The Metabolic Basis of Ecology and Evolutionary dynamics#
The struggle for existence of living beings is not for the fundamental constituents of food, but for the possession of the free energy obtained, chiefly by means of the green plant, from the transfer of radiant energy from the hot sun to the cold earth.
—Ludwig Boltzmann, 1886, “The Second Law of Thermodynamics”
Introduction#
Metabolism is fundamental to the understanding of ecological and evolutionary processes.
The laws of thermodynamics are fundamental principles describing how energy and heat behave in physical, chemical, and living systems. As Boltzmann noted [Boltzmann, 1886], the Second Law of Thermodynamics is fundamental for understanding how biological cells and systems work as well as their ecological and evolutionary dynamics:
In any spontaneous process, the total entropy of an isolated system can never decrease. Equivalently, it can stay the same (in a reversible process) or increase (in an irreversible process).
The two direct implications of this Law are directly relevant to biological systems:
The first implication is the “arrow of time” — natural processes tend to move toward greater Entropy (disorder). What makes life and biological processes unique (in contrast to purely physical or chemical processes) is that in essence, the purpose of living organisms is to counteract this law by producing order (living biomass) from disorder (photons and molecules). At the most fundamental level, this is possible because living organisms are not isolated — thay are open systems that continuously exchange energy and matter with their surroundings.
Note
While organisms maintain or increase their internal order (i.e., decrease entropy locally) by building and organizing complex molecules, they do so by taking in energy from external sources (e.g., sunlight or food) and releasing waste heat and byproducts back into their environment. This flow of energy and matter ultimately does increase the total entropy of the universe. Thus, life’s local decrease in entropy comes at the expense of a larger increase in entropy elsewhere, in accordance with the second law.
The second implication is that it is impossible to convert heat completely into work without waste (there is no perfect engine) — living organisms like any engine are subject to this limitation, but are typically designed by natural selection to maximise or optimise their energy conversion efficiency. Indeed, Lotka [Lotka, 1922] made this observation over a 100 years ago:
It has been pointed out by Boltzmann’ that the fundamental object ofcontention in the life-struggle, in the evolution of the organic world, is available energy. In accord with this observation is the principle that,in the struggle for existence, the advantage must go to those organismswhose energy-capturing devices are most efficient in directing available energy into channels favorable to the preservation of the species.
Energy and Metabolic Rate#
Energy is a measurable property that quantified an object’s capacity to perform work (in a physical sense). In living organisms, a key function is to harness and utilize energy to support growth, maintenance, and reproduction. Plants and other autotrophs capture photons through photosynthesis, transforming light energy into chemical energy stored in glucose. Heterotrophs, such as animals, obtain energy by breaking chemical bonds in organic molecules. A crucial molecule in this process is ATP (adenosine triphosphate), which serves as a primary energy carrier within cells and powers numerous biochemical reactions.
Metabolism, the set of chemical reactions within an organism, determines metabolic rate, which is the rate of energy use, often measured in J/s, kcal/day, or Watts. Organisms must balance energy consumption and expenditure:
Metabolic rates influence key life-history traits, including movement rates, development speed, lifespan, and reproductive output [Dell et al., 2011].
The metabolic theory of ecology (MTE) [Brown et al., 2004] provides a framework for understanding how metabolic processes scale across levels of biological organization and influence ecological dynamics. It combines the biomechanical constraints of cell or body size of organisms and the thermodynamic effects of temperature on biological reaction kinetics into one equation for organism-level (that of single, integral organisms, either unicells or multi-cells) metabolic rate.
We will now seperately consider the biophysical effects of size and temperature and then combine them into a unified understanding (and equations).
Importance of Size#
Metabolic rate (\(B\)) increases with body size (\(M\)) according to the allometric scaling law [Kleiber, 1947]:
or equivalently:
where \(b\) is typically close to 3/4 for multicellular euykaryotic organisms. This implies that larger organisms have higher absolute metabolic rates but lower mass-specific rates (\(B/M\)):
For example, a mouse (0.1 kg) needs approximately 12.3 kcal/day, while its mass-specific rate is 123 kcal/(kg\(\cdot\)day).
This reduction in per-mass energy use with size arises from a combination of constraints - geometric, overheating risk (Refs), and within-organism resource distribution (West et al., 2005).
Geometric foundations of the metabolic scaling law#
Historically, a geometric argument was used to suggest that metabolic rate scales like the surface area, which in turn scales as the \(\tfrac{2}{3}\) power of mass.
