# Source paper archive - Source title: Heat-induced increases in xylem vulnerability: Experimental evidence and possible mechanisms - Author(s): Link, R. M., Paligi, S. S., Abeysundara, N., Ammer, C., Dharmawardhane, J. D. K., Pierick, K., Razzak, A., Schuldt, B., Tezeh, F. R., Wighton, W. R., Johnson, K. M., Matos, I. S. - Source landing URL: https://www.biorxiv.org/content/10.64898/2026.11.05.764619v1 - PDF URL: https://www.biorxiv.org/content/10.64898/2026.11.05.764619v1.full.pdf - PDF SHA-256: bbf34bc8671f990bec74501e06b39dd9819afe272979588110d99ae58df2660e - Converter: pymupdf4llm 0.0.27 - Limits: Automated PDF text extraction; reading order, equations, tables and formatting may be incomplete or incorrect. Images are omitted, and no OCR was performed. Markup and links are neutralized for safe downloading. Consult the original PDF for the authoritative version. --- [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) **Heat-induced increases in xylem vulnerability: Experimental evidence and possible** **mechanisms** Link, Roman Mathias [1,§,*] ; Paligi, Sharath Shyamappa [2,3,§] ; Abeysundara, Namalu [4] ; Ammer, Christian [3,5] ; Dharmawardhane, Janithya Dishani Kodituwakku [4] ; Pierick, Kerstin [3] ; Razzak, Abdul [6,7] ; Schuldt, Bernhard [1] ; Tezeh, Fon Robinson [1] ; Wighton, William Robert [4] ; Johnson, Kate M [8,$] ; Matos, Ilaine Silveira [4,$ ] 1 Chair of Forest Botany, Institute of Forest Botany and Zoology, TUD Dresden University of Technology, 01737 Tharandt, Germany 2 Plant Ecology and Ecosystems Research, Albrecht von Haller Institute for Plant Sciences, University of Goettingen, 37073 Göttingen, Germany 3 Department of Silviculture and Forest Ecology of the Temperate Zones, Faculty of Forest Sciences and Forest Ecology, University of Göttingen, 37077 Göttingen, Germany 4 School of Biological Sciences, Adelaide University, Adelaide, South Australia, Australia 5 Centre for Biodiversity and Sustainable Land Use (CBL), University of Göttingen, 37075 Göttingen, Germany 6 Department of Plant Ecophysiology, Institute of Forest Ecology, Slovak Academy of Sciences, 960 01 Zvolen, Slovakia 7 Faculty of Forestry, Technical University in Zvolen, TG Masaryka 24, 96001, Zvolen, Slovakia 8 Centre for Ecological Research and Forestry Applications (CREAF), Barcelona, Spain _Correspondence_ : roman.link@plant-ecology.de § _Shared first authorship:_ Roman Mathias Link & Sharath Shyamappa Paligi $ _Shared last authorship:_ Kate M Johnson & Ilaine Silveira Matos **Running title:** Heat-induced increases in xylem vulnerability **Word count:** Abstract 197 Main body total 7333 Introduction 1473 Description 1646 Material and methods 1686 Results 616 Discussion 1912 Acknowledgements 232 Tables 1 Figures 6 Boxes 1 Supporting information 3 Figures, 4 Tables, Notes S1 **Keywords:** bubble nucleation, Gibbs free energy, gas solubility, plant hydraulics, vapour pressure deficit, dehydration timing, Cavicam [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) # 1 Abstract 2 - Hotter droughts driven by global warming are increasing tree mortality worldwide. Here, 3 we propose a theoretical framework that explains how dissolved gas affects the stability of 4 bubbles in xylem sap, and shows that in the presence of gas, sudden increases in 5 temperature can accelerate embolism formation. We then provide experimental evidence 6 that heat exposure can change xylem vulnerability to drought. 7 - Using data from seven Australian and European tree species, we assessed how elevated 8 temperature (+15 and +20°C above ambient) affects both xylem embolism resistance, 9 measured with the Optical Vulnerability Technique, and VPD-corrected drying times. 10 - We found evidence for reduced embolism resistance and accelerated dehydration in heat 11 exposed plants. On average, trees reached the water potential at 50% loss of xylem function 12 at 0.73 MPa less negative water potentials and 7.8 kPa h earlier, though responses were 13 highly species-specific, and credibly different from zero only in the Australian species. 14 - Our results are consistent with the proposed framework and suggest that embolism 15 resistance may not be a static trait, which has important repercussions for studies of plant 16 vulnerability to drought. Further studies are necessary to assess the potential effects of heat 17 induced reductions in embolism resistance on tree mortality during heatwaves. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) # 18 Introduction 19 Over the last decades, climate-change driven drought events have become a major driver of tree 20 mortality worldwide (Allen et al., 2015; Hammond et al., 2022). Particularly extreme tree 21 mortality events have been linked to global change-type and thus hotter drought conditions that 22 combine reduced soil water availability with extreme atmospheric temperatures and dryness 23 during heatwaves in Europe (Schuldt et al., 2020), Australia (Mitchell et al. 2014; Matusick et 24 al., 2018; Marchin et al., 2022), and across the world (Allen et al., 2015; Breshears et al., 2013). 25 Under such conditions, a disruption of the plant water transport system due to the formation of 26 air emboli has been identified as an important cause of tree death (Arend et al., 2021; McDowell 27 et al., 2022). 28 How exactly xylem embolism develops under drought – and particularly under combined 29 drought and heat stress – is complex and subject to debate (Lens et al., 2022). The water 30 transport from roots to leaves via the cohesion-tension mechanism (Askenasy, 1895; Dixon & 31 Joly, 1895) requires the liquid in the xylem to be under – often substantial – negative pressures. 32 Under these conditions, water is in a metastable state, and a spontaneous phase transition into 33 the gas phase is theoretically possible. However, the water potentials required to induce 34 homogeneous cavitation, i.e., the spontaneous nucleation of water vapour bubbles under 35 sufficiently negative water potentials, are orders of magnitude more negative than the water 36 potentials usually observed in the xylem (Herbert & Caupin, 2005; Stroock et al., 2014; Kanduč 37 et al., 2020). Instead, it is widely accepted that the major driver of embolism formation is the 38 spread of gas from already embolized to adjacent conduits via pit membranes, although the 39 process driving this embolism propagation is debated (Jansen et al., 2026). The classical 40 explanation for embolism propagation is a process called air seeding, which involves the 41 pressure-driven bulk flow of gas bubbles through pit membrane pores and their subsequent 42 expansion (Lewis 1988; Tyree & Sperry, 1988). Explanations for the initial source of air for air [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 43 seeding include mechanical damage to the plant (Johnson et al., 2022), and air in cells connected 44 to the xylem, such as imperforate tracheary elements, which may embolise earlier than the 45 xylem conduits (Johnson et al., 2023). As the air-seeding pressure of a pit membrane pore is a 46 function of its diameter, the embolism risk of a conduit would be determined by the size of its 47 largest membrane pore (Wheeler et al., 2005; Hacke et al., 2006; Christman et al., 2009). 48 However, newer evidence on the ultrastructure of pit membranes indicates that large enough 49 pit membrane pores are unlikely to be as common as required for this mode of embolism 50 propagation (Kaack et al., 2021). An alternative explanation for embolism propagation is that 51 air instead enters into adjacent conduits in the form of stable nanobubbles that snap off in the 52 pore constrictions of pit membranes (Schenk et al., 2015). These nanobubbles are then coated 53 by surfactants (Schenk et al., 2017), which stabilize them against expansion due to the resulting 54 dynamic surface tension (Yang et al., 2021; Ingram et al., 2021). Embolism formation happens 55 at a later stage when nanobubbles lose their stability and expand, and is thus temporally and 56 spatially separated from bubble formation (Jansen et al., 2026). Embolism spread may further 57 be aided by gas diffusion from adjacent conduits (Guan et al.; 2021; Pereira et al., 2023). 58 Cavitation events in the true sense of the word (i.e., spontaneous bubble expansion rather than 59 the growth of existing gas-filled cavities) are likely rare. Mechanistic modelling suggests that 60 they are most likely to happen in lipid bilayers present in the sap, and unlikely to be 61 quantitatively relevant at water potentials less negative than −10 MPa (Kanduč et al., 2021). 62 It has long been known that xylem sap can contain high concentrations of dissolved gas (see 63 e.g. Scholander et al., 1955). Plants take up water from the soil, which is a complex porous 64 interface where – unless fully saturated – water is always in direct contact with air over a large 65 surface area. It can therefore be assumed that soil water is more or less gas-saturated (Marion 66 et al., 2026). Inside the xylem, dissolved gas is in equilibrium with the gas phase in air-filled 67 spaces, e.g. in fibres, embolised conduits and intercellular spaces in the mesophyll. As gas 68 solubility is extremely temperature-dependent, with solubility decreasing with increasing [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 69 temperature (Box 1), conditions of supersaturation have long been assumed to happen regularly 70 driven by diurnal temperature cycles (Hammel et al., 1967; Sperry & Sullivan, 1992; Cobb et 71 al., 2007), as demonstrated experimentally by Schenk et al. (2016). Nonetheless, in a study of 72 the molecular dynamics and kinetics of cavitation processes in the xylem, Kanduč et al. (2020) 73 concluded that dissolved gases at low concentrations do not significantly modify the time to 74 cavitation events compared to pure water. However, recent analyses of the thermodynamics of 75 bubble nucleation on preexisting nuclei indicate that gas content can in fact significantly reduce 76 the energy barrier for bubble nucleation compared to bubbles consisting of pure water vapour, 77 and completely eliminate it if a bubble is supersaturated with gas compared to the equilibrium 78 state (Alamé & Mahesh, 2024). Spontaneous bubble nucleation in the xylem from gas-saturated 79 nuclei might thus require less extreme negative pressures than previously expected. 