Pipe Cavitation Parameters Reveal Bubble Embolism Dynamics in Maize Xylem Vessels across Water Potential Gradients

: Maize, a crop of international relevance, frequently undergoes xylem embolism due to water shortage, negatively impacting growth, yield, and quality. Consequently, a reﬁned comprehension of xylem embolism is vital for enhancing maize cultivation. Notwithstanding extensive research and the generation of analytical models for embolism mechanisms, prevalent models often disregard crop-speciﬁc hydraulic processes and the formation of embolisms via air bubbles in the xylem conduit. In this research, we present an inventive model applying pipe cavitation parameters to discern water potential and bubble formation in maize leaf xylem. The model integrates pivotal physiological traits of the maize–leaf count, leaf vein count, and diameter of xylem vessels—demonstrating robust correlations. Furthermore, we constructed Percent Loss of Conductivity (PLC) curve based on water potential and compared it with our model, offering interval data to observe embolization events triggered by air bubbles. Utilizing experimental data, our novel cavitation-parameter-based model effectively corresponds with observed bubble phenomena and appropriately characterizes water transport in plant xylem conduits. This method enabled us to observe the transition from bubble occurrence to cavitation embolism microscopically, which aligned with the embolism intervals provided by the model. This procedure reveals potential trends in bubble-induced embolism and deepens our knowledge of microscopic plant hydraulics and crop embolism. This work establishes a basis for understanding the generation of bubble embolisms in maize, assists in evaluating maize-plant water status for efﬁcient water supply management throughout the growth cycle, and contributes towards potential water management strategies for maize.


Introduction
Maize, also referred to as corn (Zea mays), is a versatile and extensively cultivated crop that has served as a vital food source for several millennia [1].Due to its high nutritional content and ability to be processed into various products, corn has become a staple crop that holds a significant role in global agri-food systems [2][3][4].However, maize production necessitates substantial amounts of water, and water scarcity may negatively impact plant growth and development, leading to smaller organs, inhibited flower production, and reduced seed filling [5].The maintenance of optimal yield and quality of corn relies heavily on effective water management practices [6].In this regard, assessing the water status of maize proves to be a valuable tool for managing water supply during crop growth.Among the various methods available, measuring pre-dawn leaf water potential to construct the percentage loss of conductivities (PLC) is a superior technique, both in terms of simplicity and accuracy, when compared to relative evapotranspiration or soil water potential [7].This underscores the criticality of leaf water in the growth of maize [8,9].
Plants rely on a complex network of xylem cells to transport water under negative pressure [10].This negative pressure, however, makes the water column susceptible to Agriculture 2023, 13, 1867 2 of 17 interruption by large bubbles, leading to embolism formation and the blockage of water flow within the plant [11].Factors such as limited soil water or high evaporative demand can increase tension in the water column, making embolism formation more likely [12].If water stress persists, the hydraulic conductivity of the xylem system progressively declines, eventually resulting in its failure, which can have serious consequences for the plant's photosynthetic performance and even lead to organ dieback [13].
Embolisms can form due to drought-induced tension reaching a critical threshold, where air is aspirated through pit membranes separating adjacent conduits [14], or gas bubbles spontaneously nucleate from dissolved gas in the xylem sap [15].The negative pressure inside the xylem conduits causes the gas to expand rapidly and forces water into connected neighboring conduits.Embolisms can also form following freeze-thaw events, where the crystallization of liquid water forces dissolved gas out of solution, leaving gas bubbles that expand to fill the conduit when tension is subsequently applied to the xylem sap [16][17][18].Wounding and pathogen infections can also result in air entry into the xylem network, leading to non-functional xylem conduits that disrupt the transport and distribution of water and nutrients throughout the plant [19].Despite the negative physiological impacts, most plant species live with some proportion of their xylem rendered non-functional due to embolism [20].
Extensive research into plant xylem embolism has led to the development of models to analyze the state of bubbles at the time of embolism [21].Cavitation is a process involving the formation, growth and collapse of gas-and vapor-filled bubbles in low-pressure fluids, and has been shown to be formed by the expansion of bubbles through pre-existing cracks or by the seeding of air in the pores of pit membranes [22].The cavitation number serves as a predictive index of cavitation phenomena and potential impacts, illustrating the propensity for cavitation and enabling the prediction of bubble states during cavitation [23].In scenarios of unhindered soil moisture supply, the sap flow rate correlates with solar radiation and vapor pressure disparity [24][25][26].Conversely, under drought conditions, both the stem water potential and sap flow rate experience significant reductions [27].Models grounded in field measurements have demonstrated a robust correlation between the sap flow rate and water potential [28].
Employing high-resolution micro-CT imaging, the complex three-dimensional architecture of the grape xylem network has been meticulously delineated.Simulations indicate a significant 40% augmentation in its resistance to embolism spread due to structural variability, compelling us to re-evaluate the commonly held view of how a single embolism permeates the entirety of the network [29].Concurrently, the principles of fluid dynamics in engineered pipelines, extensively applied in fields like medicine and engineering, resonate intriguingly with this natural system [23,30].The successful simulation analysis of the cavitation number, particularly in minuscule 40µm conduits, underscores its potential applicability in microscopic catheters [31].
Rising global crop mortality rates, tied to climate change and weather events, underscore the need for a better understanding of crop mortality during water stress [32].A recent breakthrough has identified xylem cavitation as a primary cause of plant mortality during drought [33].The visualization of cavitation (xylem-bubble diffusion and formation within the xylem water column) showed a varying susceptibility to cavitation between individuals, within individuals, and within tissues [34].Xylem cavitation not only results in plant death but also slows recovery from drought, underlining its significance.Contemporary hydraulic models provide valuable frameworks to study complex plant-water interactions, including factors like leaf and soil water potentials, stomatal conductance, and transpiration under a variety of environmental conditions [35][36][37].These models' efficacy hinges on the criteria established by percent loss of conductance (PLC) curves, which quantify changes in xylem hydraulic conductance as a function of water stress [38][39][40].Persisting challenges lie predominantly in the detailed portrayal of microscopic bubble conduction within leaf xylem conduits, a process that triggers changes in water potential and incites embolism.Current modelling paradigms tend to concentrate on tree-specific hydraulic mechanisms [41][42][43], while often neglecting those pertinent to crops and falling short in accommodating embolism within the micro xylem canals.In this study, we endeavor to ameliorate these gaps by unveiling a new model that meticulously accounts for bubble dynamics and water potential.We further substantiate this model with empirical observations of bubble formation and their relationships to PLC curves.The novel insights garnered from this work unquestionably enrich our understanding of microscopic crop hydraulics, while also offering a new assessment method for combating drought-induced embolism.