Let us assume an organism (or cell) is approximately spherical of radius \(l\).
Note
We will use \(l\) instead of the traditional \(l\) to denote the radius of a sphere to avoid confusion, because the latter is a comonlyu used symbol for population growth rate (which we will encounter in the next chapter).
Let’s start with the basic equations:
Surface area of sphere \(A = 4\pi l^2\)
Volumev of a sphere \(V = \tfrac{4}{3}\pi l^3\)
If the organism/cell has a constant density \(\rho\), then its mass \(m\) is proportional to its volume:
Then, the surface-to-volume ratio is
That is, as \(l\) increases, \(\tfrac{A}{V}\) decreases (\(\propto 1/l\)).
Now, in biological cells, the surface area is where exchange of nutrients, oxygen, and waste takes place, while the volume represents total metabolic demand (more “living material” inside).
So, larger cell radius \(\Rightarrow\) lower surface-to-volume ratio \(\Rightarrow\) harder to meet the metabolic needs of the cell or organism.
This leads to the classic geometric argument that metabolic rate \(B\) (e.g., oxygen consumption per unit time) is limited by the total surface available for exchange:
Using \(A = 4\pi l^2\), we get
Since \(m \propto l^3\), we have \(l \propto m^{1/3}\). Therefore,
Hence,
Including a scaling constant upo front,
which is the same as the first scaling equation above with \(b = \frac{2}{3}\).
That is, this scaling arises from the fgact that an organism’s metabolic rate scales with its ability to exchange materials through a surface (which goes as length squared), while its mass (and thus total metabolic demand) goes as length cubed.
Biological implications of the geometric scaling constraint#
Thus, small cells have a relatively large surface area for their volume. They can exchange nutrients and wastes quickly, supporting higher metabolic rates per unit volume. In contrast, larger cells (or organisms) have a relatively smaller surface area per unit volume. Consequently, they may be limited in how fast they can exchange vital substances, placing a cap on metabolic rate per unit mass or per unit volume. This is why cells are relatively small: They need large surface area relative to volume to ensure adequate exchange of nutrients/oxygen and removal of wastes for the metabolic demands of their volume.
Going beyond geometric constraints#
Th above surface-to-volume geo biophysical constraint indeed holds at the level of cells, but is not the full story.
Empirically, the scaling most multicellular organisms in fact follows a somewhat different exponent (\(\approx \frac{3}{4}\), known as Kleiber’s law. This comes from the two additonal constraints: the requirement of heat dissipation and energy distribution (circulation) thorughout the body. We will not cover these here, but if you are intersted, please have a look at the readings at the end of this chapter.
Importance of Temperature (and Thermodynamic constraints)#
Temperature profoundly impacts metabolism, as biochemical reaction rates in cells increase inexorably with temperature following the Boltzmann-Arrhenius equation:
where \(k\) is the reaction rate, \(k_0\) is a constant, \(E_a\) is the activation energy, \(k_B\) is the Boltzmann constant, and \(T\) is temperature in Kelvin. This relationship explains the characteristic thermal performance curve observed in biological processes, where rates increase with temperature to an optimum before declining due to enzyme denaturation or other stress factors (Johnson et al., 1974) (which we will consider later below).
The Arrhenius equation and its variants (often collectively referred to as the “Boltzmann-Arrhenius” equation in certain contexts) are cornerstones of chemical kinetics. These formulations describe how reaction rates vary with temperature. For enzyme kinetics, the temperature dependence can also be described via modified forms of the Arrhenius equation, sometimes combined with effects such as protein denaturation at higher temperatures.
Historically, this work can be traced back to Jacobus Henricus van ‘t Hoff (1852–1911), whose studies first quantified the temperature dependence of reaction rates. Building on van ‘t Hoff’s work, Svante Arrhenius (1859–1927) proposed a more explicit relationship that connected the rate constant \(k\) of a chemical reaction to an exponential function of temperature:
where:
\( A \) is the pre-exponential factor (or frequency factor),
\( E_a \) is the activation energy,
\( R \) is the universal gas constant, and
\( T \) is the absolute temperature in Kelvin.
