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Fitting tolerance time versus temperature to build a thermal death time curveThe high coefficients of determination found in the D. melanogaster TDT curves (Fig. 3A) are not uncommon and the exponential relation has consistently been found to provide a good fit of tolerance time vs. temperature in ectotherms3,15,20,22,23,24. Tolerance time vs. temperature data are also well fitted to Arrhenius plots which are based on thermodynamic principles (see for example15,36) and the absence of breakpoints in such plots provides a strong indication (but not direct proof) that the cause of coma/heat failure under the different intensities of acute heat stress is related to the same physiological process regardless whether failure occurs after 10 min or 10 h2,3 (but see “Discussion” section below). Despite the superior theoretical basis of Arrhenius analysis, we proceed with simple linear regressions of log10-transformed tcoma (TDT curve) as this analysis likewise provides a high R2 and is mathematically more straightforward. The physiological cause(s) of ectotherm heat failure are poorly understood37,38 but we argue that they are founded in a common process where heat injury accumulates at a temperature-dependent rate until a species-specific critical dose is attained (area below the curve and above Tc in Fig. 2). Thus, the organism has a fixed amount (dose) of thermally induced stress that it can tolerate before evoking the chosen endpoint. The experienced temperature of the animals then dictates the rate of which this stress is acquired, and accordingly when the endpoint is reached (Fig. 2) It is this reasoning that leads to TDT curves and explains why heat stress can be additive and thus also determines the boundaries of TDT curve modelling.Injury is additive across different stressful assay temperaturesIf heat stress acquired at intense and moderate stress within the span of the TDT curve acts through the same physiological mechanisms or converges to result in the same form of injury, then it is expected that injury is additive at different heat stress intensities. This hypothesis was tested by exposing flies sequentially to two static temperatures (different injury accumulation rates) and observe whether coma occurred as predicted from the summed injury (Fig. 2C). The accumulated heat injury at the two temperatures was found to be additive regardless of the order of temperature exposure (Fig. 3B,C). This finding is consistent with a conceptually similar study using speckled trout which also found strong support for additivity of heat stress at different stressful temperatures13. The exact physiological mechanism of heat injury accumulation is interesting to understand in this perspective, but it is not critical as long as the relation between temperature and injury accumulation rate is known.If injury accumulation is additive irrespective of the order of the heat exposure, we can extend the model to fluctuating temperature conditions. We have previously done this by accurately predicting dynamic CTmax from TDT parameters obtained from static assays for 11 Drosophila species (Fig. 6A, see “Discussion” section below and15). Here we extend this to temperature fluctuations that cannot be described by a simple mathematical ramp function. Specifically, groups of flies were subjected to randomly fluctuating temperatures and the observed tcoma was then compared to tcoma predicted using integration of heat injury based on TDT parameters (Fig. 4). The injury accumulation (Fig. 4C) was calculated by introducing the fluctuating temperature profiles in the associated R-script and the observed and predicted tcoma was found to correlate well (R2  > 0.94) across the 13 groups tested for each sex. These results further support the idea that injury is additive across a range of fluctuating and stressful temperatures and hence that similar physiological perturbations are in play during moderate and intense heat stress. It is important to note that in these experiments, temperatures fluctuated between 34.5 and 42.5 °C and accordingly the flies were never exposed to benign temperatures that could allow repair or hardening (see below).Figure 6adapted from Fig. 4b in15. (B) TDT parameters based on dCTmax from three dynamic tests were used to predict tcoma in static assays. Each point represents an observed vs. predicted value of species- and temperature-specific log10(tcoma). (Inset) Species values of the thermal sensitivity parameter z parameterized from TDT curves based on static assays (x-axis) or dynamic assays (y-axis). The dashed line represents the line of unity in all three panels.Conversion of heat tolerance measures between static and dynamic assays in Drosophila. Data from43. (A) Heat tolerance (dCTmax, d for dynamic assays) plotted against predicted dCTmax derived from species-specific TDT curves created from multiple (9–17) static assays. Data are presented for three different ramping rates (0.05, 0.1 and 0.25 °C min-1). Note that this graph is Full size imageIn conclusion, empirical data (present study;6,13,14,22) support the application of TDT curves to assess heat injury accumulation under fluctuating temperature conditions both in the lab and field for vertebrate and invertebrate ectotherms. Potential applications could be assessment of injury during foraging in extreme and fluctuating environments (e.g. ants in the desert39 or lizards in exposed habitat40) or for other animals experiencing extreme conditions41,42. The associated R-scripts allow assessment of percent lethal damage under such conditions if the model is provided with TDT parameters and information of temperature fluctuations (but see “Discussion” section of model limitations below).Model application for comparison of static versus dynamic dataThere is little consensus on the optimal protocol to assess ectotherm thermal tolerance and many different types of static or dynamic tests have been used to assess heat tolerance. TDT curves represent a mathematical and theoretical approach to reconcile different estimates of tolerance as the derived parameters can subsequently be used to assess heat injury accumulation at different rates (temperatures) and durations13,15,16. Here we provide R-scripts that enable such reconciliation and to demonstrate the ability of the TDT curves to reconcile data from static vs. dynamic assays we used published measurements of heat tolerance for 11 Drosophila species using three dynamic and 9–17 static measurements for each species43. Introducing data from only static assays we derived TDT parameters and subsequently used these to predict dynamic CTmax that were compared to empirically observed CTmax for three ramp rates (Fig. 6A). In a similar analysis, TDT parameters were derived from the three dynamic (ramp) experiments to predict tcoma at different static temperatures which were compared to empirical measures from static assays (Fig. 6B). Both analyses found good correlation between the predicted and observed values regardless whether the TDT curve was parameterized from static or dynamic experiments (Fig. 6). However, predictions from TDT curves based on three dynamic assays were characterised by more variation, particularly when used to assess tolerance time at very short or long durations. Furthermore, D. melanogaster and D. virilis which had the poorest correlation between predicted and observed tcoma in Fig. 6B had values of z from the TDT curves based on dynamic input data that were considerably different from values of z derived from TDT curves based on static assays (Fig. 6B inset). In conclusion TDT curves (and the associated R-scripts) are useful for conversion between static and dynamic assessment of tolerance. The quality of model output depends on the quality and quantity of data used as model input, and in this example the poorer model was parameterized from only three dynamic assays while the stronger model was based on 9–17 static assays (see also “Discussion” section below).Model application for comparison of published dataThermal tolerance is important for defining the fundamental niche of animals1,2,4 and the current anthropogenic changes in climate has reinvigorated the interest in comparative physiology and ecology of thermal limits in ectotherms. Meta-analyses of ectotherm heat tolerance data have provided important physiological, ecological and evolutionary insights5,44,45,46, but such studies are often challenged with comparison of tolerance estimates obtained through very different methodologies.Species tolerance is likely influenced by acclimation, age, sex, diet, etc.47 and also by the endpoint used (onset of spasms, coma, death, etc.27). Nevertheless, we expected heat tolerance of a species to be somewhat constrained45, so here we tested the model by converting literature data for nine species to a single and species-specific estimate of tolerance, sCTmax (1 h), the temperature that causes heat failure in 1 h (Fig. 5). The overwhelming result of this analysis is that TDT parameters are useful to convert static and dynamic heat tolerance measures to a single metric, and accordingly, the TDT model and R-scripts presented here have promising applications for large-scale comparative meta-analyses of ectotherm heat tolerance where a single metric allows for qualified direct comparison of results from different publications and experimental backgrounds. While this is an intriguing and powerful application, we caution that careful consideration should be put into the limitations of this model (see “Discussion” section below).Practical considerations and pitfalls for model interpretationAs shown above it is possible to convert and reconcile different types of heat tolerance measures using TDT parameters and these parameters can also be used to model heat