2.3  Mass Transport and Bioenergetic Considerations

F. Grant Ferris; Natalie Szponar; Brock A. Edwards

2.3 Mass Transport and Bioenergetic Considerations

Recognition of the abundance and enormous genetic diversity among prokaryotes is balanced by unavoidable limitations imposed on life by physics and chemistry. The main physical processes that impinge on prokaryotic life relate to fluid dynamics and mass transport of solutes, including essential nutrients and metabolic waste products. At the same time, there is no way for any living organism to escape the laws of chemical thermodynamics that govern the spontaneity and progress of metabolic processes, including catabolic energy yielding and anabolic biosynthetic reactions.

Because of their small size, prokaryotic cells exist under conditions where viscous forces dominate over inertial forces. The dimensionless ratio of inertial to viscous forces corresponds to Reynold’s number (R_e), which is used in fluid dynamics to predict different flow patterns as a function of fluid density (ρ), relative flow velocity (u), characteristic linear length scale (L), and dynamic viscosity (μ) as shown in Equation 13.

$\displaystyle R_{e}=\frac{\rho uL}{\mu }$

(13)

At low Reynold’s numbers characteristic of prokaryotes in the 10^-6 m size range (R_e << 1), water flows in a direction parallel to cell surfaces in a smooth laminar fashion. In effect this isolates prokaryotic cells inside a viscous boundary layer of water, even in the presence of high groundwater flow velocities (> 100 m/d) and turbulence that occurs in some karst aquifers (Shoemaker et al., 2008).

Dissolved solutes are carried along by advection at the same relative velocity and in the same direction as water in the laminar flow boundary layer around cells. Because the flow of water runs parallel to the surface of cells, direct access to essential nutrients and dispersal of metabolic waste products by means of advective mass transport is not possible. To overcome this limitation, prokaryotes rely on molecular diffusion to mediate the lateral mass transport of solutes towards and away from cells.

In accordance with Fick’s first law, the diffusive flux (F_x; moles/m²∙s) in a direction perpendicular to the surface of a cell depends on the diffusion coefficient (D; m²/s) and change in concentration of a solute with respect to distance from the cell surface (dC/dx; moles/m⁴) as indicated by Equation 14.

$\displaystyle F_{x}=D\cdot \frac{dC}{dx}$

(14)

This relationship implies that higher (steeper) concentration gradients will increase diffusional mass transfer rates for prokaryotic microorganisms.

Metabolic rates of nutrient uptake and waste excretion play an instrumental role in the development and maintenance of concentration gradients around cells. Specifically, nutrient uptake will tend to decrease cell surface solute concentrations relative to surrounding mainstream concentrations, thereby establishing a concentration gradient towards the cell. Waste excretion will have the opposite effect on solute concentrations, resulting in a concentration gradient that extends away from the cell. On the other hand, higher mainstream nutrient solute concentrations or lower waste concentrations relative to cell surface concentrations will serve to increase concentration gradients and diffusional mass transfer to the benefit of microbial cells. Conversely, a decrease in mainstream nutrient concentrations or increase in waste concentrations will have the opposite effect on concentration gradients, resulting in a cutback on diffusional mass transfer and possible metabolic malnutrition.

Adoption of a larger cell size with a greater surface area seems like it would be helpful strategy to mitigate diffusion limitations. However, the diffusion distance to the middle of spheroidal coccoid bacterial cell (the cell radius) is greater than for a rod-shaped cell of the same volume (Figure 8). Diffusion time (t) furthermore varies in proportion to the square of diffusion distance (l) and molecular diffusion coefficient (D) as shown by Equation 15.

$\displaystyle t=\frac{l^{2}}{2D}$

(15)

For a doubling in size (volume) of a coccoid cell with an initial radius of 1.0 μm, the respective diffusion distance will increase by 30 percent to 1.3 μm and the corresponding solute diffusion time will increase by almost 70 percent. Conversely, diffusion distances and times remain unchanged in a fixed radius rod-shaped cell that doubles in volume by elongation instead of expanding outwards.

Figure showing relationships between the characteristic length scales (radius) of spheroidal coccoid and rod-shaped bacterial cells with corresponding cell surface areas, cellular volumes, and surface area to volume ratios.

Figure 8 – Relationships between the characteristic length scales (radius) of spheroidal coccoid and rod-shaped bacterial cells with corresponding cell surface areas, cellular volumes, and surface area to volume ratios.

The issue of thermodynamics boils down to how different kinds of microorganisms generate and conserve energy needed for growth and reproduction by cellular division (Bethke et al., 2011; Bird et al., 2011). All forms of life, including microorganisms, depend strictly on oxidation reactions for energy generation. In these reactions, a reduced chemical substance undergoes oxidization with the transfer of electrons to another, more oxidized, chemical substance. Such reactions are spontaneous and exergonic, corresponding to a negative Gibbs energy of reaction (ΔG_r). The amount of energy that is released is directly proportional to the difference in redox potential (ΔEh) between half-cell reactions of the reduced electron donor and oxidized electron acceptor as shown in Equation 16.

ΔG_r = −nFΔEh = –nF(Eh_{electron acceptor} – Eh_{electron donor})

(16)

where:

n	=	number of electrons transferred in the reaction
F	=	Faraday constant (C/mol)

The Eh values of the electron acceptor and electron donor are given by the Nernst equation (Equation 4).