For enzyme-catalyzed reactions, temperature significantly affects the reaction velocity \(v\) or the turnover number (catalytic rate constant \(k_{\mathrm{cat}}\)). At “moderate” temperatures, higher temperature generally increases reaction rates, consistent with Arrhenius behavior. However, at higher temperatures, thermal denaturation of the enzyme reduces or abolishes its catalytic activity. Hence, more realistic models for enzyme kinetics must account for both the Arrhenius increase in rate and the loss of enzyme activity due to denaturation, which we will consider later below.
Note
All organisms on earth are dependent on temperature directly or indirectly: Poikilotherms such as unicells and tiny insects cannot thermoregulate much; Ectotherms such as plants, insects and reptiles, rely on external temperature to regulate their metabolic processes (thermoregulation); Endotherms, such as mammals and birds, generate internal heat to maintain a stable body temperature, but this comes at a significant energetic cost, and ultimately are reliant on the energy harnessed by poikilotherms and ectotherms (starting with green plants).
The rate of an enzyme-catalyzed reaction can be modelled as:
where,
\(k_\text{cat}(T)\) is strongly dependent on \(T\),
\([\text{C}]\) is the enzyme concentration,
\(f([\text{S}])\) is a function of substrate concentration (such as the Michaelis-Menten form \(\tfrac{[\text{S}]}{K_M + [\text{S}] }\)).
We can model \(k_\text{cat}(T)\) using either the Arrhenius or the Eyring Equations. However, because real enzymes denature at higher temperatures, we typically observe a bell-shaped temperature activity profile.
Classical Transition State Theory (TST)#
We will now look at classical transition state theory (TST), which provides a microscopic rationale for the temperature dependence of reaction rates, explaining both how energy barriers (enthalpy) slow down reactions and how changes in molecular order (entropy) can further modulate the speed at which reactants cross from the well of the reactants to the well of reaction products.
TST was developed in the 1930s by Henry Eyring (1901–1981) and others (in particular Meredith Evans and Michael Polanyi).
The key idea underlying the TST is that a chemical reaction proceeds through a high-energy configuration of the reactants called the transition state (or activated complex). Once formed, the transition state either proceeds forward to products or reverts back to reactants, adhering to two key assumptions:
Reagents must first form an activated complex, which is in a “quasi-equilibrium” with the reactants.
The rate of the reaction then depends on how frequently and efficiently this activated complex “crosses over” from the reactant side of the potential energy surface to the product side.
Using statistical mechanics arguments, TST yields the famous Eyring (or Eyring–Polanyi) equation, sometimes called the “absolute rate equation:
where
\(k\) is the rate constant,
\(k_B\) is Boltzmann’s constant,
\(h\) is Planck’s constant,
\(R\) is the gas constant,
\(T\) is the absolute temperature,
\(\Delta H^\ddagger\) is the activation enthalpy,
\(\Delta S^\ddagger\) is the activation entropy.
The key result of TST is the Eyring (or Eyring-Polanyi) Equation, usually written as:
where
\(\kappa\) is the transmission coefficient (often assumed to be ~1 for many simple reactions),
\(k_B\) is the Boltzmann constant,
\(h\) is Planck’s constant,
\(\Delta S^\ddagger\) is the entropy of activation,
\(\Delta H^\ddagger\) is the enthalpy of activation, and
\(R\) and \(T\) are as before.
This equation is more often called the “Eyring equation” and offers a more explicitly thermodynamics-based perspective. It reduces to an Arrhenius-like form when you group constants appropriately.
Through this equation, TST provides a molecular-level interpretation for the pre-exponential factor
\(\frac{k_B T}{h} \exp \Delta S^\ddagger / R)\)) and the exponential barrier term \(\exp(-\Delta H^\ddagger / (RT))\)) in the rate expression.
\(\Delta H^\ddagger\) is the difference in enthalpy (heat content) between the transition state (activated complex) and the reactants. It represents the energy barrier that must be surmounted for the reaction to occur. A larger \(\Delta H^\ddagger\) means a higher barrier, which tends to slow down the reaction, especially at lower temperatures.
In the factor \(\exp(-\Delta H^\ddagger / (RT))\), \(\Delta H^\ddagger\) appears in the exponential term with a \(1/T\) dependence.Consequently, the higher the activation enthalpy, the more strongly the rate constant depends on temperature (i.e., higher sensitivity in an Arrhenius-type plot).
Next, \(\Delta S^\ddagger\) is the difference in entropy between the transition state and the reactants. It reflects the change in molecular disorder when going from reactants to the activated complex. A positive \(\Delta S^\ddagger\) implies that forming the transition state is relatively more disordered than the reactants, favoring the reaction (larger rate constant). A negative \(\Delta S^\ddagger\) implies a more ordered transition state, which lowers the rate constant by decreasing the pre-exponential factor.
