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increasingly outperform simpler alternatives as the number of experimental temperatures increases, we next identi fied the ten models with the highest median AICc weights across datasets with (a) 4 (low resolution), (b) 5– 7 (moderate resolution), or (c) 8 or more dis- tinct temperatures (high resolution). Even for datasets with at least 8 distinct temperatures, the ten models with the highest average per- formance had either three or four parameters (Fig. 4c). Moreover, datasets of low, moderate, and high resolution shared five three- parameter models among their top ten. It is worth pointing out that the top ten models in the three resolution groups had median AICc weights that were not higher than 0.05 and interquartile ranges that often included values that were effectively zero (Fig.4). Certain phenomenological models consistently outperform mechanistic models, regardless of trait type . Our third and final question was to determine if mechanistic models (whose parameters describe the activity of underlying rate-limiting enzymes) exhibit bet- ter performance than phenomenological models for physiological traits. To this end, we classified each dataset as originating from (a) a physiological, (b) an emergent, or (c) an (organism-organism) inter- action trait. Physiological traits directly depend on a small set of bio- chemical pathways and can be measured asfluxes or rates (e.g., gross photosynthesis rate, aerobic respiration rate). Emergent traits are the ultimate outcome of a wide diversity of cellular biochemical pathways and processes, and are not governed by species interactions. For e x a m p l e ,t h ep o p u l a t i o ng r o w t hr a t eo fas e x u a l l yr e p r o d u c i n gs p e c i e s depends on development rate, fecundity rate, and mortality rate, each of which is, in turn, dependent on a distinct set of biochemical pro- cesses that may also vary by life stage. Lastly, traits that explicitly describe species interactions were classified as interaction traits (e.g., resource consumption rate, filtration rate). We then extracted and compared the ten best-performing models across these three groups. For physiological traits, none of the ten best- fitting models was mechanistic (Fig. 5a), whereas physiological, emergent, and interac- tion traits shared seven out of ten best-fitting models, albeit with dif- ferent rankings (Fig. 5a– c, Supplementary Fig. 3 in Supplementary Note 3). We reached similar conclusions after examining the ten best- fitting models for three vastly different traits with large numbers of thermal performance datasets across our data compilation: respiration rate (a physiological trait), population growth rate (an emergent trait), and resource consumption rate (an interaction trait). The three traits had six models in common among their ten best-fitting ones (Fig.5d– f, Supplementary Fig. 4 in Supplemen tary Note 3), whereas the inter- quartile ranges of AICc weights ranged from≈ 0 to ~ 0.13. These results show that model performance is not strongly trait-dependent and that even when measurements come from traits that are governed by specific biochemical pathways, mechanistic models do not tend to outperform simpler phenomenological alternatives. To further explore this unexpected finding, we searched the lit- erature for datasets of the performance of individual enzymes at multiple temperatures. Our search yielded 60 such datasets, 56 of which passed our filtering criteria (see Fig. 2d and the Methods sec- tion). Fitting all 83 models to the enzyme datasets 54 revealed a strik- ingly similar result (Supplementary Fig. 5 in Supplementary Note 4) to that obtained for trait datasets (Fig.3). More specifically, we detected remarkable variation in model performance across datasets, with mechanistic models not consistently outperforming popular non- mechanistic alternatives such as the Briere I model. This may help explain why the performance of the mechanistic model subset does not vary systematically between physiological, emergent, and inter- action traits (Fig. 5). Predicting the optimal TPC model Finally, we employed machine learning to uncover any rules for selecting the optimal TPC model(s) for a given thermal performance dataset. For this, we compiled 29 variables that describe the data in our study in four main aspects: (i) the type of trait measured, (ii) taxonomic information about the organism (kingdom and phylum), (iii) the shape of the TPC (e.g., its breadth, the degree of symmetry before and after the peak), and iv) information related to sampling resolution (e.g., the number of distinct temperatures before the peak of the curve). The full list of variables, along with their descriptions, is available in Supple- mentary Table 2 in Supplementary Note 5. We then randomly split the trait data into training and testing subsets (80% and 20% of the data, respectively) andfitted multi-output conditional inference regression trees 59 with all possible combinations of the four aforementioned groups of predictors using the R package partykit60 ( v .1 . 