Materials
This experiment was conducted in a maize seed plantation in Mengrun Township, Mengla County, Xishuangbanna Dai Autonomous Prefecture, at latitude and longitude 21 • 56 N and 101 • 13 E, respectively, with maize selected for experimentation in March of each year from 2021-2023 (Table 1).The maize growth cycle was divided into seven growth stages, namely the germination and seedling stage, early ear-expansion stage, late vegetative growth stage before flowering, flowering stage, lag phase, effective grain-filling stage, and late grain-filling stage.The maize specimens investigated in this study demonstrated swift growth during the initial five stages, leading to an increased susceptibility to damage due to their frail state.However, the final two stages were marked by a deceleration in growth, resulting in the plants and their foliage reaching a more durable and stable phase, optimal for experimental purposes.Consequently, our experiments focused on maize plants at the terminal grain-filling stage, including the late grain-filling stage.Cultivation of the seedlings spanned a period of 15 days, followed by the random collection of mature leaves approximately 55 days post-transplantation, in line with the optimal growth parameters of this specific maize variety.In an attempt to simulate the plant's natural environment, the harvested leaves were submerged in water and propagated in plastic receptacles.The leaves were uniformly distributed in a well-ventilated room under visible light conditions, with a regulated temperature of 20 ± 3 • C (Figure S1).To maintain their inherent physiological conditions, all experimental procedures were executed within a 48 h period following the leaf harvest.This rigorous approach facilitates a meticulous exploration of the maize growth cycle, ensuring accuracy and replicability.