80 Consequently, the presence of dissolved gas in the xylem may have a strong impact on 81 embolism formation. 82 The solubility of the four most common gases in water declines by between −52.1 and −72.8% 83 over the biologically relevant temperature range from 0 to 50 °C (Fig. 1a, Box 1). Over the 84 same range, surface tension and viscosity, two other highly relevant properties for bubble 85 nucleation and hydraulic processes in the xylem, decline by −10.2 and −40.3%, respectively 86 (Box 1, Fig 1b, 1c). The temperature rise on extremely hot days may push xylem sap into 87 conditions of strong gas supersaturation, and potentially induce bubble nucleation at less 88 negative water potentials compared to non-heated conditions. The same process can also result 89 in embolism formation in blood vessels in the human body even in the absence of negative 90 pressures, e.g. when hypothermic patients are rewarmed too rapidly (Geissler et al, 1997; 91 Mardanpour, 2026), when hyperthermic intraperitoneal chemotherapy is administered 92 (Vukadinovic et al., 2015; Mardanpour, 2026), or when blood transfusions are saturated at low 93 temperatures and rewarmed rapidly prior to infusion (Mashimo, 1980). If a similar temperature 94 driven outgassing happened in plants, it would be expected to push vulnerability curves (i.e. the [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 95 relationship between cumulative embolism and water potentials) towards less negative water 96 potentials. 97 To our knowledge, so far, there are few studies on temperature-dependent changes in 98 vulnerability curves. An increased vulnerability to embolism at higher temperatures was 99 demonstrated experimentally by Cochard et al. (2007). Further evidence comes from studies of 100 fire effects, which suggest a reduced embolism resistance (Michaletz et al, 2012; Lodge et al., 101 2018; Bär et al., 2018) and an increase in native embolism (West et al., 2016) after heat exposure 102 in water baths or simulated heat plumes, which these authors at least partially attribute to 103 declining air-seeding pressures due to the reduced surface tension at higher temperatures. For 104 the same reason, Cochard (2021) predicted a shift in the water potential at 50% loss of 105 conductivity by about +0.2% per °C (Fig. 1c). We argue that the magnitude of the changes in 106 surface tension alone is likely not sufficient to explain observed shifts in vulnerability curves, 107 and propose that the strong temperature sensitivity of gas solubility (Fig. 1a) paired with the 108 effect of dissolved gases on bubble nucleation warrant further investigation. 109 Here, we present a theoretical framework for bubble nucleation in the presence of dissolved 110 gases based on Alamé and Mahesh (2024), and highlight how rapid temperature surges may 111 drive embolism formation by destabilizing existing nuclei. We find experimental evidence for 112 increased vulnerability to xylem embolism under elevated temperature, providing potential 113 support for this mechanism. Through two experiments, conducted in Australia and Germany, 114 we investigated whether leaf embolism formation is temperature dependent. In both 115 experiments, tree saplings were subjected to heat treatments in climate chambers, and leaf 116 vulnerability curves obtained with the Optical Vulnerability Technique (Brodribb et al., 2016) 117 were compared between controls and heat treatments. In the German branch of the experiment, 118 we further experimentally supersaturated the water in the xylem with air before dehydration. 119 We used these two datasets to test the hypotheses that (i) the water potential at 12, 50 and 88% [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 120 cumulative embolism ( _P_ 12, _P_ 50 and _P_ 88, MPa), as well as the expected value of the embolism 121 pressure distribution ( _P_ ex, MPa), are shifted towards less negative values under hotter-drought 122 conditions, (ii) after correcting for differences in vapour pressure deficit at different 123 temperatures, the time to reach 12, 50 and 88% cumulative embolism ( _T_ 12, _T_ 50 and _T_ 88, kPa h), 124 as well as the expected value of the drying time distribution ( _T_ exp, kPa h), are shorter in the heat 125 treatment. # 126 Description ## 127 Bubble nucleation in pure water 128 Following the framework of classical nucleation theory (Sear et al., 2007), the Gibbs free 129 energy 𝛥𝐺 𝑡𝑜𝑡,𝑣 of a spherical cavity in pure water can be described as: 𝛥𝐺 = [4] 𝑡𝑜𝑡,𝑣 3 [𝜋(𝑝] [𝑙] [−𝑝] [𝑣] [)𝑟] [3] [ + 4𝜋𝜎] [𝑙,𝑔] [𝑟] [2] [, ] Eqn 1 130 where the indices 𝑣, 𝑙 and 𝑔 denote the vapour, liquid and gas phase, respectively, 𝑟 is the 131 bubble radius (m), 𝑝 𝑣 is the vapour pressure inside the gas bubble (Pa), 𝑝 𝑙 is the liquid pressure 132 (Pa) and 𝜎 𝑙,𝑔 is the surface tension at the liquid-gas interface (J m [−2] ). The first term describes 133 the work performed by the volume expansion, while the second term describes the free energy 134 required for creating the bubble interface (Kanduč et al., 2021). 135 By setting the first derivative of Eqn 1 with respect to 𝑟 to zero and rearranging, an equilibrium 136 solution can be found which results in the critical radius 𝑟 : 𝑐𝑟,𝑣 𝑟 = [2𝜎] [𝑙,][𝑔] 𝑐𝑟,𝑣 𝑝 𝑣 −𝑝 𝑙 , Eqn 2 137 The energy barrier 𝛥𝐺 𝑎𝑐𝑡,𝑣∗ (J) required to surpass the critical radius 𝑟 𝑐𝑟,𝑣 follows analogously 138 as: [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) ∗ 𝛥𝐺 = 𝑎𝑐𝑡,𝑣 16𝜋𝜎 3 𝑙,𝑔 3(𝑝 𝑣 −𝑝 𝑙 ) [2] [, ] Eqn 3 ## 139 Bubble nucleation in the presence of dissolved gas 140 In the presence of gas, a free bubble is stable when the expanding forces (i.e., pressure of the 141 gas in the bubble and vapor pressure) are equal to the collapsing forces (i.e., pressure of the 142 liquid and effects of surface tension): 3 𝑛 𝑅𝑇 𝑔 [, ] [+ 𝑝] [𝑣] [= 𝑝] [𝑙 ] [+ 2𝜎] [𝑙,][𝑔] 4 𝜋𝑟 [3] 𝑟 Eqn 4 143 where 𝑛 𝑔 is the mole number of the gas in the bubble (mol), 𝑝 𝑣 is the vapour pressure (Pa), 𝑅 144 is the gas constant (8.3145 J K [−1] mol [−1] ) and 𝑇 is the temperature in K. 145 For a bubble radius to be stable, the rate of change of the expanding and collapsing forces has 146 to be equal. Taking the derivative of both sides of the equation and rearranging results in the 147 following expression for the critical radius in dependence of gas content: 𝑟 𝑐𝑟 8𝜋𝜎 𝑙,𝑔 = √ [9𝑛] [𝑔] [𝑅𝑇] Eqn 5 148 Inserting this value for 𝑟 𝑐𝑟 into the Laplace equation results in the following equation for the 149 critical radius, which is known as Blake’s radius 𝑟 𝑐𝑟,𝐵𝑙 (Blake 1949a,b): 4𝜎 𝑙,𝑔 𝑟 𝑐𝑟,𝐵𝑙 = 3(𝑝 𝑣 −𝑝 𝑙,𝑐𝑟 ) [, ] 150 where 𝑝 𝑙,𝑐𝑟 (Pa) is the critical pressure. Eqn 6 151 While this is a valid description for the critical radius under equilibrium conditions, Alamé and 152 Mahesh (2024) point out that in itself it is not a complete cavitation model. These authors 153 instead used a Gibbs free energy approach to derive a more accurate expression of the free 154 energy of a bubble that takes into account the chemical equilibrium between the gas phase in 155 the bubble and the surrounding liquid (for a full derivation, see Alamé and Mahesh, 2024). [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 156 According to their framework (upon which the rest of this section is based on), the free energy 157 𝛥𝐺 𝑡𝑜𝑡,𝑣+𝑔 (J) of a spherical bubble in water containing dissolved gases is as follows: 𝛥𝐺 = [4] 𝑡𝑜𝑡,𝑣+𝑔 3 [𝜋(𝑝] [𝑙] [−𝑝] [𝑣] [)𝑟] [3] [ + 4𝜋𝜎] [𝑙,𝑔] [𝑟] [2] [ + ] 3 𝑛 𝐵𝑇 𝑔 𝑛 𝑔 𝑅𝑇 [ln (4 𝜋𝑠 (𝑝 𝑣 [∗] −𝑝 𝑙 )𝑟 [3] ~~[)]~~ [ − 1] + 𝑛] [𝑣] [𝑅𝑇 ln (𝑝] 𝑝 [𝑣] 𝑣 [∗] ~~[)]~~ [, ] Eqn 7 158 where 𝑛 𝑔 and 𝑛 𝑣 are the mole number of the gas and water vapour in the bubble (mol), 𝑝 𝑣 is 159 the current and 𝑝 𝑣∗ the saturation vapour pressure in the bubble (Pa), 𝑅 is the gas constant 160 (8.3145 J K [−1] mol [−1] ), 𝑇 is the temperature in K, and 𝑠 is the degree of gas saturation expressed 161 as the ratio of the current mole fraction of dissolved gas (𝑥 𝑑𝑔 ) to the saturated one (𝑥 𝑑𝑔∗ ), i.e. 162 𝑠= 𝑥 𝑑𝑔 /𝑥 𝑑𝑔∗ (unitless). Here, as in Eqn 1, the first and second term describe the work 163 performed by the volume expansion and the free energy required for creating the bubble 164 interface. The third and fourth terms together describe the total chemical potential energy, with 165 the third term describing the difference in chemical potential between the free gas in the bubble 166 and the dissolved gas in the liquid phase, and the fourth term describing the potential difference 167 between unsaturated and saturated vapor phase. Fig. 2a shows the relationship between 168 𝛥𝐺 𝑡𝑜𝑡,𝑣+𝑔 and bubble radius for different gas contents under the incipient regime (𝑝 𝑙 −𝑝 𝑣 < 0), 169 with a visible decline in the nucleation barrier for higher degrees of gas saturation. An 170 analogous plot for the desinent regime (𝑝 𝑙 −𝑝 𝑣 - 0) is shown in Fig. S1. 171 A critical radius 𝑟 𝑐𝑟,𝑣+𝑔 (m) for incipient cavitation (𝑝 𝑙 −𝑝 𝑣 < 0) can be derived by setting the 172 first derivative of Eqn 7 with respect to 𝑟 and 𝑛 𝑔 to zero and solving the resulting cubic equation 173 (Alamé and Mahesh, 2024). The solution can be expressed as the product of the critical radius 174 𝑟 𝑐𝑟,𝑣 in pure water and a correction factor 𝜉 𝑖𝑛𝑐 (𝑛 𝑔 ) depending on gas content: 2𝜎 𝑙𝑔 𝑟 𝑐𝑟,𝑣+𝑔 = 𝜉 𝑖𝑛𝑐 (𝑛 𝑔 ) ⋅ 𝑝 𝑣 −𝑝 𝑙 175 The correction factor 𝜉 𝑖𝑛𝑐 (𝑛 𝑔 ) is as follows: = 𝜉 𝑖𝑛𝑐 (𝑛 𝑔 ) ⋅𝑟 𝑐𝑟,𝑣 Eqn 8 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 𝜉 𝑖𝑛𝑐 (𝑛 𝑔 ) = [2] 3 [cos (1] 3 [cos] [−1] [(1 −𝑓(𝑛] [𝑔] [) ) + 1] 3 [ for 0 ≤ 𝑓(𝑛] [𝑔] [) ≤ 2 ] Eqn 9 176 Here, 𝑓(𝑛 𝑔 ) is an equation that describes the effect of gas content on the stability of the bubble: Eqn 10 ) 𝑓(𝑛 𝑔 8𝜋𝜎 ) = 2 ( [9𝑛] [𝑔] [𝑅𝑇] 𝑙𝑔 8𝜋𝜎 4 [9𝑛] [𝑔] [𝑅𝑇] 𝑙𝑔 ~~)~~ ( [3] (𝑝 𝑣 −𝑝 𝑙,𝑐𝑟 ) 4 𝜎 [3] 𝜎 𝑙𝑔 2 , 177 For pure water vapour, 𝑛 𝑔 is 0, resulting in a value of 0 for 𝑓(𝑛 𝑔 ) and hence a value of 1 for 178 𝜉 𝑖𝑛𝑐 (𝑛 𝑔 ), recovering the expression for 𝑟 𝑐𝑟,𝑣 from Eqn 2 when inserting into Eqn 8. If the 179 expanding and contracting forces in the bubble are in equilibrium, the following equality holds 180 (compare Eqn 5, 6): 9𝑛 𝑔 𝑅𝑇 4𝜎 𝑙𝑔 = 8𝜋𝜎 𝑙𝑔 3 (𝑝 𝑣 −𝑝 𝑙,𝑐𝑟 ) √ Eqn 11 181 Accordingly, when the bubble is in equilibrium, 𝑓(𝑛 𝑔 ) is equal to 2, resulting in 𝜉 𝑖𝑛𝑐 (𝑛 𝑔 ) = 182 2/3, in which case 𝑟 𝑐𝑟,𝑣+𝑔 evaluates to Blake’s radius 𝑟 𝑐𝑟,𝐵𝑙 from Eqn 6. For values of 0 < 183 𝑓(𝑛 𝑔 ) < 2, the bubble is undersaturated and for values of 𝑓(𝑛 𝑔 ) > 2 supersaturated with 184 gas compared to equilibrium conditions (Fig. 2b). Note that Eqn 9 is only defined for 𝑓(𝑛 𝑔 ) ≤ 185 2, reflecting that no critical radius exists if the bubble is supersaturated, as the activation barrier 186 required for nucleation is then reduced to 0 (see below). 