An Eyring equation based reformulation of metabolic theory of ecology
Ken Locey
Metabolism plays a role in all biological and ecological phenomena. According to the recently
formalized metabolic theory of ecology (MTE), metabolism affects ecological phenomena according to
the chemical kinetics of biochemical reactions and the fractal-like distribution of resources within
organisms. As currently formulated, MTE predicts that the exponential effect of temperature on
metabolic rate, as described by the Boltzmann factor (e-E/kT), drives an identical response in ecological
phenomena.
The Boltzmann factor comprises one of two primary components of the main MTE equation,
𝐸
𝐼 = 𝑖𝑜 𝑀𝑏 𝑒 −𝑘𝑇
Where I is the rate of a process, ioMb is a scaling relationship of mass, k is Boltzmann’s constant
(8.617343x10-5 eVK-1), and E is the activation energy of metabolism and is approximated as the slope of
the graph relating log of the rate to inverse absolute temperature. MTE makes two assumptions of E.
First, as defined by the main equation, E is independent of temperature. Second, E is confined to a
narrow range (0.6-0.7 eV) (Brown et al. 2004).
Recently, both of the above assumptions have been shown to be unrealistic. Nonlinear
Arrhenius plots, suggesting that E is not independent of temperature are common in physical,
biochemical, physiological, and ecological studies (Ulrike et al. 2009, Forrest 1967, Fabiano & Perego
2002, Chernyshev & Cukierman 2002, Converti et al. 2001, Parent et al. 2010, Algar et al 2007, McCain
and Sanders 2010, Hawkins et al. 2007). Variation in E well-outside the predicted range of 0.6-0.7 eV is
also commonly reported in MTE studies (Hawkins et al. 2007, Algar et al 2007, McCain and Sanders
2010). Hawkins et al. (2007) also reveal that the reported range and average of E has been changed
frequently since 2002, making the use of E as grounds for testing MTE problematic (e.g. Algar et al.
2007). Despite these issues, demonstration of linear Boltzmann relationships for rate-temperature and
richness-temperature relationships are commonly provided as support for MTE (Brown et al 2004, Allen
et al. 2002, Gillooly et al. 2005, Anderson-Teixeira et al. 2008).
Failure to demonstrate linear Boltzmann relationships for species-temperature relationships (i.e.
predicted exponential decrease in richness with temperature) has led some to the reject temperaturerelated chemical kinetics modeled by MTE as a primary driver of species richness (e.g. Algar et al 2007,
McCain and Sanders 2010, Hawkins et al. 2007). To maintain that the chemical kinetics behind
metabolism have predominant effects on the number of species found within large-scale geographical
areas, MTE may require a kinetics-based reformulation. One possibility is to explore adaptations of the
Eyring equation of transition state theory to the main MTE formulation. This would anticipate the
commonly observed downward curvilinear Boltzmann relationships of richness to temperature that are
common in varied ecological studies and that have caused some to reject the basic MTE premise.
The Eyring equation
A precise formulation of the rate response of systems to changing temperature is given by the
Eyring equation of transition state theory (Eyring 1936). The Eyring equation is used when
thermodynamic components of a reaction or system are of interest, or when activation energy is
anticipated to be temperature dependent (Kast et al. 1996, Converti, et al 2001, Lonhienne et al. 2000,
Low et al. 1973, Parent et al. 2010). The Eyring equation resembles the Arrhenius equation, but is a
more precise varying function of temperature:
𝐾 =𝜎
𝑘𝑇 −∆𝐺‡
𝑒 𝑅𝑇
ℎ
Where h is Planck’s constant (6.626069x10-34 Js), R is the gas constant (8.314472 JK-1mol-1), σ is the
symmetry factor and is the number of equivalent reaction paths, σkT/h is the universal frequency factor,
and ∆G‡ is the Gibbs free energy of activation and is the molar Gibbs energy change for the conversion
of reactants into an activated complex (i.e. transition state). ∆G‡ is related to E through the enthalpy
and entropy of activation ∆H‡ and ∆S‡, respectively (CITE) as:
∆𝐺 ‡= ∆𝐻 ‡ −𝑇 ∆𝑆 ‡
∆𝐻 ‡= 𝐸 − 𝑅𝑇
Hence, neither E nor ∆G‡ is independent of temperature:
𝐸 = ∆𝐺 ‡ +𝑇(𝑅 + ∆𝑆 ‡)
An Eyring-based MTE equation
Recall that in the MTE, Arrhenius, and Eyring equations, the left-hand of the equation is the rate
and the right-hand side is the product of two factors, one representing the exponential effect of
temperature and the other being an empirically derived pre-exponential factor (e.g. σkT/h , Ao, ioMb). In
the MTE equation ioMb can be interpreted as the effect of the mass scaling relationship on the
temperature-rate relationship. With these similarities in mind, we can seek to replace the Boltzmann
factor of MTE with an Eyring-based factor that anticipates downward curvilinear Boltzmann
relationships:
𝑇𝑒
−∆𝐺‡
𝑅𝑇
This produces the modified MTE equation:
𝐼 = 𝑖𝑜 𝑀𝑏 𝑇𝑒
−∆𝐺‡
𝑅𝑇
In log-linear form,
1
ln(𝐼𝑀−𝑏 ) = −∆𝐺 ‡ ( ) + ln(𝑖𝑜 𝑇)
𝑅𝑇
And without accounting for the temperature dependence of ∆G‡ (usually negligible given the
relationship ∆G‡ = ∆H‡ - T∆S‡ and that values of ∆S‡ are typically two to three orders of magnitude
smaller than ∆G‡ and ∆H‡), the modified MTE equation predicts that the graph relating the mass
corrected metabolic rate to inverse temperature should become increasingly negative with decreasing
temperature for positive values of ∆G‡. Such graphs are common in MTE studies (e.g. Algar et al 2007,
McCain and Sanders 2010, Hawkins et al. 2007). The slope of this plot is an approximation of the
standard Gibbs free energy of activation, ∆G‡. Values of ∆G‡ for ATP and tricarboxylic acid cycle
related reactions often range between 52 and 82 kJmol-1 for diverse groups (e.g. Lonhienne et al. 2000,
Bokhari et al. 2010, Banerjee et al. 2006, Greene and Frasch 2003, Hussain et al. 2009, Riaz et al. 2007,
Mnatsakanyan et al. 2009, Feller 2010, Santabarbara et al. 2009).