Theory
Plant cavitation has been extensively investigated, although our understanding of the precise mechanisms underpinning bubble behavior in xylem vessels remains incomplete.Phenomena like embolism propagation via pits and similar duct structures have been noted in earlier work [29].Moreover, previous research has dissected cavitation into two broad categories, encompassing three distinct mechanisms [44].In the multifaceted process of plant cavitation, the behavior of xylem vessels encompasses two crucial stages: the initiation of bubbles subsequent to air-seeding through xylem pit membranes, and the extension of these bubbles, culminating in cavitation occlusions [45].The dynamics of this bubble extension warrant meticulous examination due to its instrumental role in embolism For instance, in Oriental Red, empirical data elucidate that during the transition of bubbles to occlusions within unblemished xylem vessels, there is a discernible generation of acoustic or ultrasonic waves.Concurrently, water within the xylem vessel experiences surface tension, prompting a rapid change in the bubble radius and leading to the formation of a phase boundary layer, which ultimately results in a stable occlusion [46].Cavitation, based on the structural and hydrodynamic attributes of the cavitation zone, can be subcategorized into four types: wandering, stationary, vortex, and oscillatory [47].Stationary cavitation manifests post the incipient cavitation's critical state.When water is displaced from the sidewalls of a tortuous object or passage, unstable cavities form on the wall surface.These appear static to the unaided eye but are in reality continuously fluctuating.Stationary cavities may occasionally grow and refill from the rear, triggering a cyclic process of cavity collapse.Stationary cavitation is observed where the solid wall's pressure approximates the vapor pressure (or critical tensile strength), and local cavitation at this point facilitates fluid deviation and the formation of a stationary cavitation cavity.
The focus of this study is cavitation, a process that entails vapor bubble formation in a fluid's low-pressure region.Predominantly, this is instigated as alterations in system flow rates push fluid pressure levels to converge upon the fluid's saturated vapor pressure for a given temperature.This culminates in rupture conditions where the extant pressure effectively becomes the differential between the ambient pressure and the fluid's vapor pressure [48].In botanical contexts, it is more apt to delineate this as the difference between the gaseous pressure (P g ) and the absolute pressure on the surface of the xylem's liquid content (P l ) [45].To prognosticate cavitation dynamics and discern its prospective ramifications in plant systems, the cavitation number has been adopted as an instrumental metric: where P g is the gas pressure; the absolute pressure of the xylem liquid at its surface (P l ); V e is the sap flow; and ρ is a liquid density.The cavitation number can be calculated for any flow, and when cavitation begins within the flow, this number is known as the critical cavitation number or incipient cavitation number.As illustrated in Equation (1), cavitation will not occur when σ is large, which can be due to high rupture pressure or low flow velocity.By increasing flow velocity and reducing ambient pressure, the cavitation number can be decreased, and when it reaches a critical (incipient) level, cavitation bubbles start forming within the flow.The intensity of cavitation and the subsequent cavitation erosion are influenced as the cavitation number further decreases.
Our investigative endeavor into plant embolism is anchored in the foundational air-seeding theory, which finds its theoretical grounding in the distribution of pit sizes.Air-seeding has emerged as a crucial phenomenon, initiated when the hydrostatic pressure within the water column recedes to levels incapable of opposing the encircling atmospheric pressure.This leads to air infiltration, colloquially termed "seeding", into the water column.This entrapped air consequently acts as a nidus for cavitation bubble generation, which possesses the potential to metamorphose into embolisms over time.Within the purview of xylem's sap flow and its water-potential dynamics, our research pivoted towards assimilating the intricate nuances of bubble dynamics, thereby harmonizing principles drawn from the realm of microchannel cavitation.This synthesis facilitated the discernment of the cavitation number parameters, encapsulating the journey from nascent bubble origination to prolific embolism culmination.In our quest to decode the intricate relationship between varying rupture pressures and the dynamics of xylem sap flow, we rigorously examined cavitation numbers, specifically concentrating on the notable values: 1, 0.5, 0.1, and 0.05.Guided by the velocity of the sap flow and its corresponding water potential, we categorized post air-seeding events into four pivotal stages reflecting the progression of cavitation bubbles: the emergence of incidental cavitation bubbles, the onset of cavitation bubble formation, the stabilization of these bubbles, and the eventual surge of cavitation bubbles.It is precisely these proliferating, stable bubbles that herald the commencement of plant embolism.During cavitation, a smaller initial cavitation number implies a lower likelihood of occurrence, while a larger initial number suggests a higher probability of cavitation due to bubble formation.At the moment of bubble inception, a smaller cavitation number is associated with increased bubble production and more pronounced cavitation, while a larger number signifies lesser bubble generation and a less intense cavitation.Theoretically, bubble generation should occur when σ ≤ 1, with stable bubbles forming when σ ≤ 0.5 [49].
Upon the reperfusion of embolized xylem vessels, we observed that minute air bubbles frequently re-form and adhere to hydrophobic fissures in the vessel walls, a behavior strongly modulated by the spatial distribution and biophysical characteristics of xylem pits and their associated membranes [50].The xylem pressure increases, extracting air from the fissures and resulting in the regeneration of bubbles.According to Henry's Law and Fick's Law, if the gas pressure within a bubble exceeds the atmospheric pressure, the air must dissolve in the neighboring water and disperse [51].Based on the empirical formula by Yang and Tyree [52,53], it takes over 10 h to fully recover conductivity when the xylem pressure is at 0 kPa.In consonance with prior investigations, the involvement of the injection of water into glass capillaries yielded heterogeneity in the bubble behavior; most notably, we observed instances of persistent air bubbles, which exhibited a stable adherence to the interior walls of the capillaries for a period extending to a minimum of three days [45].This finding elucidates that air dissolution exerts a comparatively negligible influence on cavitation within xylem vessels as opposed to the dynamics of bubble expansion.We contend that air dissolution in water does not serve as a pivotal mechanism affecting bubble formation in xylem vessels' cavitation.Specifically, we posit that the molar quantity (n) of the gas remains invariant throughout the process of bubble expansion.To streamline our analysis [50], we operated under the assumption that bubble expansion occurs isothermally.Consequently, within the scope of our analytical framework, the hydrostatic pressure (P) is singularly contingent upon the dynamically variable bubble radius (r), which itself is circumscribed by the anatomical constraints imposed by the radius of the xylem vessels.
when a bubble reaches equilibrium, its rupture pressure (2k/r) is offset by the pressure differential between the gas pressure (P g ) and the absolute pressure of the xylem liquid at its surface (P l ).
One study discussed the observation of saplings growing on the edge of the Fraxino-Aceretum forest in western Germany, where a model was designed to directly link sap flow rates to water potential [54].A strong correlation between the measured sap flow rates and simulated sap flow rates was also confirmed, enabling the model to be employed for data interpretation between the sap flow rates and water potential in xylem conduits (1).
where v(%) is the percentage of maize sap flow; Ψ w is the maize water potential.
In accordance with liquid flow monitoring assessments executed from the tassel stage through to maturity, we derived a mean maximum sap flow (Vmax) of 90.3 g h −1 .In the expansive corpus of research devoted to plant sap flow, a variety of quantification methodologies have been rigorously explored and elucidated [55][56][57].Notably, the unit g h −1 has been empirically deployed in the specific literature to serve as a viable metric for sap flow [28].To establish a relationship between the water potential within the maize leaf and the xylem-vessel sap flow, we employed a liquid flow rate-water potential model.Observing the sap flow of the vessels, we integrated these data into Equation (1), yielding the following expression: where σ is the cavitation number; k = 0.073 Nm −1 signifies the coefficient of water surface tension at a temperature T = 293 K [46]; r is the bubble radius; V max is the maximum daily stem flow rate; ρ is the sap flow density; and Ψ w is the maize water potential.Drawing upon the theoretical foundation of air-seeding and its subsequent cavitation dynamics, the equation aptly delineates the intricacies of bubble formation within the xylem vessel of maize, modulated by the sap flow rate and pressure (Figure 1).
where σ is the cavitation number; k = 0.073 Nm −1 signifies the coefficient of water surface tension at a temperature T = 293 K [46]; r is the bubble radius; Vmax is the maximum daily stem flow rate; ρ is the sap flow density; and Ψ is the maize water potential.Drawing upon the theoretical foundation of air-seeding and its subsequent cavitation dynamics, the equation aptly delineates the intricacies of bubble formation within the xylem vessel of maize, modulated by the sap flow rate and pressure (Figure 1).