187 A stable radius 𝑟 𝑒𝑞 (m) for incipient cavitation (𝑝 𝑙 −𝑝 𝑣 < 0) can be derived analogously: 2𝜎 𝑙𝑔 𝑟 𝑒𝑞 = 𝜉 𝑒𝑞 (𝑛 𝑔 ) ⋅ 𝑝 𝑣 −𝑝 𝑙 = 𝜉 𝑒𝑞 (𝑛 𝑔 ) ⋅𝑟 𝑐𝑟,𝑣, 188 With the correction following correction factor 𝜉 𝑒𝑞 (𝑛 𝑔 ): 𝜉 𝑒𝑞 (𝑛 𝑔 ) = [2] 3 [cos (1] 3 [(1 −𝑓(𝑛] [𝑔] [) + 4𝜋] 3 ~~[)]~~ [ + 1] 3 [ for 0 ≤ 𝑓(𝑛] [𝑔] [) ≤ 2 ] Eqn 12 Eqn 13 189 The equilibrium concentration is 0 for pure water vapour and rises with 𝑓(𝑛 𝑔 ) until reaching 190 the same value as 𝑟 𝑐𝑟,𝑣+𝑔 at 𝑓(𝑛 𝑔 ) = 2, again resolving to Blake’s radius (Fig. 2b). This can be [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 191 seen in Fig. 2a, where 𝑟 𝑒𝑞 corresponds to a local minimum in free energy for 0 < 𝑓(𝑛 𝑔 ) < 2. 192 For 𝑓(𝑛 𝑔 ) = 2, the minimum disappears, and the free energy has a saddle point at Blake’s 193 radius. 194 The energy barrier 𝛥𝐺 𝑎𝑐𝑡,𝑣+𝑔∗ (J) required to surpass the critical radius 𝑟 𝑐𝑟,𝑣+𝑔 can also be solved 195 analytically (Alamé and Mahesh, 2024): 𝛥𝐺 𝑎𝑐𝑡,𝑣+𝑔∗ = 𝜉 𝑎𝑐𝑡 (𝑛 𝑔 ) ⋅ 16𝜋𝜎 3 𝑙,𝑔 ∗ 3(𝑝 𝑣 −𝑝 𝑙 ) [2] [= 𝜉] [𝑎𝑐𝑡] [(𝑛] [𝑔] [) ⋅𝛥𝐺] [𝑎𝑐𝑡,𝑣], Eqn 14 196 where 𝜉 𝑎𝑐𝑡 (𝑛 𝑔 ) is a correction factor for the effect of gas content that is defined as follows: 𝜉 𝑎𝑐𝑡 (𝑛 𝑔 ) = [1] 9 [((6 cos(𝜃) + 3 cos(2𝜃) −2][√][3sin(𝜃) + ][√][3 sin(2𝜃))] + [4] 3 [𝑓(𝑛] [𝑔] [) ln (1 −cos(𝜃) + ] 1 + 2 cos(𝜃) [√][3 sin(𝜃) ] ~~)~~ 3 𝑔 ) ≤ 2, ) for 0 ≤𝑓(𝑛 Eqn 15 197 where 𝜃= 13 [cos] [−1] [ (1 −𝑓(𝑛] [𝑔] [))][. Accordingly, the energy barrier ] [𝛥𝐺] [𝑎𝑐𝑡,𝑣+𝑔] ∗ is equal to the where 𝜃= 1 198 value of 𝛥𝐺 𝑎𝑐𝑡,𝑣∗ for pure water vapour from Eqn 3 and declines with increasing gas content, 199 reducing to zero for equilibrium conditions (𝑓(𝑛 𝑔 ) = 2). For supersaturated conditions, 200 𝛥𝐺 𝑎𝑐𝑡,𝑣+𝑔∗ is not defined (Fig. 2b). 201 For desinent conditions when negative pressure subsides (𝑝 𝑙 −𝑝 𝑣 - 0), Alamé and Mahesh 202 (2024) provide a solution for a new equilibrium radius 𝑟 𝑒𝑞,𝑑𝑒𝑠 (m): 2𝜎 𝑙𝑔 𝑟 𝑒𝑞,𝑑𝑒𝑠 = 𝜉 𝑑𝑒𝑠 (𝑛 𝑔 ) ⋅ 𝑝 𝑣 −𝑝 𝑙 = 𝜉 𝑑𝑒𝑠 (𝑛 𝑔 ) ⋅𝑟 𝑐𝑟,𝑣 Eqn 16 203 Here, the correction factor 𝜉 𝑑𝑒𝑠 (𝑛 𝑔 ) for the effect of gas content is defined as follows: 𝜉 𝑑𝑒𝑠 (𝑛 𝑔 ) = { 23 [cos (1] 3 [cos] [−1] [(𝑓(𝑛] [𝑔] [) −1) ) −1] 3 [ if 0 ≤ 𝑓(𝑛] [𝑔] [) ≤ 2 ] [2] [2] 3 [cosh (1] 3 [cosh] [−1] [(𝑓(𝑛] [𝑔] [) −1) ) −1] 3 [ otherwise ] Eqn 17 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 204 Accordingly, for pure water vapour, the equilibrium radius under a desinent regime is equal to 205 0, and increases with higher gas concentrations into the supersaturated range (Fig. 2d, Fig. S3). 206 Hence, the presence of gas prevents complete bubble dissolution, and the more moles of gas a 207 bubble contains, the higher its stable equilibrium radius. ## 208 Implications for xylem embolism formation 209 The model of bubble nucleation by Alamé and Mahesh (2024) has several important 210 implications for the study of xylem embolism formation, most importantly, a) in a bubble with 211 gas content under equilibrium conditions, the energy barrier for expansion is effectively 212 reduced to zero (Fig. 2c), b) the critical radius for bubble expansion is highest for pure water 213 vapour bubbles, decreases with gas content and is not defined for bubbles in supersaturated 214 conditions, as under those conditions there is no energy barrier for expansion (Fig. 2b), and c) 215 if they contain gas, bubbles below the critical radius shrink to their stable equilibrium radius 216 𝑟 𝑒𝑞 rather than dissolving completely (Fig. 2b), even after the negative pressure subsides (Fig. 217 2d, Fig. S1). 218 While it is possible that bubbles with a non-zero gas content form and expand spontaneously, 219 even in the presence of considerable dissolved gas it is much more likely that expansion 220 happens from pre-existing nucleation sites. Potential nucleation sites in xylem sap are 221 hydrophobic surfaces, surfactant micelles or lipid bilayers (Kanduč et al., 2021), and, 222 importantly, stable nanobubbles present in the sap (Schenk et al., 2015, 2017). While the 223 spontaneous expansion of nanobubbles is a potential explanation of embolism formation, the 224 factors that cause this expansion are not yet fully resolved (Jansen et al., 2026). High 225 concentrations of dissolved gas are assumed to stabilize nanobubbles against dissolution 226 (Schenk et al., 2015, 2016), which is confirmed by the existence of non-zero stable radii for 227 bubbles below the critical radius derived above. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 228 However, our framework also indicates that existing bubbles may be destabilized when 229 temperature rises. On the short term, a rise in temperature will increase 𝑓(𝑛 𝑔 ) directly because 230 it scales linearly with temperature and to the third power with inverse surface tension (Eqn 10). 231 In consequence, rapid surges in temperature can cause a change in the equilibrium state that 232 results in previously stable bubbles now containing enough gas to push them over their critical 233 radius. Over longer time scales, higher temperatures can moreover increase the gas content 𝑛 𝑔 234 in the bubble itself due to diffusion driven by the reduced gas solubility of the sap (Fig. 1a, Eqn 235 B1.3). Even on most regular days, it is likely that xylem sap goes through periods of slight 236 supersaturation during warm periods (Schenk et al., 2016). During extreme heatwaves, the 237 effect of strong temperature differentials and the resulting highly supersaturated conditions may 238 drive the spontaneous expansion of bubbles and result in xylem embolism. # 239 Materials and methods ## 240 Plant material 241 Both experiments were conducted on potted saplings, with a total of eight angiosperm tree 242 species assessed, one of which had to be discarded due to data quality issues. 243 In Adelaide, South Australia, five _Eucalyptus_ tree species ( _E. argophloia_ Blakely – subtropical 244 to mediterranean climate, _E infera_ A.R.Bean – subtropical climate, _E. macrocarpa_ (Maiden) 245 Maiden – Mediterranean climate, _E. occidentalis_ Endl. – temperate climate, _E. camaldulensis_ 246 Dehnh. – broad distribution across Mediterranean, temperate, tropical and sub-tropical 247 climates) were grown from seeds purchased from the Australian Tree Seed Centre or were 248 obtained as tube stocks from local tree nurseries. All saplings were grown in 10 L native pot 249 mix soil (Scotts Osmocote, Bella Vista, NSW, Australia) under well-watered conditions and 250 natural sunlight, and fertilized every six months (Native Scotts Osmocote Controlled Release 251 Fertiliser, Bella Vista, NSW, Australia) to achieve maximum growth. Once saplings were about [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 252 1–2 years old (Jan–Feb 2026), five replicates per species per treatment were selected for the 253 xylem vulnerability measurements, prioritising saplings with similar height and healthy 254 looking leaves. Before the measurements, the pots were watered to full capacity, and the whole 255 saplings were covered with a dark plastic bag and rehydrated overnight to full turgor. 256 In Göttingen, Germany, we used each 10 saplings of the three temperate broadleaved tree 257 species _Alnus glutinosa_ (L.) Gaertn., _Castanea sativa_ Mill. and _Prunus avium_ L. In 2026, two 258 year-old saplings were sourced from local tree nurseries and potted into 12 L pots using a soil 259 mixture of medium-textured peat mix and medium grained fine sand, and then grown in full 260 sun in the Experimental Botanical Garden Göttingen until their harvest in July 2026. Before 261 measurement, we bare-rooted the saplings and washed their rootstock to remove all remaining 262 soil substrate to speed up dry-down. We then hydrated the plants overnight at 15 °C in a dark 263 room with water that we saturated with gas by bubbling it with compressed air for at least 2 h 264 to ensure supersaturation of the water in the xylem. After hydration, we towel-dried their 265 rootstock and moved them to the climate chambers where the dry-down experiment took place. ## 266 Image acquisition 267 Vulnerability to xylem embolism was assessed in all samples using the Optical Vulnerability 268 Technique (OVT; Brodribb et al., 2016). 269 _Australia:_ saplings were bench-dried inside climate chambers which were either set to a 270 temperature of 25 °C and a vapour pressure deficit (VPD) of ~1.9 kPa (for the control treatment) 271 or to 45 °C and ~6.7 kPa (for the heat treatment). One leaf per plant was fitted with an optical 272 imaging set-up composed of a Raspberry Pi computer and a digital Raspberry Pi Camera 273 Module V2 (Cavicam Co., Hobart, Tas., Australia; cavicam.com.au). Another cavicam was 274 fitted in the closest available stem segment to monitor stem shrinkage over time. The cavicams 275 also have a sensor that recorded the air temperature and humidity at 5-minute intervals, from [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 276 which we calculated the ambient VPD. Once the cavicam system was set up, the sapling 277 aboveground tissue was cut from the root, the cut was then sealed with white glue and images 278 of both the target leaf and stem were captured every 5 minutes as the samples dried. 