Finally, \(\exp \Delta S^\ddagger / R)\) multiplies the basic frequency factor \(\frac{k_B T}{h}\). Therefore, even if \(\Delta H^\ddagger\) is moderate, a large positive \(\Delta S^\ddagger\) can significantly enhance the reaction rate, whereas a negative \(\Delta S^\ddagger\) can suppress it.
Overall Temperature Dependence of the Rate#
By combining \(\Delta H^\ddagger\) and \(\Delta S^\ddagger\), TST clarifies how and why reaction rates depend on temperature:
The \(\exp(-\Delta H^\ddagger / (RT))\) term governs the primary exponential sensitivity of \(k\) to changes in temperature. The \(\exp\)\Delta S^\ddagger / R)$ term influences the intrinsic magnitude of the rate constant at any given temperature by encoding the “positional” or “configurational” freedom of the transition state.
Experimentalists often plot \(\ln(k/T)\) versus \(1/T\), known as an Eyring plot:
Putting it together: temperature, size scaling, and metabolic rate#
Thus, metabolism drives growth rates and influences population dynamics through its scaling with size and temperature (Brown et al., 2004). Larger organisms exhibit slower per-mass metabolic rates but greater absolute energy use (Savage et al., 2004). Temperature affects metabolic rates exponentially, shaping life-history strategies and population dynamics (Dell et al., 2011).
Combining these gives MTE’s fundamental equation:
Where \(B'_0\) is a size- and temperature-independent (normalization) constant.
Ecological implications#
We can now consider how the temperature dependence of enzyme kinetics and basic cellular physiology leads to a temperature‐dependent rate of cell division, and ultimately to exponential population growth. We will use microbes as a model because of the relative simplicity of their growth process, which relies on simple cell division.
From intracellular reaction rates to cell growth#
A living cell’s growth and division require a multitude of coupled enzyme reactions (for nutrient uptake, biosynthesis, energy generation, etc.). MTE assumes that one key rate‐limiting step (or a small set of them) dominates the overall pace of cell growth and division. If those rate‐limiting reactions follow Arrhenius‐type kinetics, then the cellular physiological rate (e.g., biomass production rate, cell cycle progression rate) will inherit a similar temperature dependence:
Hence, as \(T\) increases (within a tolerable range), enzyme‐catalyzed processes speed up, and so does the cell’s overall metabolism and ability to replicate.
From single‐cell growth to division#
Cell division is a stochastic process#
To move from enzyme kinetics to cell‐level dynamics every cell must Replicate all essential components (DNA, proteins, membranes, etc.), Reach a certain “completion threshold” in biomass or chromosome replication, and Divide into two daughter cells.
Because cellular biochemical events occur stochastically (randomly), the exact division time of any one cell can vary. However, in an exponential (log) growth phase with abundant nutrients, the average cell‐division rate \(\lambda\) (divisions per unit time) is fairly constant (Jafarpour et al 2020). This average division rate is ultimately tied to the cumulative rate of intracellular biochemical and biosynthetic reactions, and thus inherits their temperature dependence.
(Average) Doubling time#
Let \(\tau_d\) be the average doubling time (the mean time for one cell to become two). Then,
From the Arrhenius perspective, we can write
so that the cell’s doubling time decreases with increasing temperature (again, within the physiological limit). Equivalently, the division rate \(\lambda(T) = 1/\tau_d(T)\) would increase with temperature:
From cell division rate to population growth#
Let’s consider a well‐mixed microbial population in batch or chemostat conditions (nurients are flowing through the environment continuously), focusing on their exponential growth phase (AKA log phase). If each cell divides (on average) at rate \(\lambda(T)\), then the expected number of cells \(N(t)\) follows the exponential growth law:
This is simply the standard Malthus‐type (or exponential) growth equation, whose solution is an exponential in time, with the growth rate \(\lambda(T)\) now an Arrhenius‐like function of temperature.
Growth Rate vs Doubling Time#
Microbiologists typically define the specific growth rate \(\mu(T)\) via
so that \(\mu(T)\equiv \lambda(T)\). Equivalently, the doubling time is
Thus, you will often see microbial growth curves written as
where
So, in a nutshell,
\( \to\) Reaction rates speed up with temperature according to Arrhenius or Eyring equations.