Physiological Indicators
Maize, an angiosperm with slender leaves and well-structured transport tissues, exhibits an orderly arrangement of vascular bundles.Its main xylem is comprised of uniform vessels.Given its propensity to undergo cavitation under moderate water stress [58,59], maize serves as an exemplary subject for observing the bubble-producing cavitation process.
The daily average maximum sap flow rate in maize, as identified by the measurements Qs, represents the average sap flow rate in the stem's xylem.This foundational metric was then calibrated to illuminate the sap flow dynamics within individual leaf xylem vessels.For the purpose of our investigation into the complex phenomena of plant embolism, we conceptualized the leaf xylem vessels, cylindrical vessels that, despite displaying variations in dimensional attributes, exhibit consistent functional properties, punctuated by a spatially intermittent distribution of xylem pits and nuanced pit membrane characteristics.This model necessitated the identification of several geometrical parameters, including the number of leaves (n1), the number of veins per leaf (n2), and the radius of the xylem in vein (R).Utilizing the equation Q = A × V , where Ql, Al, and Vmax denote the flow rate, cross-sectional area, and maximum velocity in individual leaf xylem vessels [60].Respectively, we reformulated the cross-sectional area in terms of radius as A = π × (2R) and translated Qs to Ql via Q = Q × n × n .Through these Figure 1.Bubbles within the xylem vessels of maize leaves.V denotes the flow rate of sap within the xylem, Ψ w signifies the pressure of the surrounding water potential, and r is the radius of the resulting bubble.P g represents the gas pressure inside the air pie, P w indicates the water surface tension, and P 1 is the pressure of the xylem sap, separated by the phase boundary layer.

Sample Processing 2.3.1. Physiological Indicators
Maize, an angiosperm with slender leaves and well-structured transport tissues, exhibits an orderly arrangement of vascular bundles.Its main xylem is comprised of uniform vessels.Given its propensity to undergo cavitation under moderate water stress [58,59], maize serves as an exemplary subject for observing the bubble-producing cavitation process.
The daily average maximum sap flow rate in maize, as identified by the measurements Qs, represents the average sap flow rate in the stem's xylem.This foundational metric was then calibrated to illuminate the sap flow dynamics within individual leaf xylem vessels.For the purpose of our investigation into the complex phenomena of plant embolism, we conceptualized the leaf xylem vessels, cylindrical vessels that, despite displaying variations in dimensional attributes, exhibit consistent functional properties, punctuated by a spatially intermittent distribution of xylem pits and nuanced pit membrane characteristics.This model necessitated the identification of several geometrical parameters, including the number of leaves (n 1 ), the number of veins per leaf (n 2 ), and the radius of the xylem in vein (R).Utilizing the equation Q l = A l × V max , where Q l , A l , and V max denote the flow rate, cross-sectional area, and maximum velocity in individual leaf xylem vessels [60].Respectively, we reformulated the cross-sectional area in terms of radius as Through these mathematical manipulations, we derived an equation that succinctly captures the interplay between flow rate, conduit diameter, and fluid velocity: Sampling was conducted over three consecutive years: 2021, 2022, and 2023.Each year, seven maize plants were randomly chosen for sampling from the experimental field as illustrated in Figure 2a, and the quantity of leaves on each maize plant was recorded.In the manner shown in Figure 2b, a total of 21 maize plants were randomly selected across the three years.From each plant, five leaves were randomly picked.Post-sampling, leaves were immediately immersed in water.Images of the leaf veins were then captured using a Canon D550 camera, and the number of prominent leaf veins per leaf was manually tallied.
Below, we show manual cross-sections of the lateral veins of maize leaves (Figure 2c).As in Figure 2c, the xylem cross-sectional area radius of the leaf veins was counted in the unstained state within 48 h, at 20°, maintaining the physiological activity of the maize leaves, and the sections were observed.We observed two large xylem vessels (V) in the lateral veins, each with a diameter close to 40 µm.Above the two large xylem vessels were protoxylem (arrows), which were approximately 20 µm in diameter (Figure 2c: unstained cross-section of lateral veins of maize leaves.Scale bar = 50µm).
We have also present hand cross sections of the lateral vein of a maize leaf below (Figure 2c).We observe two large metaxylem vessels (V) in the lateral vein, each with a diameter near 40 µm, and one protoxylem lacuna (arrow), with a diameter of approximately 20 µm.(Figure 2c