279 _Germany:_ whole plants of the three study species were laid on benches within the climate 280 chambers (York Industriekälte GmbH & Co. KG, Germany), which were set to either 20 °C 281 (control treatment) or 35 °C (heat treatment) at 50% relative humidity, resulting in a VPD of 282 1.17 and 2.81 kPa, respectively. One leaf per plant was fitted with a custom-built optical 283 imaging set-up composed of a Raspberry Pi computer and a digital Raspberry Pi Camera 284 Module V2 (9132664, Raspberry Pi Ltd., Cambridge, UK) embedded in a custom-built 3D 285 printed clamp [(www.cavicams.com). Images were captured every 5 minutes as the samples](http://www.cavicams.com/) 286 dried. ## 287 Image processing 288 All leaf image sequences from both Australia and German datasets were assessed using the 289 well-established semi-automated image-processing pipeline for the OVT 290 [(www.opensourceov.org). Briefly, images were processed using FIJI software v2.16.0](http://www.opensourceov.org/) 291 [(Schindelin et al., 2012) and the OSOV toolbox plugin (www.opensourceov.org). An image](http://www.opensourceov.org/) 292 subtraction was performed, taking each image away from the one before to reveal the changes 293 between the images (and therefore the embolism events). A series of steps was then undertaken 294 to remove ‘noise’, i.e. the pixel signals from other changes that occur during drying, such as 295 leaf movement and shrinkage, thereby isolating the embolism events. Embolism throughout the 296 sequence was then expressed as a percentage of the total amount of pixels classified as part of 297 embolism events observed as the sample dried, and plotted against water potential to construct 298 vulnerability curves. All samples of _E. argophloia_ and two samples from _Prunus avium_ had to 299 be removed from the dataset due to poor data quality resulting from incomplete and inconsistent 300 detection of embolism events in the major veins. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) ## 301 Water potential measurements 302 In both experiments, water potentials were measured with the pressure chamber method. Before 303 excision, we bagged the leaves with aluminium foil for at least 30 min to ensure equilibration 304 with the stem water potential (Rodriguez-Dominguez et al., 2022). We then measured water 305 potentials using a Scholander pressure chamber (PMS Instruments, Corvallis, OR, USA). The 306 first measurement was performed directly after placing the samples into the climate chambers. 307 The subsequent measurements were done at different intervals in each branch of the experiment. 308 In the Australian branch, water potentials were initially obtained in 15–30 min intervals for the 309 first few hours, which was increased to 1–3 h at more negative water potentials (no overnight 310 water potential measurements were conducted). Typically, 6–15 (mean = 9) water potential 311 measurements were obtained per treatment, except for _E. camaldulensis_ in the heat treatment, 312 where only 3–5 water potential measurements were obtained per sample, as this species dried 313 down much faster. The stem diameter shrinkage and the corresponding leaf water potential 314 measurements were then used to create a temporal series of water potential data for the entire 315 experimental period using leaf optical dendrometry (Bourbia & Brodribb, 2023) assuming 316 linear calibration relationships. The R² of these leaf-specific calibrations varied from 0.71 to 317 0.99 with an average of 0.91 (Fig. S2). 318 In the German branch, water potentials were initially measured in intervals of 30–60 min, which 319 were consecutively increased at more negative water potentials, resulting in median time 320 differences between measurements of 1:33 h in the hot and 3:55 in the cold treatment. At more 321 negative water potentials, we sampled two instead of one leaf to account for the higher 322 variability between leaves in more dried-out plants. To interpolate water potentials for the 323 vulnerability curves from Germany, we used shape constrained additive models based on `R` 324 package `scam` v1.2-22 (Pya & Wood, 2015; Pya, 2026). Natural log-transformed (absolute) 325 water potential was expressed as a function of time with a monotonically increasing P-spline, [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 326 resulting in models with an R [2] between 0.801 and >0.999 with an average of 0.967 (Fig. S3). 327 Water potentials for all measurement times were then interpolated based on the back 328 transformed predictions from the additive models with a correction for retransformation bias 329 (Smith, 1993). ## 330 Data analysis 331 All data handling and statistical analyses were performed in `R` v4.5.2 ( `R` Core Team, 2025) in 332 the framework of the `tidyverse` (v2.0.0, Wickham et al., 2019). 333 VPD-corrected drying times (kPa h) for each sample were calculated as the cumulative sum of 334 the product of the VPD for each time interval with its duration, thus weighting the elapsed time 335 by the evaporative demand during that period (Blackman et al., 2019). 336 To analyse differences between water potentials associated with 12, 50 and 88% of cumulative 337 embolism ( _P_ 12, _P_ 50 and _P_ 88, respectively) between treatments (control vs. heat) in both datasets 338 (Australian and German), we extracted these values from the raw observed vulnerability curves 339 from both datasets, using the `approx()` function of base R to interpolate if required and 340 averaging water potentials if more than one water potential value was observed for that 341 particular value of cumulative embolism. In addition, we computed the expected embolism 342 pressure ( _P_ ex ) as the weighted average of the water potentials associated with embolism events 343 weighted by the number of registered pixels in each event. This quantity can be considered the 344 expected value of the distribution of embolism pressures described by the vulnerability curve, 345 and has advantageous properties over _P_ 12, _P_ 50 and _P_ 88 because it quantifies a property of the 346 entire curve instead of a single percentile (see Notes S1.1). Analogously, we extracted the VPD 347 corrected drying time until reaching 12, 50 and 88% percent cumulative embolism ( _T_ 12, _T_ 50 and 348 _T_ 88, respectively), and the expected value of the drying time distribution ( _T_ ex ). Average 349 vulnerability curves and cumulative embolism-drying time relationships were calculated by 350 interpolating quantiles in regular intervals and averaging them. To quantify the uncertainty of [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 351 average curves, we computed 95% parametric confidence intervals assuming approximate 352 normality. 353 As _P_ 12, _P_ 50, _P_ 88 and _P_ ex are related properties determined from the same curves, they are highly 354 correlated. To account for this non-independence, we analysed them with Bayesian multi 355 response mixed models using treatment, sampling location and their interaction as fixed effects, 356 with residual correlations and treatment-specific random effects for species to account for the 357 heterogeneity in the heat effects between species. We compared this model against a simplified 358 model without site differences to estimate the overall treatment effect. Drying times were 359 analysed with analogous Bayesian multi-response mixed models with _T_ 12, _T_ 50, _T_ 88 and _T_ ex as 360 response variables. The models were fitted with the `R` package `brms` v.2.23.0 (Bürkner 2017; 361 2018) using moderately informative priors. Sampling was performed on 4 chains for 10000 362 iterations each, 5000 of which were discarded as warm-up. Convergence was evaluated based 363 on the potential scale reduction factor 𝑅̂ (Vehtari et al., 2021) and inspection of trace plots. 364 Model adequacy was tested based on plots of posterior predictive checks (Gabry et al., 2019) 365 and leave-one-out cross-validation based on `R` package `loo` v. 2.9.0 (Vehtari et al., 2017, 2025). 366 Model comparisons were based on the expected log pointwise predictive density (ELPD) from 367 leave-one-out cross-validation. We assumed parameter estimates to be credibly different from 368 zero if their 95% credible intervals excluded zero. For details about model structure and prior 369 specifications and derived quantities, see Notes S1.2 – S1.4. # 370 Results ## 371 Observed vulnerability curves and effect of drying time 372 The average _P_ 50 of the curves differed between −2.91 ± 0.13 MPa for _Castanea sativa_ and 373 −6.43 ± 0.22 MPa for _Eucalyptus camaldulensis_ (mean ± standard error), and was slightly less 374 negative in the heat treatment compared to the control treatment for all studied species, though [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 375 with a large variability within species (Fig. 3, Table 1). _P_ 12, _P_ 88 and _P_ ex showed analogous 376 patterns (Fig. 3, Table 1). 377 The average drying time until reaching the _P_ 50 differed greatly between species, ranging from 378 12.25 ± 1.28 kPa h for _Castanea sativa_ to 51.86 ± 8.78 kPa h for _Eucalyptus infera_ (Table 1). 379 In contrast to the critical water potentials, for the drying times there was no consistent pattern 380 in the treatment averages (Fig. 4), with consistently lower drying times at higher temperatures 381 only observed in the samples from Australia (Table 1). ## 382 The effect of heat on embolism thresholds 383 We found an effect of heat exposure on the water potentials at 12, 50 and 88% loss of 384 conductivity and the expected embolism pressure, with the plants in the heat treatment reaching 385 _P_ 12, _P_ 50, _P_ 88 and _P_ ex at credibly less negative water potentials than the control plants (Fig. 5, 386 Fig. 6). On average, plants in the heat treatment had a _P_ 12, _P_ 50, _P_ 88 and _P_ ex that differed from 387 the control treatment by +0.79 MPa, +0.73 MPa, +0.84 MPa and +0.75 MPa, respectively 388 (Table S1). 389 However, this effect was almost exclusively driven by the _Eucalyptus_ species from the 390 Australian branch of the study, while there was no conclusive evidence for a heat effect for the 391 German plants (Fig. 6). The effect sizes for Australian species were even higher than in the 392 pooled dataset, with increases in _P_ 12, _P_ 50, _P_ 88 and _P_ ex by +1.08 MPa, +0.89 MPa, +0.90 MPa 393 and +0.87 MPa, respectively (Table S2). For the experiments in Germany, the heat treatment 394 effect on critical water potentials neither differed credibly from zero nor from the effect size in 395 Australia. In consequence, the pooled model had a slightly better fit to the data than the model 396 with different means for Australia and Germany with a ΔELPD of 3.6, though with high 397 standard error of 4.2, indicating that the two models were approximately equivalent. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) ## 398 The effect of heat on drying time 399 The time until reaching 12, 50 and 88% loss of conductivity as well the expected value of the 400 drying time distribution on average were all credibly lower for the plants in the heat treatment 401 compared to the plants in the control treatment (Fig. 5, Fig. 6). On average, plants in the heat 402 treatment reached their _T_ 12, _T_ 50, _T_ 88 and _T_ ex earlier by −7.2 kPa h for _T_ 12, by −7.8 kPa h for _T_ 50, 403 −8.6 kPa h for _T_ 88 and by −7.7 kPa h for _T_ ex (Table S3). 