stress under fluctuating field conditions. Modelling and discussion of TDT predictions beyond the boundaries of the input data has recently gained traction (see examples in48,49) but we caution that the potent exponential nature of the TDT curve requires careful consideration as it is both easy and enticing to misuse this model.Input dataThe quality of the model output is dictated by the input used for parameterization. Accordingly, we recommend TDT parameterization using several ( > 5) static experiments that should cover the time and temperature interval of interest, e.g. temperatures resulting in tcoma spanning 10 min to 10 h, thus covering both moderate and intense heat exposure. Such an experimental series can verify TDT curve linearity and allows modelling of temperature impacts across a broad range of temperatures and stress durations13,15,22. It is tempting to use only brief static experiments (high temperatures) for TDT parameterization, but in such cases, we recommend that the resulting TDT curve is only used to describe heat injury accumulation under severe heat stress intensities. Thus, the thermal sensitivity factor z represents a very powerful exponential factor (equivalent to Q10 = 100 to 100,000;15) which should ideally be parametrized over a broad temperature range (see below). We also include a script that allows TDT parametrization from multiple ramping experiments and again we recommend a broad span of ramping rates to cover the time/temperature interval of interest. A drawback of ramping experiments is the relatively large proportion of time spent at benign temperatures where there is no appreciable heat injury accumulation. Thus, dynamic experiments can conveniently use starting temperatures that are close to the temperature where injury accumulation rate surpasses injury repair rate (see “Discussion” section of “true” Tc below, in Supplemental Information and19 for other considerations regarding ramp experiments).A final methodological consideration relates to body-temperature in brief static experiments where the animal will spend a considerable proportion of the experiment in a state of thermal disequilibrium (i.e. it takes time to heat the animal). To avoid this, we recommend direct measurement of body temperature (large animals) or container temperature (small animals), and advise against excessive reliance on data from test temperatures that results in coma in less than 10 min.ExtrapolationMost studies of ectotherm heat tolerance include only a single measure of heat tolerance which is inadequate to parameterize a TDT curve. However, a TDT curve can still be generated from a single measure of tolerance (static or dynamic) if a value of z is assumed (see Supplemental Information). As z differs within species and between phylogenetic groups (Table S115,20), choosing the appropriate value may be difficult and discrepancies between the ‘true’ and assumed z represent a problem that should be approached with care. In Fig. 7A we illustrate this point in a constructed example where a single heat tolerance measurement is sampled from a ‘true’ TDT curve (full line; tcoma = 40 min at 37 °C). Along with this ‘true’ TDT curve we depict the consequences for model predictions if the assumed value of z is misestimated by ± 50%. Extrapolation from the original data point is necessary if an estimate of the temperature that causes coma after 1 h is desired, however due to limited extrapolation (from 40 to 60 min), estimation of sCTmax (1 h) values based on the ‘true’ and z ± 50% are not very different ( 6 h) between heat exposure disrupted additivity, suggesting that injury is repaired at benign temperature50. Injury repair rate is largely understudied but repair rate is generally increasing with temperature51,52,53. It is therefore an intriguing and promising idea to include a temperature-dependent repair function in more advanced modelling of heat injury. Until such repair processes are introduced in the model, we recommend that additivity of heat injury is evaluated critically if it involves periods at temperatures both above and below Tc (i.e. over consecutive days, see also13). An alternative, but not mutually excluding, explanation of increased heat resilience in split-dose experiments relates to the contribution of heat hardening as it is likely that the first heat exposure in a series can induce hardening responses that increase resilience (and thus change the TDT parameters) when a second heat exposure occurs. Such issues of repeated thermal stress have been discussed previously54 but for the purpose of the present study the main conclusion is that simple TDT curve modelling is not applicable to fluctuations bracketing Tc unless this is empirically validated. Future studies could address this issue as inclusion of repair functions would add further promise to the use of TDT curves in modelling of the impacts of temperature fluctuations. More