Temperature dependence of performance and precision:
Ectotherms versus homeotherms
When various performance rates of ectothermic organisms are measured as a function of environmental temperature, the data very often have the following general characteristics: at low temperatures the slope increases with temperature before beginning to decrease (at the "optimum" temperature) faster than the initial phase. Various equations have been used to fit such behavior (temperature performance curves, TPC), sometimes based on some mechanistic model. This behavior is observed at various level of biological organization, for example, at the molecular level in the photosynthesis rate in plants, at the organismal level in the maximum running speed of lizards, and at the population level in the rate of population increase in insects.
The initial rising slope is often attributed, at least in part, to most chemical reactions, including those catalyzed by enzymes, thermal vibrations must overcome an energy barrier. Some enzymes must also undergo a confirmational change, again with an energy barrier. Ritchie et al. (2018) attributed some of the increase with temperature in enzymatic reactions to increased rates of diffusion of substrates to or products away from the reaction site. At low temperatures, enzymes are often not because the hydrophobicity of amino acids side chains in the core is not strong enough to prevent water from entering, effecting folding and the active site. Enzymatic activity increases with temperature because hydrophobicity increases.
A variety of reasons have been suggested for the fast decrease in performance above the optimum temperature. In the case of enzymatic reactions, this could do to increase rates of unfolding and degradation. In the case of organism activity, it may be that at the higher temperatures the rate at which energy can be produced cannot keep up with the energy necessary to maintain the activity at a high level.
Despite all the possible mechanisms that result in this kind of behavior, Arnoldi et al. (2025) have proposed that almost all TPCs can be fitted by a relatively simple equation with three parameters, the Universal Temperature Performance Curve (UTPC). The figure at right illustrates a typical TPC, in this case the hopping speed of a frog. Topt is the temperature T for optimum speed, Tc is the temperature at which speed falls to zero, and Pfmax. I cannot use sub- or superscripts because the web editor I am using does not make provision for them. Brackets will be used to indicate the power to which e is raised.
Labeling the temperature at which performance is maximum may be something of a misnomer. When a organism or in vitro system is transferred from its customary temperature to and equilibrates to the maximum performance temperature, it is already in the range where degradation process(es) are already considerably counteracting the exponentially-related processes. Since arriving at this temperature can already have resulted in some (possibly irreversile) unfolding or degradation of critical components, this would not seem to very optimal for the organism.
They first express the temperature (in Celsius) and performance (Pf) as a dimensionless variables, tau, τ, equal to (T-Topt)/273, where 273 is 0 degree Celsius in Kelvin units.
, and the relative performance RPf, Pf divided by Pfmax. They reasoned that a good fit might be obtained by using the following equation:
RPf = e[τ/ε] X g(τ/ε) ,
where ε is parameter in temperature units to be determined in the fit. The function g would have to vary slower than exponential part at temperatures considerably below Topt, but its decrease must dominate at temperatures above Topt. If we express the equation in terms of x = (τ/ε), the equation is now:
RPf = e[x] X g(x), and the slope (first derivative) at any x is given by
d(RPf)/dτ = e[τ/ε] X (dg/dτ + g/ε) .
The slope τ = 0 must be 0. The exponential term will be 1 at τ = 0, so the right had term must itself be 0. The simplest equation with this property is:
g(x) = 1 - x .
Plugging in the definition of x, results in:
ε = τc - τopt .
This is a surprising result in that ε is determinative (in part) of both the exponential rising part of the curve below Topt and the rapidly decreasing part above Topt. Arnoldi discussed ε in relationship to an activation energy, Ea, through the Arrhenius equation for the exponentially-increasing rate of biological reaction with increasing temperatures, resulting in the equation:
ε = (k - 273)/Ea ,
where k is the Boltzmann constant. It might be strange to discuss a single activation energy when, as discussed above, any particular TPC curve may result from a variety of processes. Perhaps it could be understood as the result of many different processes with exponentially-increasing rates, with some working to enhance performance and others to hinder.
Arnoldi illustrated (their Figure 3) how TPCs from many different species and activities can be well-fitted with the UTPC equation, for growth rate in 131 proteobacteria specie, rate of photosynthesis in 43 plant species, maximum running speed in 51 lizard species, and population growth rate in 47 insect species.
Arnoldi also discusses the finding that the behavior of these complex systems can be expressed with just a few parameters relates to work on so-called sloppy models, that was initially used to model a variety of complex biological systems (Guttenkunst et al. 2007). This earlier worked found the even when the models had a large number of parameters, the range in which particular parameters could vary without the model no longer coming close to replicating the observed behavior varied over orders of magnitude. Only a relatively few parameters (or combinations of parameters) needed to be well characterized for the model to reproduce the observed behavior.
The nature of the UTPC equation dictates that any particular system in ectotherms can operate near the maximum rate over a relatively small range of temperatures. In the case of many enzymatic reactions, many ectotherms have more than one version of the enzyme (isozymes), each which operates near maximum over different temperatures ranges.
Precision
What does all the above discussion of ectotherms have to do with endotherms and homeotherms and the topics of interest to this site? Endotherms and homeotherms have the ability to control the level of their activities despite large changes in environmental temperature. This ability opens up numerous opportunities for evolutionary innovation. Consider that genes for a group of isozymes. In the long evolution from ectothermy to homeothermy, eventually only a single version would be necessary to accomplish the original function. The redundant genes would be free to change function in some way, perhaps to broaden the substrates the enzyme could handle, or to retain the same function but evolve more complicated patterns of expression.
In the latter case, not only the rate of function would be important, but ability to more precisely perform a function, such as delivering a differentiation signal at a precise time and place during embryonic development. In Exploration 2, I examine whether sloppy modeling can be used to study the effect of homeothermy in providing such precision.
The above discussion concerns understanding of average temperature behavior in a species. In Exploration 2, the focus will be on how mutational events might affect the precision by which each individual accomplishes crucial events and the resultant effects on individual's survival and reproduction.