Measurement of Stem Flow Rate and Leaf Water Potential in Maize
To achieve an intricate understanding of both the stem flow rate and leaf water potential in maize, we employed a robust and technologically advanced approach.For the quantification of the stem flow rate, a highly precise Sartorius BSA 1245 analytical balance was employed, which underwent calibration prior to each measurement session to guarantee methodological rigor.Freshly cut maize stem segments were placed in a naturally air-dried experimental apparatus designed to simulate embolism conditions, thereby promoting gravitationally influenced xylem water flow.Concurrently, the leaf water potential was assiduously quantified employing a PMS 1505D-EXP Portable Plant Water Potential Pressure Chamber from the U.S.-based PMS Instrument Company.Calibration steps were performed in accordance with the manufacturer's stipulations to eliminate any potential measurement biases.Leaf samples were extracted from maize plants during the Below, we show manual cross-sections of the lateral veins of maize leaves (Figure 2c).As in Figure 2c, the xylem cross-sectional area radius of the leaf veins was counted in the unstained state within 48 h, at 20 • , maintaining the physiological activity of the maize leaves, and the sections were observed.We observed two large xylem vessels (V) in the lateral veins, each with a diameter close to 40 µm.Above the two large xylem vessels were protoxylem (arrows), which were approximately 20 µm in diameter (Figure 2c: unstained cross-section of lateral veins of maize leaves.Scale bar = 50 µm).
We have also present hand cross sections of the lateral vein of a maize leaf below (Figure 2c).We observe two large metaxylem vessels (V) in the lateral vein, each with a diameter near 40 µm, and one protoxylem lacuna (arrow), with a diameter of approximately 20 µm.(Figure 2c: lateral veins of maize leaves without staining in cross section.Scale = 50 µm).

Measurement of Stem Flow Rate and Leaf Water Potential in Maize
To achieve an intricate understanding of both the stem flow rate and leaf water potential in maize, we employed a robust and technologically advanced approach.For the quantification of the stem flow rate, a highly precise Sartorius BSA 1245 analytical balance was employed, which underwent calibration prior to each measurement session to guarantee methodological rigor.Freshly cut maize stem segments were placed in a naturally air-dried experimental apparatus designed to simulate embolism conditions, thereby promoting gravitationally influenced xylem water flow.Concurrently, the leaf water potential was assiduously quantified employing a PMS 1505D-EXP Portable Plant Water Potential Pressure Chamber from the U.S.-based PMS Instrument Company.Calibration steps were performed in accordance with the manufacturer's stipulations to eliminate any potential measurement biases.Leaf samples were extracted from maize plants during the early morning hours to sidestep the confounding influence of diurnal water-potential fluctuations.Once harvested, the leaves were expeditiously transferred into the pressure chamber and their water potential was determined.

Quantifying Hydraulic Conductivity Loss for Xylem Vulnerability-Curve Generation
Leaf vulnerability curves and xylem anatomy: Utilizing a natural air-drying technique [14], we collected the lower leaves of maize cobs in the early morning.These leaves were isolated and placed in a chamber to undergo dehydration for varying durations to establish different water potential gradients.Leaf vulnerability curves were then derived by measuring the hydraulic conductivity corresponding to each water potential.The functional leaf intended for measurement was enveloped in tin foil to maintain equilibrium in the whole leaf's water potential.The xylem water potential was estimated by examining the leaf's end part, with water conductivity measured after a 15 min equilibration period.We then cut the leaf end under water for a length of 15 cm and measured its water potential.Following this, we cut the leaf's middle underwater, ensuring it was 7 cm long, trimmed the vein segment to a width of 0.7 cm, and maintained a water head height between 40-50 cm.We defined the specific leaf hydraulic conductivity (Kl) as the ratio of the initial hydraulic conductivity to the leaf area, and calculated the hydraulic conductivity per unit pressure gradient (Kh, MPa m −1 ) using the formula Kh = F/(d P/dx), where F represents the water flow (kg s −1 ) and d P/dx is the hydraulic conductivity per unit stem length (MPa m −1 ).
We determined the percentage loss of hydraulic conductivity (PLC) as (1 − Kh/Kmax) × 100%, with the relationship between PLC and negative xylem pressure forming the leaf xylem vulnerability curve.We fitted the correlation between PLC and negative xylem pressure in stems using the Weibull distribution function: In this mathematical formulation, the parameter xc designates the absolute magnitude of the negative xylem pressure that correlates with a 64.2% reduction in hydraulic conductivity.The variable d serves as a metric of the curve's steepness at a xylem pressure antithetical to −xc, with larger d values signifying a more precipitous curve.Concurrently, A signifies the asymptotic maximum of the percentage loss of hydraulic conductivity (PLC) under conditions of extreme negative xylem pressure.The parameter k acts as a sophisticated scaling factor, meticulously fine-tuning the xylem pressure variables (x) to ensure precise concordance with the observed fluctuations in hydraulic conductivity.