2 - 1 7 ) .T h i sm e t h o da i m sa tp r e d i c t i n gm u l t i p l er e s p o n s ev a r i a b l e s simultaneously (here, the AICc weights of all models) based on binary splits using the values of predictor variables, selected through non- parametric tests. It ultimately yields data subsets (leaf nodes) that differ considerably in the composition of AICc weights, with the maximum number of such subsets being equal to two raised to the depth of the tree. For practical purposes, we set the maximum tree Taylor-Sexton Analytis-Kontodimas Tomlinson-Phillips Eubank Ashrafi II Gaussian Atkin Mitchell-Angilletta second-order polynomial Janisch I Janisch I 0 0.01 0.04 0.1 AICc weight TPC model 4 distinct temperatures (685 datasets) a Mitchell-Angilletta simplified Cardinal Temperature with inflection skew-normal Thomas I β-type Gaussian Analytis-Kontodimas Taylor-Sexton Atkin 0 0.01 0.04 0.1 AICc weight ≥ 8 distinct temperatures (1,036 datasets) c Eubank second-order polynomial simplified Briere I simplified β-type Taylor-Sexton Gaussian Atkin Analytis-Kontodimas Mitchell-Angilletta 0 0.01 0.04 0.1 AICc weight b 5-7 distinct temperatures (1,018 datasets) Parameter count 3 4 Weibull Fig. 4 | Comparison of model performance across datasets that differ in their sampling resolution.The ten best-fitting models across datasets of low (a;4d i s - tinct temperatures), moderate (b;5 – 7 distinct temperatures), or high resolution (c; 8 or more distinct temperatures). Circles represent the median AICc weight, whereas horizontal bars stand for the interquartile range. Three- and four- parameter models are shown in brown and green, respectively. Note that values along the horizontal axes do not increase linearly. Source data are provided as a Source Data file. Article https://doi.org/10.1038/s41467-024-53046-2 Nature Communications| (2024) 15:8855 5 depth to four, allowing the algorithm to generate predictive rules for up to 16 different groups of thermal performance datasets. We also grouped rare levels of categorical predictors (those with fewer than 10 datasets) as ‘other’ to facilitate training. We determined the best- performing tree based on theR 2 value achieved on the training set and also evaluated its performance across the separate testing subset. Obtaining a tree that could accurately predict the AICc weights for a given dataset proved impossible as the best- fitting tree (Supple- mentary Fig. 6 in Supplementary Note 6) hadR2 values of only 0.12 and 0.1 on the training and testing data subsets, respectively (Fig. 6a, b). More worryingly, the maximum predicted AICc weight by the tree was not greater than 0.22 (Fig.6b). When we examined how accurately the tree predicted the AICc weights separately for each model, we found that the tree achieved its highestR 2 values (below 0.19) for models with am e d i a nA I C cw e i g h to f≈ 0( F i g .6c). In other words, the tree was slightly better able to predict the models that generally do not repre- sent experimental data well, rather than the models that often provide good fits. Increasing the maximum tree depth to, e.g., 10 (which would raise the number of leaf nodes to 1024) or developing a more advanced predictive method based on deep learning could possibly yield more accurate predictions, but at the cost of limited interpretability. Recommendations All our results converge on the same conclusion, i.e., that there are no simple and well-de fined scenarios in which particular TPC models would be consistently favoured over others. This highlights the importance of comparing the performance of multiple alternative models across a given dataset, rather than selecting a single model in an arbitrary manner. An anticipated objection to this recommendation might be that researchers often choose a particular model because it includes a parameter of interest for their study. We recommend that model selection still be considered because multiple models may share t h es a m ep a r a m e t e r .E v e nw h e nam o d e ld o e sn o ti n c l u d et h ep a r a - meter of interest, it may be possible to obtain its value based on other model parameters or from the fitted curve itself. Obtaining estimates of the same parameter using different models additionally allows AICc- weighted model-averaged estimates to be calculated. In cases where none of the fitted models stands out, such estimates should be much more reliable than estimates obtained from a single model. Careful attention should be spent, however, on ensuring that the parameter estimates to be averaged are fully equivalent across models, otherwise the results obtained from averaging will be meaningless. Besides parameter estimates, model averaging can also be applied to the entire TPC 61, enabling a more objective quantification of its shape. We should clarify that we do not expect (or recommend) that all possible models be fitted to each new thermal performance dataset. Instead, one could examine the dendrogram that we estimated in the present study (Fig.3) to choose a subset of sufficiently distinct models (i.e., those branching far from each other) for fitting to a dataset. By using our dendrogram as a guide, researchers no longer need to compare the equations of different models manually, which can often be quite challenging in our experience. For