The two main types of energy-generating pathways are respiration and fermentation. Respiration involves the transfer of electrons through a chain of metabolic intermediates, ultimately ending with a terminal electron acceptor. At various steps through the electron transport chain, released energy is captured in biochemical form as adenosine triphosphate (ATP). In fermentation, ATP is formed when a mixed oxidation state chemical compound is split in two, with one part being oxidized and the other reduced. Compared to respiration, the energy yield of fermentation tends to be lower. For this reason, respiration ranks as the preferred energy generation pathway of most organisms on Earth.

Aerobic respiration using molecular oxygen as a terminal electron acceptor yields the greatest amount of energy. This is because of the high standard redox potential (Eh = 1.23 V) of the oxygen-water half-cell reaction relative to that of chemical substances that may serve as electron donors (Table 1). But what really sets microorganisms (especially prokaryotes) apart from other forms of life is the vast array of other terminal electron acceptors that are used when oxygen is not available. This is called anaerobic respiration.

Table 1 – Standard potentials of some electron acceptor and electron donor half-cell reactions.

Electron Acceptors		Eh⁰ (V)
	$1/4\ \textup{O}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/2\ \textup{H}_{2}\textup{O}$	1.23
	$\displaystyle 1/5\ \textup{NO}_{3}^{-}+6/5\ \textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/10\ \textup{N}_{2}+3/5\ \textup{H}_{2}\textup{O}$	1.24
	$1/2\ \textup{MnNO}_{2}+2\ \textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/2\ \textup{Mn}^{2+}+\textup{H}_{2}\textup{O}$	1.22
	$\textup{Fe}(\textup{OH})_{3}+3\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =\textup{Fe}^{2+}+3\textup{H}_{2}\textup{O}$	1.07
	$1/8\ \textup{SO}_{4}^{2-}+9/8\ \textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/8\ \textup{HS}^{-}+1/2\ \textup{H}_{2}\textup{O}$	0.25
	$1/8\ \textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/8\ \textup{CH}_{4}+1/4\ \textup{H}_{2}\textup{O}$ (methane)	0.17
Electron Donors		Eh⁰ (V)
	$\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/2\ \textup{H}_{2}$	0
	$1/6\ \textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/6\ \textup{CH}_{3}\textup{OH}+1/6\ \textup{H}_{2}\textup{O}$ (methanol)	0.03
	$1/4\ \textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/4\ \textup{CH}_{2}\textup{O}+1/4\ \textup{H}_{2}\textup{O}$ (formaldehyde)	−0.07
	$1/2\ \textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/2\ \textup{HCOOH}$ (formic acid)	−0.20
	$\textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/2\ \textup{C}_{2}\textup{H}_{2}\textup{O}_{4}$ (oxalic acid)	−0.48
	$1/6\ \textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/12\ \textup{C}_{2}\textup{H}_{4}+1/3\ \textup{H}_{2}\textup{O}$ (ethylene)	0.07
	$1/6\ \textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/12\ \textup{C}_{2}\textup{H}_{5}\textup{OH}+1/4\ \textup{H}_{2}\textup{O}$ (ethanol)	0.09
	$1/8\ \textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/8\ \textup{CH}_{3}\textup{COOH}+1/4\ \textup{H}_{2}\textup{O}$ (acetic acid)	0.12
	$1/4\ \textup{CO}_{2}+\textup{H}^{+}+\textup{e}^{-}$ $\displaystyle =1/24\ \textup{C}_{6}\textup{H}_{12}\textup{O}_{6}+1/4\ \textup{H}_{2}\textup{O}$ (glucose)	−0.01

The conventional way of explaining anaerobic respiration uses the iconic imagery of steps down an energy ladder, where each downwards interval corresponds to a terminal electron acceptor half-cell reaction with a lower standard redox potential. This gives the often cited – and well worth memorizing – sequence of terminal electron acceptors for anaerobic respiration as nitrate, followed by Mn(IV), Fe(III), sulfate, and finally carbon dioxide. The corresponding conjugate reductants of the terminal electron acceptors are nitrogen, Fe(II) and Mn(II), sulfide, and methane/acetic acid.

At the same time, the image of a redox ladder for anaerobic respiration in several ways does not do justice to what happens in nature. First, the ladder is more like a slide, or continuum, because ΔG_r does not depend on Eh⁰ alone (Figure 9); instead, Eh is a critical variable as determined by actual concentrations of oxidants and conjugate reductants based on the Nernst equation (Equation 4). Second, the conventional list of terminal electron acceptors does not fully reflect the vast pool of potential oxidants available for anaerobic respiration in natural environments. These include various inorganic and organic substances, some of which are toxic to other organisms. As a third point, energy harvesting may occur simultaneously at different “rungs” of the redox ladder within a given environmental setting, by different groups of microorganisms or by a single microbial community shifting between energy sources.

Figure showing range of oxidation-reduction (Eh) potentials for common terminal electron acceptors from pH 6.0 (upper values) to pH 8.0 (lower values).

Figure 9 – Range of oxidation-reduction (Eh) potentials for common terminal electron acceptors from pH 6.0 (upper values) to pH 8.0 (lower values). The Eh values were calculated using the Nernst equation (Equation 4) on the basis of half-cell reactions listed in Table 1. A concentration of 10^-5 M was assumed for all dissolved chemical species other than H⁺, and unity activity for solid mineral phases and water with atmospheric pCO₂ = 10^–^3.50, pO₂ = 10^–^0.68, and pN₂ = 10^–^0.11 atm.

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