Small Flow Method
We measured the diameters of vascular conduits within the veins of maize leaves from the spring season of 2023.We formed three distinct experimental groups, each composed of nine leaves selectively chosen due to their vascular conduit diameters spanning between 60 to 65 µm.These leaves were subjected to a succession of experimental treatments to emulate the varying stages of primary leaf cavitation during their growth.
In the Control Group (CG), the basal portions of the maize leaves were submerged in water, and immediate measurements were conducted upon exposure to ambient air, thereby simulating the water potential under physiologically normal conditions.The Before Embolization Group (BE) involved a brief air exposure of the leaves for a minute postimmersion, imitating the onset of embolization within leaf vein conduits when minuscule air bubbles are created by air pressure.Lastly, the After Embolization Group (AE) allowed the leaves to remain exposed to air for an extended duration to enable the close monitoring of the progression of embolization in these conduits.These distinct treatments were assigned randomly, and water potential measurements of individual leaves in each group were conducted three times to ensure reproducibility.
To quantify the water potential within leaf vein conduits at differing bubble formation stages, ten perforations in the upper third of each leaf were created using a 1 cm hole puncher.The perforated leaf discs were then soaked in a sucrose solution and placed in a 2 mL test tube containing a sucrose solution gradient.This mixture was thoroughly stirred and methyl blue was added.After aspirating 0.1 mL of this blend, it was reintroduced into the original sucrose solution tube.The final resting position of the leaf disc within the sucrose solution provided an approximation of the osmotic potential equivalence between the leaf and the solution at this stage.ψw = ψπ = −CiRT (7) where ψw refers to the osmotic potential, ψπ represents the leaf water potential, C denotes the small-scale current-stopped sucrose concentration in mol/L, R is the gas constant (0.008314 L-Pa/mol-K), T is the absolute temperature (293 K), and i is the dissociation coefficient, with the sucrose solution taking a value of 1.

Physiological Parameters of Maize
Conducted between 2021 and 2023, our research involved the annual sampling of seven randomly selected maize plants from our experimental field (Figure 2a).The leaf count of these specimens was duly noted.
During this triennial span, the leaf count for mature maize plants fluctuated mildly between 9.29 and 10.57, with the highest average appearing in 2021 (Table 2).The consistency of the data is striking; throughout the years, the leaf count for each mature maize plant remained steady, typically close to ten, with no significant yearly deviations.A robust positive correlation emerged between maize leaves and leaf vein counts across all categories (r = 0.82, p < 0.05), pointing towards effective photosynthesis, regular intra-plant transport exchange, and beneficial growth conditions for maize [61].Physiological indicators, encompassing leaf and vein counts, and the diameters of leaf vein conduits demonstrated the varying trends across the years under differing environmental conditions.The peak values for these parameters were recorded in 2021, with an average leaf count of 10.5, a vein count of 43, and a mean leaf vein conduit diameter of 58 µm.The diameter of xylem vessels remained somewhat uniform, falling within 30.27 and 40.08 µm from 2021 to 2023.However, within the 2022 samples, the diameter of xylem vessels presented a wide range from 18.43 to 46.19 µm.This variability likely arose from differing growth conditions within the maize sections from where the samples were randomly selected.Leaf length, a critical factor for estimating leaf area, maintained general stability over the three years, with lengths between 57.80 and 67.20 cm.On average, the diameters of mature maize leaves exhibited substantial overlap throughout the three years.A comprehensive analysis of maize samples from 2021, 2022, and 2023 revealed relative uniformity in the physiological indicators-leaf count, vein count, average leaf diameter, and leaf length-across the three years, affirming their appropriateness as parameters (Table 2).