example, our dendrogram shows that the Johnson-Lewin model 29 and a common modi fication thereof (the simpli fied Johnson-Lewin model; see Supplementary Note 7) yield effectively identical fits, consistent with the conclusions of Yin62. It is also worth pointing out that the fits of the mechanistic Johnson-Lewin model (and its variants) are typically highly similar to Analytis-Kontodimas Taylor-Sexton Eubank Mitchell-Angilletta Ashrafi I Gaussian Atkin second-order polynomial Ashrafi II Janisch I Janisch I Janisch I Janisch I Janisch I Janisch I 0 0.015 0.06 0.13 TPC model physiological traits (477 datasets)a Analytis-Kontodimas Taylor-Sexton Eubank Mitchell-Angilletta Gaussian Ashrafi I Atkin second-order polynomial Ashrafi II 0 0.015 0.06 0.13 AICc weight TPC model respiration rate (351 datasets)d Warren-Dreyer Tomlinson-Phillips Eubank simplified β-type Analytis-Kontodimas Taylor-Sexton Atkin Gaussian Mitchell-Angilletta 0 0.015 0.06 0.13 interaction traits (251 datasets)c Tomlinson-Phillips Ashrafi II second-order polynomial Analytis-Kontodimas Atkin simplified β-type Taylor-Sexton Gaussian Mitchell-Angilletta 0 0.015 0.06 0.13 AICc weight f resource consumption rate (112 datasets) Eubank second-order polynomial simplified Briere I simplified β-type Gaussian Analytis-Kontodimas Taylor-Sexton Atkin Mitchell-Angilletta 0 0.015 0.06 0.13 emergent traits (2,011 datasets)b Thomas I Weibull simplified Briere I Gaussian simplified β-type Mitchell-Angilletta Atkin Analytis-Kontodimas Taylor-Sexton 0 0.015 0.06 0.13 AICc weight e population growth rate (1,174 datasets) Fig. 5 | Comparison of model performance across trait groups. The ten best- fitting models across thermal performance datasets from the three main trait groups (a– c) and for one representative trait per group (d– f). Circles represent the median AICc weight, whereas horizontal bars stand for the interquartile range. Note that values along the horizontal axes do not increase linearly. Source data are provided as a Source Data file. Article https://doi.org/10.1038/s41467-024-53046-2 Nature Communications| (2024) 15:8855 6 the fits of the phenomenological asymmetric Janisch model (Janisch II; Fig. 3), one of the two earliest developed TPC models. Perhaps the most unexpected result of our study was that phe- nomenological models generally outperform mechanistic alternatives, even across thermal performance datasets of individual enzymes. These two types of models serve two different purposes and hence occupy different niches in terms of their utility. On the one hand, mechanistic TPC models were developed to estimate key thermo- dynamic and biochemical kinetic parameters (e.g., enzyme activation energy from the rising part of the TPC; E 63). Estimates of these para- meters may be necessary to compare the temperature dependence of processes within and across scales of biological organisation 9,15,38,42,46,64– 66 (e.g., whole-organism metabolic rate or community-levelfluxes). Conversely, phenomenological models seek to deliver the most statistically reliable characterisation of the shape of as p e c ific TPC, without considering the underlying mechanisms. Because of this, phenomenological models may be better suited for quantifying parameters related to the shape of the TPC, such as critical thermal limits ( T min and Tmax). Therefore, we would argue that mechanistic models should not be abandoned due to their (relatively) lower performance, but that their application (or exclusion) should be justified by the aims of a given research study. Moreover, systematic and strategic model comparison is necessary for theoretical advancement because understanding why certain models perform better than others can accelerate the development of more general theoretical frameworks or the refinement of those existing ones. No matter which models are selected for fitting to a speci fic dataset, it is essential that the data allow sufficient statistical power to yield reliable estimates. In particular, in the present study, to avoid overfitting, we only fitted a model to a given dataset if its parameter count was lower than the number of distinct temperatures. In addition to this, researchers should ensure that the TPC is sampled sufficiently in temperature ranges relevant to their parameter(s) of interest. For example, if the goal is to estimate the thermal optimum, there should be numerous data points before and after the peak of the TPC. If this is not the case, the parameter estimates may be imprecise and biased to various degrees 43. The number of experimental replicates per tem- perature is another matter that merits consideration, especially for traits for which precise measurements are dif ficult to obtain (e.g., behavioural traits). The above points do not constitute an exhaustive list but represent some common guidelines that should be relevant to most thermal biology studies. In conclusion, our study shows that for describing the shape of thermal performance curves, there is no model“to rule them all”.T h i s 0.00 0.25 0.50 0.75 1.00 Trait Taxonom y TPC shape Sampling resolution Trait

  • Taxono my Trait
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  • Sampling resolution Taxonomy