404 As for the vulnerability curve parameters, the overall heat effect on the drying times resulted 405 from the large effect sizes for the plants of the Australian branch of the study, while drying 406 times nearly did not differ between treatments for the German plants (Fig. 6). The effect sizes 407 for Australia were substantial, with _T_ 12, _T_ 50, _T_ 88 and _T_ ex reduced by −11.3 kPa h for _T_ 12, by 408 −13.3 kPa h for _T_ 50, −16.8 kPa h for _T_ 88 and by −13.6 kPa h for _T_ ex (Table S4). Unlike the model 409 for vulnerability curve parameters, in the model of drying times, the magnitude of the heat 410 effect was credibly higher for the Australian species. Here, the grouped model had a better fit 411 than the pooled model with a ΔELPD of 5.5 ± 6.3, although the large standard error again did 412 not allow to rule out equivalence of the models. # 413 Discussion ## 414 Heat exposure results in reduced embolism resistance 415 Our results provide evidence for a shift in vulnerability curves in a subset of species exposed 416 to higher temperatures that supports our first hypothesis, with heat-exposed plants on average 417 reaching critical levels of embolism at less negative water potentials. This is consistent with the 418 proposed mechanism of temperature changes that shift the thermodynamic equilibrium in 419 previously stable bubbles in the xylem sap to an unstable state that favours their expansion. 420 However, there are other possible explanations for the observed decrease in embolism 421 resistance. Based on the classical air-seeding concept, higher temperatures would be expected [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 422 to reduce air-seeding pressures due to decreases in surface tension (Cochard, 2021). However, 423 the observed effect size in our dataset is much larger than the +0.2% change per °C predicted 424 by Cochard (2021). For the Australian species with an average _P_ 50 of −6.29 MPa, the increase 425 in _P_ 50 of +0.89 MPa for a 20 °C increase in temperature was over 3.5 times higher than the 4% 426 (+0.25 MPa) predicted based only on surface tension effects on air-seeding pressure. For the 427 German species, which were subjected to a temperature difference of 15 °C, the mismatch in 428 effect size was similar (+0.36 MPa instead of the expected 3% or +0.11 MPa), albeit here the 429 credible intervals of the effect included zero. Accordingly, the effect of a temperature-driven 430 decrease in surface tension on air-seeding pressures alone is not sufficient to explain the 431 observed shift in vulnerability curves. 432 On the other hand, the temperature change alone (20 and 15 °C above ambient in the Australia 433 and Germany, respectively) would increase 𝑓(𝑛 𝑔 ), the effect of gas content on bubble stability 434 (Eqn 10), by 22% percent in the Australian and 16% in the German experiment. This could 435 push pre-existing nuclei in the sap, such as stable nanobubbles, into a state where no energy 436 barrier exists for their expansion (Fig. 2), assuming that they were in equilibrium before heat 437 exposure. Over a longer timescale, the reduced gas solubility at higher temperatures would 438 promote gas diffusion into existing nuclei, further driving them into an unstable state. In xylem 439 sap, nanobubbles are assumed to be coated with surfactant shells that promote their stability via 440 dynamic surface tension (Schenk et al., 2015, 2017; Yang et al., 2020; Ingram et al., 2021). 441 This means that bubble expansion leads to an increased surface tension due to the reduced 442 density of surfactants on the larger surface, which then results in a new and larger stable radius 443 (Schenk et al., 2017). This process is likely to provide a safety mechanism against the 444 spontaneous bubble expansion that otherwise would be likely to happen after steep temperature 445 increases due to the resulting changes in bubble stability. However, surfactant-coated bubbles 446 are not exempt from the process described here, and radius-dependent surface tension only 447 protects against bubble expansion until reaching values close to the surface tension of pure [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 448 water (Yang et al., 2020). Extreme temperature changes may therefore also promote bubble 449 nucleation in the presence of surfactants. 450 The changes in water viscosity related to the treatment are unlikely to have affected our results, 451 as viscosity does not play a direct role in bubble nucleation (compare Eqn 1–18). While an 452 increased hydraulic conductance resulting from lower viscosity (Cochard et al., 2000) to a 453 certain degree may offset the loss of hydraulic conductance due to embolism, this does not play 454 a role during the experimental dry-down of plants for the optical vulnerability technique 455 because during the dry-down actual flows in the xylem are likely negligible, and because the 456 method is not based on conductance measurements. 457 We observed a difference in the heat effect on vulnerability curves between the German and 458 Australian branch of the experiment, with smaller effect sizes and no credible overall effect in 459 Germany. A possible explanation is that the experimental supersaturation of xylem sap for the 460 plants in Germany did not affect embolism formation, and that the larger temperature 461 differential and higher end temperature in Australia had a larger impact. Since xylem sap is 462 likely close to full gas saturation most of the time (Schenk et al., 2016), a low impact of artificial 463 saturation is not surprising. However, we cannot exclude that the high temperatures of 45 °C in 464 the Australian heat treatment may have caused leaf tissue damage resulting in an earlier failure 465 of leaf veins that contributed to the observed heat effects. As the affected species come from 466 regions in Australia that can be exposed to temperatures equal or even higher than 45 °C, and 467 no leaf scorching was observed in our experiment, we are confident that our proposed 468 mechanism at least partly explains the observed heat effects on the vulnerability curves, despite 469 the varying results between the experiments. ## 470 Heat exposure can accelerate drying time beyond the effect of VPD 471 The observed temperature-driven shift in vulnerability curves towards less negative water 472 potentials was associated with a reduction in average VPD-corrected drying times. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 473 Accordingly, the plants in the heat treatment would have reached critical thresholds of 474 cumulative embolism earlier if VPD was held constant. This difference was only driven by the 475 _Eucalyptus_ species, with no evidence for a difference in the German part of the experiment. 476 Therefore, our results only partially support our second hypothesis, and suggest that heat effects 477 on plant dehydration time are context-dependent. Assuming a constant rate of water loss per 478 kPa h between treatments, the observed change in vulnerability to embolism after heat exposure 479 would result in reduced VPD-corrected drying times. However, the heat treatment reduced _T_ 50 480 by an average of −41% in the Australian and under −1% in German species, while _P_ 50 increased 481 by +14% in the Australian and +9% in German species. The disproportionally larger heat effect 482 on drying time in Australian species indicates that differences in water potential thresholds 483 alone are not sufficient to explain the observed differences in the time to reach these thresholds. 484 A potential explanation for the larger-than-expected reduction in drying times in Australian 485 samples could be temperature-dependent increases in cuticular conductance (Cochard et al., 486 2021). This process could lead to higher residual water loss if exceeding the temperature 487 associated with a conformation change of cuticular waxes, and has been demonstrated across a 488 wide range of species (Schreiber, 2001; Riederer, 2006; Schuster et al., 2016). Another 489 explanation could be stomatal reopening to permit evaporative cooling (Diao et al., 2024), 490 which has been demonstrated to occur at higher temperatures in several species of _Eucalyptus_ 491 independent of water access (Marchin et al., 2023) and hence could have accelerated the dry 492 down of the Australian species. Meanwhile, for the plants from the German experiment, the 493 lower maximum temperature of 35 °C makes it less likely that temperature thresholds were 494 crossed that result in phase changes in the cuticle or trigger stomatal reopening. Rather, here 495 the heat exposure led to earlier and more complete stomatal closure compared to the control 496 treatment, resulting in no detectable difference in VPD-corrected drying time. 497 In recent years, a growing number of studies has assessed the factors that determine the timing 498 to critical levels of dehydration (Blackman et al., 2019; Paligi et al., 2024, 2026; Waite et al. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 499 2024, 2026). The optical vulnerability technique is a useful tool for studying drying times, as it 500 results in high-resolution data of temporal changes in both water potential and the degree of 501 embolism, and can be used across multiple plant compartments simultaneously (Harrison Day 502 et al., 2024; Rodriguez-Dominguez et al., 2018). The differences in results between water 503 potential thresholds and the times to reach them indicate that focusing on the temporal aspect 504 of the dry-down instead of only reporting water potentials allows drawing a more complete 505 picture of plant drought responses. ## 506 Consequences for future studies of plant vulnerability to embolism 507 Our framework and experimental evidence indicate that critical water potential thresholds like 508 _P_ 12, _P_ 50 and _P_ 88 may not be static traits, but instead may vary dynamically based on 509 measurement conditions such as temperature and the gas content of the xylem sap. This has 510 important repercussions for vulnerability curve measurements. Firstly, our findings highlight 511 that for comparability of measurements obtained with the same method, measurement 512 temperatures should be kept constant and should be reported together with the results. At least 513 for the flow-centrifuge method, temperature-related changes in viscosity are known to be an 514 issue affecting data quality (Wang et al., 2014; Burlett et al., 2022). However, we are not aware 515 of studies that pointed out mechanisms how temperature could affect embolism formation itself 516 with the exception of Cochard et al. (2021). Secondly, our findings indicate that the gas content 517 of the liquid in the xylem can affect measurements. For that reason, methods where embolism 518 formation happens in natural xylem sap, such as the optical vulnerability technique (Brodribb 519 et al., 2016), classical bench dehydration (Sperry et al., 1988), micro-CT (Cochard et al., 2015), 520 or the pneumatic method (Pereira et al., 2016) may come to systematically different results than 521 methods that use a filtered and often degassed artificial measurement solution, such as the flow 522 centrifuge method (Cochard et al., 2005), or the air injection method (Cochard et al., 1992; 523 Ennajeh et al., 2011). [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 524 An important consequence is that water potential thresholds obtained under laboratory 525 conditions may not always reflect the behaviour of plants under natural conditions. During 526 extreme heatwaves, plants may die _before_ they actually reach the critical water potentials 527 measured in the laboratory at room temperature. This can be relevant, for example, in modelling 528 settings where embolism resistance measurements from global databases, like the Xylem 529 Functional Trait Database (Choat et al., 2012) or TRY (Kattge et al., 2020), are used as static 530 thresholds for modelling drought-induced plant mortality under rising temperatures. If the 531 mechanism outlined here—in which embolism develops at lower water potentials than 532 previously expected—is confirmed in further studies, this would have significant implications 533 for assessing the drought tolerance of tree species in a changing climate. 534 While our experimental evidence is in line with the idea that rapid temperature changes can 535 reduce the stability of bubbles in xylem sap and thus lead to increased xylem vulnerability, we 536 are aware of the pilot-study character of our work. We acknowledge that our study combines 537 results from two small experiments with limited sample sizes and differing in design, and that 538 more rigorous testing of the mechanism in larger studies is necessary. Future studies should test 539 for temperature effects on xylem vulnerability curves using a larger number of species and 540 larger sample sizes, using consistent heat treatments and a fully-factorial design for evaluating 541 the effect of experimental supersaturation. As direct evidence of embolism formation in the 542 xylem is nearly impossible to obtain, this should be accompanied with mechanistic modelling 543 of bubble nucleation processes in the xylem that take into account the influence of gas content. ## 544 Conclusions 545 The theoretical framework presented here highlights the importance of dissolved gas for bubble 546 nucleation processes in xylem sap, describes the conditions under which gas-filled nuclei can 547 remain stable, and shows how sudden temperature changes can result in the spontaneous 548 expansion of formerly stable nuclei. The effect of gas content on bubble stability adds an [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 549 important new piece to the puzzle of the mechanisms of xylem embolism formation. Our 550 experimental evidence suggests that in line with the proposed mechanism, heat exposure during 551 drought can result in reduced embolism resistance and a faster dry-down even after correcting 552 for differences in VPD, at least in a subset of species and at large enough temperature 553 differences. Our framework may thus help explaining the excessive tree mortality observed 554 following hotter-drought events. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) # 555 Acknowledgements 556 This study was supported by the ARC-DECRA 2026 Fellowship (DE260100224), the Bert and 557 Aileen Kollosche Bequest research grant 2025 from the Australian Plant Society of South 558 Australia, and by the Thomas Davies Research Grant for Marine, Soil and Plant Biology 2026 559 from the Australian Academy of Sciences awarded by ISM. KMJ was funded by the European 560 Union on a Marie Skłodowska-Curie Actions (MSCA) Postdoctoral fellowship, 101107177— 561 IVERdrought. Views and opinions expressed are, however, those of the author(s) only and do 562 not necessarily reflect those of the European Union or the MSCA. Neither the European Union 563 nor the granting authority can be held responsible for them. KMJ acknowledges the travel grant 564 provided by the University of Gottingen’s faculty of forest sciences and forest ecology which 565 facilitated collaboration including the experimental work in Germany. SSP and CA 566 acknowledge the funding support which enabled the establishment of the pot experiment in 567 Göttingen, Germany through grant number FKZ 67KIA4059E under the KI-recover project. 568 AR was supported by the DAAD fellowship to visit the department of Plant Ecology and 569 Ecosystems Research and Silviculture and Forest Ecology of Temperate Zones, University of 570 Göttingen, Germany. We gratefully acknowledge Regina Helbig, Heinz Coners and Bharath 571 Kumar Haralalli Nagarajappa for technical support in establishing and maintaining the potted 572 saplings at the Experimental Botanical Garden in Göttingen, Germany. RML and KP wish to 573 thank BS (R.I.P.) & G for moral support. # 574 Author contributions 575 RML and ISM developed the original idea of the project, which was then considerably modified 576 in collaboration with KMJ and SSP. RML developed the theoretical framework. ISM planned 577 the Australian experiment and collected the Australian dataset together with NA, JDKD, and 578 WRW.SSP, KMJ and RML planned the German experiment and collected the German dataset [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 579 together with KP, AR and FRT. RML wrote the original draft with help from SSP, KMJ and 580 ISM, which was then revised and approved by all other authors. # 581 Data availability statement 582 Upon publication, the dataset and R code underlying the analysis in this paper will be made 583 available as a public repository on zenodo.org. # 584 ORCID numbers 585 Roman Mathias Link: 0000-0003-0588-3757 586 Sharath Shyamappa Paligi: 0000-0003-2139-2018 587 Christian Ammer: 0000-0002-4235-0135 588 Kerstin Pierick: 0000-0002-1313-4601 589 Abdul Razzak: 0009-0001-2016-426X 590 Bernhard Schuldt: 0000-0003-4738-5289 591 Fon Robinson Tezeh: 0000-0002-6859-690X 592 Kate M Johnson: 0000-0002-3039-6339 593 Ilaine Silveira Matos: 0000-0001-5557-5133 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. 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The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) # 836 Boxes [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) # 871 Tables 872 **Table 1** Observed water potential at 12, 50 and 88% percent cumulative embolism ( _P_ 12, _P_ 50 873 and _P_ 88, respectively) and VPD-corrected drying time until reaching the water potential at 12, 874 50 and 88% percent cumulative embolism ( _T_ 12, _T_ 50 and _T_ 88, respectively). Shown are mean ± 875 standard error for the control (25 and 20 ºC for the Australian and German part of the 876 experiment, respectively) and heat treatment (45 and 35 ºC for the Australian and German part 877 of the experiment, respectively). 878 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) # 879 Figures 880 881 **Figure 1** Relative change in a) gas solubility, b) surface tension and c) viscosity of pure water 882 over the temperature range from 0 to 50 °C compared to their reference value at 20 °C. In a), 883 the change in solubility is shown for the 4 most common gases in the atmosphere besides water 884 vapour. The values were calculated according to the equations in Box 1, using values from 885 Sander et al. (2023) for the computation of gas solubility. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 886 887 **Figure 2** a) Change in Gibbs free energy with bubble radius for different gas contents 𝑛 𝑔 under 888 the incipient regime. Shown are the (relative) values of 𝛥𝐺 𝑡𝑜𝑡.𝑣+𝑔 following Eqn 7 coloured by 889 𝑓(𝑛 𝑔 ) (an indicator of the degree of gas saturation of the gas in the bubble as a function of gas 890 content, ranging from 0 for pure water vapour over 2 for equilibrium saturation to > 2 for 891 supersaturated conditions; see Eqn 10). Highlighted radii are the critical radius 𝑟 𝑐𝑟,𝑣 following 892 Eqn 2 for pure water vapour (gas content of 0) and Blake’s radius 𝑟 𝑐𝑟,𝐵𝑙 following Eqn 6 for 893 equilibrium conditions. b) Change in the critical radius 𝑟 𝑐𝑟,𝑣+𝑔 for incipient cavitation and the 894 equilibrium radius 𝑟 𝑒𝑞 with 𝑓(𝑛 𝑔 ) relative to the critical radius 𝑟 𝑐𝑟,𝑣 for pure water vapour. For [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 895 𝑓(𝑛 𝑔 ) = 2, both radii are identical and recover Blake’s radius 𝑟 𝑐𝑟,𝐵𝑙 . For values of 𝑓(𝑛 𝑔 ) > 2 896 (supersaturation), no critical radius is defined. c) Relative change in the activation energy 897 needed for bubble expansion Δ𝐺 𝑎𝑐𝑡,𝑣+𝑔∗ compared to pure water vapour plotted against 𝑓(𝑛 𝑔 ). 898 For supersaturated conditions, no energy barrier is defined. d) Change in the equilibrium radius 899 𝑟 𝑒𝑞 for the desinent regime with 𝑓(𝑛 𝑔 ) relative to the critical radius 𝑟 𝑐𝑟,𝑣 for pure water. Here, 900 values are also defined for the supersaturated range: more dissolved gas permits higher stable 901 bubble radii. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 902 903 **Figure 3** Vulnerability curves for the seven studied species in the control (blue) and heat 904 treatment (red). Shown are the observed vulnerability curves obtained with the Optical 905 Vulnerability Technique (faint lines) overlaid with the average curve (thick lines) with 906 parametric 95% confidence intervals (shaded areas). Temperatures were set to 25 ºC (Australia) 907 and 20 ºC (Germany) in the control and to 45 (Australia) and 35 ºC (Germany) in the heat 908 treatment. 