Establishing the Bubble Radius-Sap Flow Rate-Water Potential Model
The formation of cavitation, or the explosive creation of air bubbles leading to embolism, is significantly influenced by the interplay of sap flow and water potential in maize leaf xylem [62].Employing physiological indicators, the sap flow in maize xylem was converted, with the leaf count set to 10, the vein count to 40, and the average leaf diameter to 62.43 µm, to yield optimal preliminary xylem sap-flow parameters, according to Equation ( 5), which was then incorporated into Equation ( 4).
We incorporated smaller cavitation numbers (0.1 and 0.05) to examine the initial stage of bubble generation preceding embolism, according to the model.This stage corresponds to the generation of numerous small bubbles and intense cavitation.The cavitation process occurs in tandem with variations in the bubble radius, leading to alterations in the sap flow rate and water potential within the xylem.This phenomenon occurs intensely at a cavitation number of 0.05, leading to bubble accumulation within maize leaf vascular conduits, subsequently transitioning to stable embolism as the bubbles enlarge and the cavitation coefficient increases beyond 1.It should also be noted that bubble formation is more likely under conditions of high sap flow and a low water potential (Figure 3a).The analysis, based on cavitation numbers, revealed that there are three bubble states preceding embolism during cavitation, namely, incidental cavitation bubbles (Ib), the generation of cavitation bubbles (Gb), stable cavitation bubbles (Sb), and a large number of cavitation bubbles (Lb).These states correspond to cavitation numbers within the ranges of σ ≥ 1, 1 ≥ σ ≥ 0.5, 0.5 ≥ σ ≥ 0.1, and 0.1 ≥ σ ≥ 0.05, respectively.The moment bubble accumulation triggers cavitation, few bubbles are produced in the Ib state, making it difficult for embolism to form within the vascular conduits.The Gb state, however, begins to produce active bubbles, enabling bubble accumulation within this range.The Sb state is characterized by bubble stabilization, where an increased number of bubbles begin to connect via the phase boundary layer, naturally occurring in the Lb state.Here, large-radius bubbles accumulate, and cavitation generally occurs within the xylem, leading to the formation of one cavity after another, eventually expanding and merging to form larger cavities.This process results in the embolization of xylem vessels and the loss of the water-transport function.
This model suggests that the conditions for cavitation are present at sap flow rates between 8-140 g h −1 .Cavitation exists at varying water potentials and sap flow rates, with the Ib, Gb, Sb, and Lb sections defined by the cavitation number overlap, indicating that cavitation occurs at each sap flow rate and water potential.The bubbles produced are in various states and have different radii, suggesting that cavitation is a multi-core gas nucleus, with bubbles expanding at varying rates depending on their radii and the speed of their formation (Figure 3c).As more bubbles are produced, the sap flow rate decreases to the point immediately after bubble generation and then further drops until it reaches the region below the Gb level.At this point, cavitation reaches its endpoint, with the gas nucleus expanding to form a cavity, and the sap flow rate and water potential within the maize leaf xylem gradually stabilize, signifying the occurrence of a stable embolism (Figure 3b).During cavitation, the in-xylem sap flow of different leaf veins with an identical water potential, larger air bubbles require a higher water potential when cavitation occurs in xylem with larger radii, implying that larger conduits are less susceptible to blockage (Figure 3d).Depending on the cavitation number, the region most likely to produce bubbles ranges from L to S to G to I, suggesting that smaller bubble radii are closer to the L region when cavitation occurs (Figure 3d).
region below the Gb level.At this point, cavitation reaches its endpoint, with the gas nucleus expanding to form a cavity, and the sap flow rate and water potential within the maize leaf xylem gradually stabilize, signifying the occurrence of a stable embolism (Figure 3b).During cavitation, the in-xylem sap flow of different leaf veins with an identical water potential, larger air bubbles require a higher water potential when cavitation occurs in xylem with larger radii, implying that larger conduits are less susceptible to blockage (Figure 3d).Depending on the cavitation number, the region most likely to produce bubbles ranges from L to S to G to I, suggesting that smaller bubble radii are closer to the L region when cavitation occurs (Figure 3d).

Comparison of Cavitation Emergence in Leaf PLC and Model Predictions
Figure 4a presents the correlation between the percentage loss of conductivity (PLC) and negative xylem pressure (Ψ s ) in XueTian7401 leaf samples, utilizing the Weibull function.The analysis quantified the plant cavitation resistance by assessing the relationship between the water potential and the corresponding degree of xylem conduit embolism.PLC was employed to explore the circumstances within the xylem conduits at each phase, compared to the previously constructed model view (Figure 4b).During the initial bubblegeneration stage (Lb), as PLC began to rise to 27%, a marked decrease in hydraulic conductivity within the maize leaf vascular ducts was observed, with the concurrent formation of numerous small bubbles.This stage aligns with the Lb range of water potential, spanning from Ψs1 to Ψs2.As the PLC further increased to 34%, a deceleration in the bubble growth rate was observed, with the phase boundary layer stabilizing.Upon a further PLC increase to 50%, reaching the Gb range, corresponding to the water potential between Ψs3~Ψs4, an accumulation of bubbles within the conduit was noted.This occurrence led to a significant increase in conduit PLC, a substantial decrease in hydraulic conductivity, the cessation of sap flow due to water scarcity, and air bubbles occupying the substantial conduit space, disrupting plant homeostasis.At this stage, the accumulation of bubbles began to disrupt the phase boundary layer with each newly formed air nucleus.As PLC continued to rise beyond 58%, when the water potential dropped below Ψs4, air pockets within the maize leaf xylem started to interconnect, expanding into large sections of the duct cavity.Consequently, water transport ceased entirely, with air dominating the conduit space.mation of numerous small bubbles.This stage aligns with the Lb range of water potential, spanning from Ψs1 to Ψs2.As the PLC further increased to 34%, a deceleration in the bubble growth rate was observed, with the phase boundary layer stabilizing.Upon a further PLC increase to 50%, reaching the Gb range, corresponding to the water potential between Ψs3~Ψs4, an accumulation of bubbles within the conduit was noted.This occurrence led to a significant increase in conduit PLC, a substantial decrease in hydraulic conductivity, the cessation of sap flow due to water scarcity, and air bubbles occupying the substantial conduit space, disrupting plant homeostasis.At this stage, the accumulation of bubbles began to disrupt the phase boundary layer with each newly formed air nucleus.As PLC continued to rise beyond 58%, when the water potential dropped below Ψs4, air pockets within the maize leaf xylem started to interconnect, expanding into large sections of the duct cavity.Consequently, water transport ceased entirely, with air dominating the conduit space.