\( \to\) The cell division cycle (DNA replication, protein synthesis, etc.) is governed by these enzymatic rates, so the cell division rate \(\lambda(T)\) similarly increases with temperature (up to an optimum).
\( \to\) When each cell divides at rate \(\lambda(T)\), the population follows exponential growth \(N(t) = N(0) \exp[\lambda(T) t]\). The specific growth rate \(\mu(T)\) is effectively the cell division rate and inherits the same temperature dependence.
There are also a few key caveats to keep in mind:
Real microbes have an optimal growth temperature. At high temperatures, enzymes denature, and growth drops off. Thus, beyond a certain point, Arrhenius‐type behavior breaks down.
Many enzymatic steps contribute to growth; the overall temperature dependence can be more complicated than a simple exponential. Empirical models like the Ratkowsky model (\(\sqrt{\mu}\) vs. \(T\)) or the Cardinal model with minimum, optimum, and maximum temperatures, are often used by biologists instead of explicitly thermodynamic equations.
Below or near a certain threshold temperature, the maintenance energy requirement may exceed the energy‐generating reactions, effectively stopping growth.
In single‐cell studies (e.g., time‐lapse microscopy), you see that individual cells have variable interdivision times. However, at the population level, the average behavior typically follows the exponential law (Jafarpour et al 2019).
Incorporating Cell Size Scaling#
Metabolic Theory of Ecology (MTE)’s equation for organism’s metabolic rate (see above) can be written as:
where:
\(M\) is the organism’s mass (for microbes, think cell mass),
\(E\) is an effective activation energy for metabolic reactions,
\(k_B\) here is Boltzmann’s constant (not to be confused with the same symbol in TST; context determines which constant is meant).
A common assumption in MTE is that the growth rate of an organism is roughly proportional to its mass‐specific metabolic rate:
Thus,
For microbial cells of mass \(M\), the MTE‐inspired prediction for the specific growth rate \(\mu(T, M)\) becomes:
where \(\mu_0\) absorbs constants like \(B_0\).
The \(M^{b-1}\) factor, *in the case of multi-cellular organisms, reflects how smaller cells typically have a faster mass‐specific metabolic rate (and thus can grow/divide faster), whereas larger cells would have a somewhat lower per‐mass metabolic rate.
Note
The scaling of metabolic rate of microbial cells is very different (it is not allomteric, but iso-metric or super-linear; see DeLong et al 2010).
Next, plugging \(\mu(T, M)\) into the exponential growth equation:
Hence, from MTE, the population’s exponential growth rate now depends explicitly on both Temperature (\(\exp[-E/(k_B T)]\) factor and organismal size (\(M^{1-b}\) factor).
In simpler microbial physiology models (which ignore size variation), one might set \(M\approx\text{constant}\). But if comparing cells (or species) of different typical cell sizes, or tracking how cell size changes under various conditions, the \(M^{b-1}\) dependence becomes relevant.
So, in summary,
Intracellular metabolic/biochemical rates follow an Arrhenius‐ or Eyring‐ increase with \(T\).
The cell division (doubling) rate \(\lambda(T)\) (or \(\mu(T)\)) is governed by these enzyme‐limited processes.
In MTE, it also scales with cell mass as \(M^{-1/4}\).
When every cell divides at rate \(\mu(T,M)\), the population satisfies \(\tfrac{dN}{dt} = \mu(T,M) N\). Thus, we get exponential growth at a temperature‐ and size‐dependent rate.
Thus, MTE unifies the thermal dependence of growth rates with * Size scaling*, providing a more holistic understanding of organismal growth across different temperatures and organismal sizes across the tree of life.
A more general relationship#
In general, because larger multicellular organisms have lower mass-specific metabolic rates, enabling longer lifespans and greater reproductive investment (Savage et al., 2004). That is, population growth rate scales negatively with body size:
Therefore, population density decreases with size (Damuth’s law). Carnivores are rarer than herbivores due to energy loss across trophic levels (Colinvaux, 1978). We will address these phenomena later.
From exponential to logistic growth#
Next, let’s look at logistic growth, once cells and individuals have moved from the purely exponential growth regime (valid at low population density) to a density‐dependent (logistic) regime, where “carrying capacity” plays a role.
Classic Logistic Equation#
The logistic equation introduces a carrying capacity \(K\), which is the maximum population (or density) sustainable by the environment:
where,
\(r\) is the (intrinsic) per‐capita growth rate in the absence of density‐dependent limitation,
\(K\) is the carrying capacity.