909 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 910 911 **Figure 4** Change in cumulative embolism with VPD-corrected dehydration times for the seven 912 studied species in the control (blue) and heat treatment (red). Shown are the observed values 913 (faint lines) overlaid with the average curve (thick lines) with parametric 95% confidence 914 bounds (shaded areas). Temperatures were set to 25 ºC (Australia) and 20 ºC (Germany) in the 915 control and to 45 (Australia) and 35 ºC (Germany) in the heat treatment. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 916 917 **Figure 5** Water potential at 12, 50, 88% percent cumulative embolism and expected embolism 918 pressure ( _P_ 12, _P_ 50, _P_ 88, and _P_ ex, respectively) and VPD-corrected drying time until reaching 919 those water potentials ( _T_ 12, _T_ 50, _T_ 88, and _T_ ex ) against treatments. Shown are the observed values 920 coloured by location overlaid with the posterior means and 95% credible intervals for the 921 marginal predictions of the models with pooled and site-wise treatment effects. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 922 923 **Figure 6** Effect sizes from models of vulnerability curve results (left) and drying times (right). 924 Shown are the magnitude of the heat effects (posterior mean with 95% credible intervals) for 925 the different species, as well as the estimates of the total effect for the two study locations, and 926 pooled over both locations. Species-wise predictions are conditional effects of the site-specific 927 model, total effects are based on the marginal effects of the site-specific (totals for Australia 928 and Germany) and pooled model (pooled total). Symbols indicate credible differences on a 0.1 929 (.), 0.05 (*), 0.01 (**) and 0.001 level (***). 930 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) # 931 Supporting information ## 932 Notes S1. Supplementary methods 933 In this document, we first provide a derivation of the expected embolism pressure _P_ ex as a metric 934 for comparing vulnerability curves (S.1.1), and then describe the structure of the multivariate 935 model of different water potential thresholds ( _P_ 12, _P_ 50, _P_ 88 and _P_ ex ) in dependence of heat 936 treatment and study location (S1.2 – S1.4). The models for VPD-corrected drying time ( _T_ 12, 937 _T_ 50, _T_ 88 and _T_ ex ) followed an analogous structure and are hence not described separately. 938 _S1.1 Expected embolism pressure P_ ex 939 To describe differences in embolism resistance, it is customary to compute point estimates of 940 water potential thresholds such as the _P_ 12, _P_ 50 or _P_ 88, and then compare how they differ between 941 curves. However, the thresholds of 12, 50 and 88% cumulative embolism are largely arbitrary, 942 and if the curves are rather asymmetric, as is usually the case for the optical vulnerability 943 technique, slight differences in the percentage of cumulative embolism can result in large 944 differences in water potential. It would therefore be advantageous to have a metric to compare 945 the location of curves across their entire range. 946 Vulnerability curves describe the change of cumulative embolism with decreasing water 947 potential, and can thus intuitively be interpreted as cumulative distribution functions of the 948 underlying distribution of embolism pressures. In the following, assume two arbitrary 949 vulnerability curves 𝐹 1 (Ψ) and 𝐹 2 (Ψ) that both map water potentials Ψ onto the observed 950 fraction of cumulative embolism 𝑝. A reasonable metric of distance over the entire curves is 951 the expected value E[𝑑] of the water potential distance 𝑑 between the two curves over their 952 entire range. Using the definition of the expected value of a continuous function 𝑓(𝑥) over the 1 𝑏 953 interval [𝑎, 𝑏] as 𝑏−𝑎 [∫𝑓(𝑥) d𝑥] 𝑎, E[𝑑] can be calculated as the integral over the difference of interval [𝑎, 𝑏] as 𝑏−𝑎 [∫𝑓(𝑥) d𝑥] 𝑎 1 954 their quantile functions 𝐹 1−1 (𝑝) and 𝐹 2−1 (𝑝): −1 (𝑝) and 𝐹 2 their quantile functions 𝐹 1 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) E[𝑑] = ~~|~~ Eqn S1.1 Eqn S1.2 1 𝑥= 01 𝐹 2−1 (𝑥) −𝐹 1−1 (𝑥) d𝑥| = |∫ 𝑥= 01 𝐹 2 −1 (𝑥) −𝐹 1 1−0 [∫] 𝑥= 0 𝐹 2 𝑥= 01 𝐹 2−1 (𝑥) −𝐹 1−1 (𝑥) d𝑥|, −1 (𝑥) −𝐹 1 955 The integral in Eqn S1.1 can be split into two components, resulting in: 1 E[𝑑] = |∫ 𝑥= 0 𝐹 2 𝑥= 01 𝐹 1−1 (𝑥) d𝑥|, 𝑥= 01 𝐹 2−1 (𝑥) d𝑥 − ∫ 𝑥= 01 𝐹 1 956 It can be shown that the integral over a quantile function is equal to the expected value of the 957 corresponding distribution. Accordingly, assuming that 𝑋 1 and 𝑋 2 are random variables 958 following the cumulative distribution functions 𝐹 1 (Ψ) and 𝐹 2 (Ψ), the expected water potential 959 distance between curves reduces to: E[𝑑] = |E[𝑋 2 ] − E[𝑋 1 ]|, Eqn S1.3 960 This means that the expected value of the water potential difference between two vulnerability 961 curves is simply the difference between the expected values of the embolism pressure 962 distribution described by the two curves. It follows that comparing differences in E[𝑋] is 963 analogous to comparing the expected water potential difference E[𝑑] calculated over the entire 964 range of the vulnerability curves. In this work, the quantity E[𝑋] is denoted as 𝑃 𝑒𝑥 . 965 For symmetrical models of vulnerability curves such as the commonly used exponential 966 sigmoid model of Pammenter & Vander Willigen (1998), 𝑃 𝑒𝑥 and 𝑃 50 are identical, so that 967 comparisons in 𝑃 50 are consistent with comparing water potential differences along the entire 968 curves. For the comparison of metrics from curves obtained with different statistical models, or 969 for the highly asymmetric curves obtained by the optical method, this is not necessarily the 970 case. Here, comparisons of 𝑃 𝑒𝑥 instead of conventional thresholds like _P_ 12, _P_ 50 or _P_ 88 have the 971 vast advantage of being equivalent to a comparison over the entire curve without actually 972 having to evaluate the integral in Eqn S1.1. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 973 For the optical vulnerability technique, 𝑃 𝑒𝑥 can easily be computed as the weighted average of 974 the water potentials Ψ for embolism event 𝑖 weighted by the number of pixels 𝑝𝑥 𝑖 that were 975 classified as embolized during this embolism event: 𝑛𝑖=1 𝑝𝑥 𝑖 ⋅Ψ 𝑖 𝑃 𝑒𝑥 = [∑] ∑ 𝑛𝑖=1 𝑝𝑥 𝑖 Eqn S1.4 976 _S1.2 Model structure_ 977 In the following, we will use the subscript 𝑥 to mark statements that are valid analogously for 978 all water potential thresholds considered in the model, e.g. _P_ _x_ to signify all water potential 979 thresholds ( _P_ 12, _P_ 50, _P_ 88 and _P_ ex ). In our model, we described the joint distribution of the water 980 potential thresholds 𝑃 𝑥[𝑖] for observation 𝑖 with a multivariate normal distribution with mean 981 𝜇 𝑥[𝑖] and 4 × 4 covariance matrix Σ 𝑟𝑒𝑠 : res Eqn S1.5 ), Σ ), 𝑃 12[𝑖] 𝑃 50[𝑖] 𝑃 88[𝑖] 𝑃 𝑒𝑥[𝑖] ) ( ~ MVN ( ( 𝜇 12[𝑖] 𝜇 50[𝑖] 𝜇 88[𝑖] 𝜇 𝑒𝑥[𝑖] 982 The mean 𝜇 𝑥[𝑖] for each water potential threshold for species 𝑗 and location 𝑘 (Australia or 983 Germany) was expressed as follows: 𝜇 𝑥[𝑖] = 𝛼 0𝑥[𝑘] + (𝛽 0𝑥[𝑘] + 𝛽 𝑥[𝑗] ) ⋅𝑇𝑟𝑡 [𝑖] + 𝛼 𝑥[𝑗], Eqn S1.6 984 where 𝛼 0𝑥[𝑘] is the location-specific intercept, 𝑇𝑟𝑡 [𝑖] is the heat treatment (control or heat) for 985 observation 𝑖, 𝛽 0𝑥[𝑘] is the location-specific heat effect, and 𝛼 𝑥[𝑗] and 𝛽 𝑥[𝑗] are species-specific 986 random intercepts and slopes, respectively. 987 We additionally fitted a simplified version of the model without location-specific differences: ∗ 𝜇 𝑥[𝑖] = 𝛼 0𝑥 + (𝛽 0𝑥 + 𝛽 𝑥[𝑗] ) ⋅𝑇𝑟𝑡 [𝑖] + 𝛼 𝑥[𝑗], Eqn S1.7 988 The random intercepts and slopes were described by a multivariate normal distribution centred 989 around zero: [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅ ̅̅̅̅̅̅ ( [𝛼] 𝛽 𝑥[𝑗] [𝑥[𝑗]] ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅ ̅̅̅̅̅̅ Eqn S1.8 0 [), Σ] [𝑅𝐸𝑥] [),] ) ~ MVN ( ( [0] ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅ ̅̅̅̅̅̅ 990 where Σ 𝑅𝐸𝑥 is a 2 × 2 covariance matrix describing the joint distribution of the random effects, 991 containing the random effects standard deviations 𝜏 𝛼𝑥 and 𝜏 𝛽𝑥 and the random effects 992 correlation matrix Ω RE : Σ 𝑅𝐸𝑥 = diag(𝜏 𝛼𝑥, 𝜏 𝛽𝑥 ) Ω RE diag(𝜏 𝛼𝑥, 𝜏 𝛽𝑥 ), Eqn S1.9 993 Analogously, the covariance matrix Σ 𝑟𝑒𝑠 of the residuals consists of the residual standard 994 deviations 𝜎 𝑥 and the residual correlation matrix Ω res : Σ res = diag(𝜎 12, 𝜎 50, 𝜎 88, 𝜎 𝑒𝑥 ) Ω res diag(𝜎 12, 𝜎 50, 𝜎 88, 𝜎 𝑒𝑥 ) Eqn S1.10 995 _S1.3 Choice of priors_ 996 The location intercepts 𝛼 0𝑥[𝑘] (i.e. the average of the water potential threshold 𝑃 𝑥 in the control 997 treatment at the corresponding location) were loosely centered around the observed average 998 𝑃 𝑥[𝑘] ̅̅̅̅̅̅ in the control treatment with a normal prior with a standard deviation of 2: 𝛼 0𝑥[𝑘] ~ Normal( 𝑃 𝑥[𝑘] ̅̅̅̅̅̅, 2), Eqn S1.11 999 which puts 95% of the prior probability mass on values of approximately ± 4 MPa around the 1000 average in the control treatment. For the analogous models of drying time, due to the different 1001 scale of the response, the normal prior on the location intercepts was centered around the 1002 observed average 𝑇 𝑥[𝑘] ̅̅̅̅̅̅ in the control treatment with a standard deviation of 10, which puts 95% 1003 prior probability mass on values of approximately ± 20 kPa h around the average in the control 1004 treatment. For the simplified version of the model without location differences, we used the 1005 pooled average 𝑃 𝑥 ̅ of the control treatment instead of 𝑃 𝑥[𝑘] ̅̅̅̅̅̅. 