  • TPC shape Taxono my
  • Sampling resoluti on TPC shape
  • Sampling resolution Trait
  • Taxono my
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  • Taxono my
  • Samplin g resolution Trait
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  • Sam pling resolution Taxono my
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  • Taxono my
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  • Sampli ng resolution R2 (training subset) a R2 = 0.10 N = 548 x 83 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Predicted AICc weight (testing subset) Observed AICc weight (testing subset) b 0.00 0.05 0.10 0.15 0.20 0 0.0005 0.002 0.005 0.009 Median observed AICc weight per TPC model (testing subset) R2 per TPC model (testing subset) c Fig. 6 | Conditional inference trees with a maximum depth of four cannot accurately predict the AICc weights of TPC models.Panel (a) shows theR2 values reached by alternative candidate trees (with different sets of predictor variables) across the subset of the data on which they were trained. The tree with the highest R 2 is highlighted in green. Note that R2 values vary very little across trees (ranging from 0.0946 to 0.1227), with no tree exhibiting sufficiently high performance. Panel (b) shows the predictions of the best tree (see Supplementary Fig. 6 in Supple- mentary Note 6) versus the observed AICc weights across the testing subset (AICc weights of 83 models for 548 thermal performance datasets). The dashed line is the one-to-one line. Panel (c) shows the model-specific R 2 values achieved by the tree across the testing subset against the median observed AICc weight of each model. Source data are provided as a Source Data file. Article https://doi.org/10.1038/s41467-024-53046-2 Nature Communications| (2024) 15:8855 7 makes it imperative that researchers adopt multi-model selection approaches to identify the most app ropriate one(s) for a given class (e.g., trait type) of data and research question. Such a strategy is likely to yield multiple benefits, such as bridging thermal biology“silos” (e.g., plant physiologists versus environmental microbiologists), improving the rigour and reproducibility of the study, and advancing theory. Ultimately, this will accelerate progress towards achieving a common goal across biologicalfields: identifying general mathematical models rooted in mechanistic, biophysical principles that can predict the effects of environmental temperature on complex biological systems. Methods Sources of thermal performance datasets Our data compilation includes thermal performance datasets of (i) population growth rate of bacteria, archaea, and phytoplankton 66,67, (ii) photosynthesis and respiration rate of algae, aquatic, and terrestrial plants65, (iii) metabolic rate of zooplankton 64,68,69 and other hetero- trophic ectotherms 70, (iv) development rate of damsel flies71 and helminths72, (v) overall dynamic body acceleration of elasmobranchs73, and (vi) numerous other biological traits included in the Biotraits database

Compilation of a TPC model catalogue To compile our catalogue of TPC models, we queried Google Scholar using terms such as “thermal performance curve model ”, “thermal performance curve equation”, “thermal reaction norm model”, “tem- perature response curve model”, etc. From the resulting hits, we pro- gressively added models that were not mathematically equivalent to any model that was already on the list. For example, the equation of the Johnson-Lewin model 29,46 is equivalent to that of a particular 4-parameter variant of the Sharpe-Schoolfield model44,63. Fitting TPC models to thermal performance datasets Given that many of the models in our catalogue could not support non- positive trait measurements (e.g., negative population growth rate), we first removed such values from the datasets. To reduce the possi- bility of overfitting, we only fitted a model to a thermal performance dataset if the parameters of the model were fewer than the number of distinct experimental temperatures. Forfitting the models separately to each dataset, we used the R package nls.multstart 61,75 (v. 1.2.0). To find the optimal combination of parameter values that maximises the fit of a given TPC model to a given dataset, nls.multstart repeatedly fits the model using the Levenberg-Marquardt algorithm from different starting values. We specified seemingly realistic starting values for each model in various ways, depending on its parameters. For example, for models that include the T min and Tmax parameters (see Fig. 1), we set their starting values to the minimum and maximum experimental temperatures in the dataset, respectively. In some cases, we speci fied starting values by performing linear regression on a subset of the dataset (e.g., to estimate the slope of the rising part of the TPC) or by using parameter estimates from the literature. We then fitted each model to each thermal performance dataset 1000 times. In every iteration, we used a different starting value per parameter, sampling from a uniform distribution with bounds equal to 0.5 and 1.5 times our initial starting value. To further facilitate the overall process, we fitted models to trait measurements on the natural