If \(r\) and \(K\) are constants, the solution of this equation is well known (and which you will derive later in the course):
which saturates at \(N(t)\to K\) as \(t\to\infty\).
2.2 Temperature & Size Dependence in Logistic Growth#
To generalize, we can let the intrinsic growth rate \(r\) be temperature‐ and cell size‐dependent, i.e.,
Thus, the logistic equation becomes:
Depending on the complexity of the system, one might also consider \(K\) to be T‐dependent, \(M\)‐dependent, or both—especially if resource supply, waste removal, or cell size changes systematically with temperature or over time.
When \(N \ll K\), the term \(\Bigl(1 - N/K\Bigr)\approx 1\), and:
Hence, the early dynamics are effectively exponential, with the rate
As \(N\) approaches \(K\), resource depletion or waste accumulation slows growth:
Eventually, \(N(t)\) saturates at \(K\). No matter how fast the intrinsic rate \(r(T,M)\) is (e.g., at high temperature for small cells), the population cannot surpass the environment’s carrying capacity.
Temperature and Size Dependence of \(K\)#
In many real ecological or bioprocess contexts, the carrying capacity \(K\) might also depend on temperature (through effects on nutrient availability, oxygen solubility, etc.) or on cell size (larger cells might require more resources, reducing the total cell number sustainable). A simple extension might be:
where \(f(T)\) and \(g(M)\) are functions that capture how resource utilization or environment capacity scales with temperature and cell size. Then,
Although analytical solutions become more complicated, the qualitative behavior remains logistic—exponential growth at low \(N\), saturating at an upper limit.
Population dynamics#
Putting all these comonents into a single differential equation:
When \(N(t)\ll K\), \(dN/dt \approx r_0 M^{-1/4} \exp[-E/(k_B T)] N\).
As \(N(t)\to K\), \(dN/dt\to 0\).
Or the more general form of \(K\) also depends on \(T\) and \(M\):
A potentuially useful Integrated Model is:
(optionally letting \(K = K(T,M)\)) succinctly unifies temperature dependence, cell size scaling, and density limitations.
Hence, you get a temperature‐ and size‐dependent logistic equation that captures both the initial exponential ramp (driven by temperature‐enhanced metabolism in smaller or larger organsims / indoviduals) and the ultimate saturation governed by the ecological/physical constraints of the environment.
Biological Interpretations#
Temperature and cell‐size effects dominate, so a small organism at an optimal temperature quickly ramps up in population.
Eventually, the environment or culture conditions impose a limit \(K\). Growth slows down, and the population saturates (e.g., nutrient depletion in a batch culture).
Smaller individuals often have higher mass‐specific metabolic rates, thus a steeper initial exponential growth. However, if \(K\) is in terms of total number of cells, smaller cells might sustain a higher \(K\) (because each cell needs fewer resources). Conversely, if \(K\) is in terms of total biomass, the absolute number of larger cells may be lower at carrying capacity.
Real systems show an optimal temperature for growth. Extremely high \(T\) leads to enzyme denaturation; extremely low \(T\) slows reaction rates. Thus, the functional form \(r(T,M)\) will be more complex than a simple Arrhenius.
Readings#
Arrhenius, S. (1889). Über die Reaktionsgeschwindigkeit bei Inversion von Rohrzucker durch Säuren. Z. Phys. Chem. 4, 226–248.
van ‘t Hoff, J. H. (1884). Études de dynamique chimique. Frederik Muller & Cie.
Eyring, H. (1935). The Activated Complex in Chemical Reactions. J. Chem. Phys. 3(2), 107–115.
Polanyi, M. (1935). Z. Phys. Chem. B 28, 309–318.
Johnson, K. A., & Goody, R. S. (2011). The Original Michaelis Constant: Translation of the 1913 Michaelis–Menten Paper. Biochemistry, 50(39), 8264–8269.
Arrhenius Plots are generated by taking the natural log on both sides and plotting \(\ln(k)\) vs. \(1/T\) should yield a straight line with slope \(-E_a/R\) and intercept \(\ln(A)\).
Eyring Equation: A common linear form for plotting is:
Thus, \(\ln(k/T)\) vs. \(1/T\) should be linear with slope \(-\Delta H^\ddagger / R\) and intercept \(\ln(k_B/h) + \Delta S^\ddagger / R\).