1006 The location-specific heat effects 𝛽 0𝑥[𝑘] were assigned normal priors with mean 0 and a 1007 standard deviation of 2: [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 𝛽 0𝑥[𝑘] ~ Normal( 0, 2), Eqn S1.12 1008 which centers the expected heat effects around zero, putting 95% prior probability mass for the 1009 difference between treatments on values of approximately ± 2 MPa. For the analogous models 1010 of drying time, the normal prior on the heat effects was centered around 0 with a standard 1011 deviation of 10, which puts 95% prior probability mass on values of approximately ± 20 kPa h. 1012 The standard deviations of the random effects 𝜏 𝛼𝑥 and 𝜏 𝛽𝑥 were assigned gamma priors with 1013 shape 2 and rate 1 and 4, respectively: 𝜏 𝛼𝑥 ~ Gamma( 2, 1) 𝜏 𝛽𝑥 ~ Gamma(2, 4), Eqn S1.13 1014 Gamma( 2, ⋅) priors are boundary avoiding priors (cf. Gelman et al., p316) that discourage 1015 very low values for the random effect standard deviations. The priors put 95% probability mass 1016 on values between 0.24 and 5.57 with an expected value of 2.00 for 𝜏 𝛼𝑥 and on values between 1017 0.06 and 1.39 MPa with an expected value of 0.50 MPa for 𝜏 𝛽𝑥, reflecting a higher plausible 1018 variability between species than between treatments. For the analogous models of drying time, 1019 we used a Gamma(2, 0.25) prior for 𝜏 𝛼𝑥 and a Gamma(2, 1) prior for 𝜏 𝛽𝑥, which have expected 1020 values of 8.0 and 2.0 kPa h and put 95% of the probability mass on values from 1.0 – 22.3 kPa 1021 h and from 0.2 – 5.6 kPa h, respectively. The use of boundary-avoiding priors for 𝜏 𝛼𝑥 and 𝜏 𝛽𝑥 1022 was motivated by the low number of seven species in the dataset, which can affect the stability 1023 of the estimates of random effect standard deviations. 1024 The residual standard deviations 𝜎 𝑥 were assigned half-normal priors with a mean of 0 and a 1025 standard deviation of 2: 𝜎 𝑥 ~ half-normal( 0, 2) Eqn S1.14 1026 This prior puts 95% probability mass on values of 𝜎 𝑥 below 3.3 MPa. For the analogous model 1027 of drying time, 𝜎 𝑥 were assigned half-normal priors with a mean of 0 and a standard deviation 1028 of 10, which puts puts 95% probability mass on values of 𝜎 𝑥 below 16.4 kPa h. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 1029 Finally, the correlation matrices Ω RE and Ω res were assigned LKJ(1) priors (Lewandowski et 1030 al., 2009), which are flat priors on correlation matrices that do not favour any specific values 1031 of correlations: Ω 𝑅𝐸𝑥 ~ LKJ(1) Ω 𝑟𝑒𝑠 ~ LKJ(1) Eqn S1.15 1032 1033 _S1.4 Model fitting, validation and inference_ 1034 The models were fitted in `R` v4.5.2 ( `R` Core Team, 2025) using `R` package `brms` v.2.23.0 1035 (Bürkner 2017; 2018). For all models, we used 4 parallel chains with 10000 iterations each. 1036 The first 5000 iteration were discarded as warm-up, resulting in a total of 20000 posterior draws 1037 for each model parameter. For all models, we set the target acceptance rate to 0.99 and used a 1038 maximum tree depth of 12. 1039 Convergence was evaluated based on the potential scale reduction factor R (Vehtari et al., 2021) 1040 and inspection of trace plots. Model adequacy was tested based on plots of posterior predictive 1041 checks (Gabry et al., 2019) and leave-one-out cross-validation based on R package loo v. 2.9.0 1042 (Vehtari et al., 2017, 2025), setting `re_loo = TRUE` to guarantee reliable estimates of the 1043 expected log pointwise predictive density (ELPD), since a low number of observations resulted 1044 in Pareto _k_ values of above 0.7. Model comparisons were based on the ELPD from leave-one 1045 out cross-validation. 1046 As a Bayesian equivalent to the frequentist p-value, we computed the probability of direction 1047 𝑝𝑑 for all fixed effects. The probability of direction 𝑝𝑑 is defined as the proportion of the 1048 posterior distribution that has the same sign as its median (Makowski et al., 2019). 𝑝𝑑 = max{Pr(𝜃< 0), Pr(𝜃> 0)} Eqn S1.16 1049 Based on 𝑝𝑑, we then calculated a two-sided Bayesian pseudo-p-value 𝑃 𝐵𝑎𝑦𝑒𝑠 as follows: 𝑃 𝐵𝑎𝑦𝑒𝑠 = 2 ⋅(1 −𝑝𝑑) Eqn S1.7 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 1050 In a model with flat priors, 𝑃 𝐵𝑎𝑦𝑒𝑠 is identical to the two-sided frequentist p-value (Makowski 1051 et al., 2019). 1052 1053 _S1.5 Supplementary references_ 1054 **Bürkner P-C** . **2017** . `brms` : An R package for Bayesian multilevel models using Stan. _Journal_ 1055 _of Statistical Software_ **80** : 1–28. 1056 **Bürkner P-C** . **2018** . Advanced Bayesian multilevel modeling with the `R` package `brms` . _The_ 1057 _R Journal_ **10** : 395–411. 1058 **Gabry J, Simpson D, Vehtari A, Betancourt M, Gelman A** . **2019** . Visualization in Bayesian 1059 workflow. _Journal of the Royal Statistical Society: Series A (Statistics in Society)_ **182** : 389– 1060 402. 1061 **Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB, Christensen R,** 1062 **Johnson W, Branscum A, Hanson TE,** _**et al.**_ **2014** . _Bayesian Data Analysis, Third Edition_ . 1063 Boca Raton, FL: Taylor & Francis Group, LLC. 1064 **Lewandowski D, Kurowicka D, Joe H** . **2009** . Generating random correlation matrices based 1065 on vines and extended onion method. _Journal of Multivariate Analysis_ **100** : 1989–2001. 1066 **Makowski D, Ben-Shachar MS, Chen SHA, Lüdecke D** . **2019** . Indices of Effect Existence 1067 and Significance in the Bayesian Framework. _Frontiers in Psychology_ **10** . 1068 **Pammenter NW, Vander Willigen C** . **1998** . A mathematical and statistical analysis of the 1069 curves illustrating vulnerability of xylem to cavitation. _Tree Physiology_ **18** : 589–593. 1070 **R Core Team** . **2025** . _R: a language and environment for statistical computing_ . Vienna, Austria: 1071 `R` Foundation for Statistical Computing. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 1072 **Vehtari A, Gabry J, Magnusson M, Yao Y, Bürkner P-C, Paananen T, Gelman A** . **2025** . 1073 `loo` : Efficient leave-one-out cross-validation and WAIC for Bayesian models. `R` package 1074 version 2.9.0. URL: `https://mc-stan.org/loo/` 1075 **Vehtari A, Gelman A, Gabry J** . **2017** . Practical Bayesian model evaluation using leave-one 1076 out cross-validation and WAIC. _Statistics and Computing_ **27** : 1413–1432. 1077 **Vehtari A, Gelman A, Simpson D, Carpenter B, Bürkner P-C** . **2021** . Rank-Normalization, 1078 Folding, and Localization: An Improved 𝑅̂ for Assessing Convergence of MCMC (with 1079 Discussion). _Bayesian Analysis_ **16** : 667–718. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) ## 1080 Supplementary figures 1081 1082 **Figure S1** Change in Gibbs free energy with bubble radius for different gas contents 𝑛 𝑔 under 1083 the desinent regime. Shown are the (relative) values of 𝛥G 𝑡𝑜𝑡,𝑣+𝑔∗ following Eqn 7 coloured by 1084 𝑓(𝑛 𝑔 ) (an indicator of the degree of gas saturation of the gas in the bubble as a function of gas 1085 content, ranging from 0 for pure water vapour over 2 for equilibrium saturation to > 2 for 1086 supersaturated conditions; see Eqn 10). For pure water vapor, the equilibrium radius is zero. 1087 For all other gas contents, there exists an equilibrium radius (i.e., a local minimum), which is 1088 exactly half of Blake’s radius 𝑟 𝑐𝑟,𝐵𝑙 (Eqn 6) under equilibrium conditions (𝑓(𝑛 𝑔 ) = 2). 1089 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 1090 1091 **Figure S2** . Calibration of water potentials for the Australian samples via leaf dendrometry for 1092 the control (DO1 – DO5) and heat (DH1 – DH5) treatments and four _Eucalyptus_ species: _E._ 1093 _camaldulensis_ – EUC_CAM, _E. infera_ – EUC_INF, _E. microcarpa_ – EUC_MIC, _E._ 1094 _occidentalis_ – EUC_OCC. The models explained on average 91% of the variance in water 1095 potential, ranging from 71 to 99%. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 1096 1097 **Figure S2** . Relationship between (uncorrected) drying time and observed water potentials for 1098 the samples from Germany. Shown are the observed water potentials (points, coloured by 1099 treatment) and the predictions from the shape-constrained additive models (lines). Panel titles 1100 are sample IDs ( _Alnus glutinosa_ – ALGL, _Castanea sativa_ – CASA, _Prunus avium_ – PRAV). 1101 The models explained on average 96.7% of the variance in water potential, ranging from 80.6 1102 to > 99.9%. [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) ## 1103 Supplementary tables 1104 **Table S1** Results of the multi-response mixed model of _P_ 12, _P_ 50, _P_ 88, and _P_ ex pooled over both 1105 locations. See Notes S1.2 for model structure. Shown are the posterior means of the parameter 1106 estimates with their MCMC standard error, 95 % credible interval, potential scale reduction 1107 factor 𝑅̂, bulk and tail effective sample sizes and a Bayesian pseudo-p-value (only shown for 1108 fixed effects, see Notes S1.4). Symbols indicate parameters credibly different from zero on the 1109 0.1 (.), 0.05 (*), 0.01 (**) and 0.001 level (***). 1110 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 1111 **Table S2** Results of the multi-response mixed model of _P_ 12, _P_ 50, _P_ 88, and _P_ ex with separate 1112 parameters for Australia and Germany. See Notes S1.2 for model structure. Shown are the 1113 posterior means of the parameter estimates with their MCMC standard error, 95 % credible 1114 interval, potential scale reduction factor 𝑅̂, bulk and tail effective sample sizes and a Bayesian 1115 pseudo-p-value (only shown for fixed effects, see Notes S1.4). Symbols indicate parameters 1116 credibly different from zero on the 0.1 (.), 0.05 (*), 0.01 (**) and 0.001 level (***). 1117 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 1118 **Table S3** Results of the multi-response mixed model of _T_ 12, _T_ 50, _T_ 88, and _T_ ex pooled over both 1119 locations. See Notes S1.2 for model structure. Shown are the posterior means of the parameter 1120 estimates with their MCMC standard error, 95 % credible interval, potential scale reduction 1121 factor 𝑅̂, bulk and tail effective sample sizes and a Bayesian pseudo-p-value (only shown for 1122 fixed effects, see Notes S1.4). Symbols indicate parameters credibly different from zero on the 1123 0.1 (.), 0.05 (*), 0.01 (**) and 0.001 level (***). 1124 [bioRxiv preprint doi: https://doi.org/10.64898/2026.11.05.764619; this version posted November 12, 2026. The copyright holder for this](https://doi.org/10.64898/2026.11.05.764619) preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in [perpetuity. It is made available under a CC-BY 4.0 International license.](http://creativecommons.org/licenses/by/4.0/) 1125 **Table S4** Results of the multi-response mixed model of _T_ 12, _T_ 50, _T_ 88, and _T_ ex with separate 1126 parameters for Australia and Germany. See Notes S1.2 for model structure. Shown are the 1127 posterior means of the parameter estimates with their MCMC standard error, 95 % credible 1128 interval, potential scale reduction factor 𝑅̂, bulk and tail effective sample sizes and a Bayesian 1129 pseudo-p-value (only shown for fixed effects, see Notes S1.4). Symbols indicate parameters 1130 credibly different from zero on the 0.1 (.), 0.05 (*), 0.01 (**) and 0.001 level (***). 1131