Species-temperature relationships
Metabolic theory predicts that species richness, like rates of biological and ecological
phenomena, is related to temperature through the Arrhenius/Boltzmann relationship (Brown et al.
2004, Gillooly et al. 2001, 2005). However, tests of the species-temperature relationship often
demonstrate curvilinear relationships (e.g. Algar et al 2007, McCain and Sanders 2010, Hawkins et al.
2007), which are not predicted by the Arrhenius relationship but are predicted by the Eyring
relationship. Here, we demonstrate that the reformulated metabolic theory equation captures the
change in species richness, S, with temperature when richness is used in place of mass corrected
metabolic rate, as seen in previous studies (Brown et al. 2004, Gillooly et al. 2001, 2005, Algar et al 2007,
McCain and Sanders 2010, Hawkins et al. 2007, Anderson-Teixeira et al. 2008). Hence,
1
ln(𝑆/𝑇) = −∆𝐺 ‡ ( ) + ln(𝑖𝑜 )
𝑅𝑇
Data were gathered from previously published studies that have concluded that the observed
species-temperature relationships were often curvilinear and were not in agreement with the log-linear
prediction of the Boltzmann relationship of the original MTE formulation (Hawkins et al. 2007…and
coming).
(The following are graphs of the above log-linear relationship with data taken from Hawkins et al. 2007.
Graphs with ‘breakpoint’ indicate a partial dataset from the range of points where richness/temperature
increases with temperature. These are just simple Excel regressions).
North American Trees
0
y = -55.532x + 21.901
R² = 0.5679
ln(richness/T)
-1
-2
-3
-4
-5
-6
0.4
0.41
0.42
0.43
0.44
0.45
0.46
0.47
1/RT
European Frogs
-2
y = -51.618x + 18.384
R² = 0.3983
ln(richness/T)
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
0.4
0.41
0.42
0.43
1/RT
0.44
0.45
0.46
European Snakes
-2
y = -68.304x + 24.966
R² = 0.5258
-2.5
Axis Title
-3
-3.5
-4
-4.5
-5
-5.5
-6
0.4
0.41
0.42
0.43
0.44
0.45
0.46
Axis Title
European Lizards
-2
ln(richness/T)
-2.5
y = -73.438x + 27.208
R² = 0.6045
-3
-3.5
-4
-4.5
-5
-5.5
-6
0.4
0.41
0.42
0.43
1/RT
0.44
0.45
0.46
ln(richness/T)
North American Snakes
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
y = -112.95x + 44.85
R² = 0.8411
0.4
0.41
0.42
0.43
0.44
0.45
0.46
1/RT
North American Lizards
-2
y = -90.507x + 34.405
R² = 0.5087
ln(richness/T)
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
-6
0.4
0.41
0.42
0.43
1/RT
0.44
0.45
North American Frogs
-2
-2.5
y = -63.999x + 23.821
R² = 0.7973
ln(richness/T)
-3
-3.5
-4
-4.5
-5
-5.5
-6
-6.5 0.4
0.41
0.42
0.43
0.44
0.45
0.46
0.47
1/RT
ln(richness/T)
UK plants
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0.422
y = -79.697x + 34.496
R² = 0.5049
0.424
0.426
0.428
1/RT
0.43
0.432
0.434
0.436
Western Palearctic Eupelmids
0
-0.5
y = -7.1161x + 1.0327
R² = 0.0491
Axis Title
-1
-1.5
-2
-2.5
-3
-3.5
0.39
0.4
0.41
0.42
0.43
0.44
0.45
0.46
Axis Title
WPal Eupelmids, Breakpoint
0
-0.5
Axis Title
-1
y = -55.409x + 21.997
R² = 0.7136
-1.5
-2
-2.5
-3
-3.5
0.415
0.42
0.425
0.43
0.435
0.44
Axis Title
0.445
0.45
0.455
0.46
North American Tiger Beetles
-2
ln(richness/T)
-2.5
y = -51.339x + 18.563
R² = 0.5378
-3
-3.5
-4
-4.5
-5
-5.5
-6
0.4
0.41
0.42
0.43
0.44
0.45
0.46
0.47
1/RT
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