of various gas-exchange mechanisms in burrows (including gas diffusion as discussed in Withers's paper), this team constructed an artificial burrow. After placing sensors inside, the burrow was supplied with a slow stream of carbon dioxide to simulate occupancy, conditions were varied to exclude convective or diffusive effects, and data were collected. The team documented a previously unknown mechanism of wind-driven gas exchange in multi-entranced burrows which functions even in burrows with little or no elevation difference between entrances. They also confirmed Withers's finding that diffusion alone is sufficient to maintain livable gas concentrations for a smaller rodent (the simulated burrow was that of a Sundevaal's jird, a rodent weighing ~100g).

References

Withers, P.C. "," The American Naturalist. vol 112, pp. 1101-1112, 1978

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