logarithm scale, increased the stringency of the convergence criteria by modifying the Levenberg-Marquardt parametersftol, ptol, maxiter,a n dmaxfev, set appropriate bounds for each TPC parameter where possible, and applied other checks (e.g., we forced T pk to be >Tmin and <Tmax). The source code for fitting the models is available from GitHub and also archived on Zenodo76. We calculated the AICc values of all modelfits with anR2 of at least 0.5. In a limited number of thermal performance datasets, the AICc value of at least one model was equal to −∞, which is indicative of severe overfitting. When this was the case, we progressively removed the most parameter-rich models and examined the AICc values for that dataset again. If we still obtained values of −∞ even when only three- parameter models were included, we excluded the corresponding dataset from our analyses. After this process, we ended up with 2739 datasets out of the initial 3598. Next, we constructed a table of AICc weights per model and per dataset. If a model could not converge on a specific dataset or was excluded for any of the aforementioned rea- sons, we assigned an AICc weight of 0. Inferring a dendrogram of TPC models To systematically compare the models in our catalogue, we first generated the curve for each model fit with an R2 value greater than or equal to 0.5, across the 2739 thermal performance datasets. This was done from the minimum experimental temperature of each dataset to the maximum temperature, with a step size of 0.1 °C. We then constructed a matrix of pairwise Euclidean distances among curves (and, hence, models) per dataset, based on all aforemen- tioned temperatures. Given that datasets varied considerably in the magnitude of their trait values, we performed maximum normalisation 77 (also known as “l-infinity normalisation ”) to ensure that pairwise distances were comparable across datasets. More specifically, we corrected each dataset-speci fic distance matrix by dividing its values by the maximum height of all corresponding curves. We summarised the resulting distance matrices into a single (average) matrix by taking the median of each cell across all matrices where possible. Finally, based on the average matrix, we calculated a dendrogram of TPC models using hierarchical clus- tering with average linkage, i.e., with the UPGMA 78 algorithm, which has also been used for distance-based phylogenetic tree inference. Models that were placed close to each other in the dendrogram would generally produce very similar fits. Reporting summary Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article. Data availability The thermal performance datasets included in this study are available from Figshare 54: https://doi.org/10.6084/m9.figshare.24106161.v3. These datasets were collected from numerous previous studies64,66– 73 or from the Biotraits database 74. Source data for all figures are pro- vided in this paper as a Source Data File. Code availability T h es o u r c ec o d ef o rt h em a i na n a l y s e so ft h i ss t u d y76 is available at https://github.com/dgkontopoulos/Kontopoulos_et_al_83_TPC_ models_2024 and archived at https://doi.org/10.5281/zenodo. 12608191. References

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  66. Gu, Z. Complex heatmap visualization. iMeta 1, e43 (2022). Acknowledgements A.S. was supported by the ANR project EcoTeBo (ANR-19-CE02- 0001-01) from the French Nationa lR e s e a r c hA g e n c y( A N R ) .S . P .w a s funded by the Leverhulme Research Fellowship RF-2020-653\2 and UK national NERC grants NE/M020843/1 and NE/S000348/1. We are grateful to Alejandro Isla, Alex ander Kazhdan, Perng Lee, Olivia Morris, Richard Sheppard, and Joss Thomas for providing some of the thermal performance dat a s e t su s e di nt h i ss t u d y . Author contributions Conceptualisation: D.G.K., A.S., A.I.D., and S.P. Data curation: D.G.K. Formal analysis: D.G.K. Methodology: D.G.K., A.S., A.I.D., and S.P. Project administration: D.G.K. Resources: A.S., M.D., N.G., A.I.D., and S.P. Soft- ware: D.G.K. Visualisation: D.G.K. Writing– original draft: D.G.K. Writing– review & editing: all authors. Competing interests The authors declare no competing interests. Additional information Supplementary informationThe online version contains supplementary material available at https://doi.org/10.1038/s41467-024-53046-2. Correspondenceand requests for materials should be addressed to Dimitrios - Georgios Kontopoulos. Peer review informationNature Communicationsthanks Martina Doblin, Zachary Stahlschmidt and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is avail- able. Reprints and permissions informationis available at http://www.nature.com/reprints Publisher’s note Springer Nature remains neutral with regard to jur- isdictional claims in published maps and institutional affiliations. Article https://doi.org/10.1038/s41467-024-53046-2 Nature Communications| (2024) 15:8855 10 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visithttp://creativecommons.org/ licenses/by/4.0/. © The Author(s) 2024 Article https://doi.org/10.1038/s41467-024-53046-2 Nature Communications| (2024) 15:8855 11
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