Observations of Leaf Conditions at Different Water Potentials during Cavitation
Given the invariability in the diameters of vascular bundle conduits in maize leaves from 2021 to 2023, the 2023 cohort was selected for analysis to guarantee leaf freshness during cavitation occurrence.The examination of the vascular bundle conduit diameters, using the OLYMPUS BX51 microscope, yielded an average diameter parameter of 62.43 μm for the simulation model.Consequently, conduits ranging from 60 to 65 μm in diameter were selected for the study.
Three conditions were simulated regarding the water potential (Ψs) in maize leaves during cavitation: the Control Group (CG), Before Embolism Group (BE), and After Embolism Group (AE).Random images from each of these conditions were selected for presentation.The water potential values for these categories were aligned with the

Observations of Leaf Conditions at Different Water Potentials during Cavitation
Given the invariability in the diameters of vascular bundle conduits in maize leaves from 2021 to 2023, the 2023 cohort was selected for analysis to guarantee leaf freshness during cavitation occurrence.The examination of the vascular bundle conduit diameters, using the OLYMPUS BX51 microscope, yielded an average diameter parameter of 62.43 µm for the simulation model.Consequently, conduits ranging from 60 to 65 µm in diameter were selected for the study.
Three conditions were simulated regarding the water potential (Ψ s ) in maize leaves during cavitation: the Control Group (CG), Before Embolism Group (BE), and After Embolism Group (AE).Random images from each of these conditions were selected for presentation.The water potential values for these categories were aligned with the modeled values corresponding to XueTian7401 leaf Percentage Loss of Conductivity (PLC) within an acceptable margin of error.To preserve leaf activity, all measurements were performed within 24 h of leaf collection.Leaves in the CG were maintained through water immersion.The BE consisted of leaves observed within 1 min of air exposure, with water potential Ψ ranging from −0.72 MPa to −0.87 MPa, corresponding to the Lb region in the model.In the BE, small bubbles began to form in the conduit and the sap flow started to be influenced by the gas presence, corresponding to the Sb region in the model.Notably, large bubble sections were also observed, corroborating the model's prediction of multiple bubble states coexisting at the same water potential.The AE group comprised leaves exposed to air for more than 3 min, at which point the extended air bubbles in the xylem conduit blocked sap flow, with water potential ranging from −1.16 MPa to −1.29 MPa.This stage corresponds to the Gb region in the model.
Comparing the CG, BE, and AE groups, a rapid decrease in leaf water potential was observed upon air exposure, with xylem cavitation ensuing.According to the experimental results, the water potential change caused by cavitation in maize leaves (AE) did not significantly differ from the trend in CG and BE.This discrepancy may be attributed to the atmospheric pressure in the in vitro environment being less than when cavitation occurs within the xylem ducts of maize leaves under natural conditions.These changes in water potential due to leaf cavitation significantly impact the entire water transport network in maize plants, particularly under extreme water-potential stress (Figure 5).We found a mean decrease of 0.24 MPa in the water potential when small bubbles started to form within the ducts of the maize leaf xylem.The water potential dropped a further 0.18 MPa when bubble accumulation led to cavitation within the ducts, falling from −0.80 MPa before cavitation to −1.22 MPa after cavitation under normal physiological conditions (Figure 5).This transition marked a shift from the Lb to Gb interval within the maize leaf xylem ducts, denoting the establishment of a stable cavitation state within the ducts.
xylem conduit blocked sap flow, with water potential ranging from −1.16 MPa to −1.29 MPa.This stage corresponds to the Gb region in the model.
Comparing the CG, BE, and AE groups, a rapid decrease in leaf water potential was observed upon air exposure, with xylem cavitation ensuing.According to the experimental results, the water potential change caused by cavitation in maize leaves (AE) did not significantly differ from the trend in CG and BE.This discrepancy may be attributed to the atmospheric pressure in the in vitro environment being less than when cavitation occurs within the xylem ducts of maize leaves under natural conditions.
These changes in water potential due to leaf cavitation significantly impact the entire water transport network in maize plants, particularly under extreme water-potential stress (Figure 5).We found a mean decrease of 0.24 MPa in the water potential when small bubbles started to form within the ducts of the maize leaf xylem.The water potential dropped a further 0.18 MPa when bubble accumulation led to cavitation within the ducts, falling from −0.80 MPa before cavitation to −1.22 MPa after cavitation under normal physiological conditions (Figure 5).This transition marked a shift from the Lb to Gb interval within the maize leaf xylem ducts, denoting the establishment of a stable cavitation state within the ducts.

Discussion
Plant embolism has intrigued researchers for over a century, with microstructural analyses in recent years unveiling the crucial relationship between air bubbles and embolism formation [63,64].Nevertheless, harmonizing macroscopic observations with microstructural findings using formulas has posed a considerable challenge.In response, we devised a novel formulation of the Bubble Radius-Sap Flow Rate-Water Potential Model

Discussion
Plant embolism has intrigued researchers for over a century, with microstructural analyses in recent years unveiling the crucial relationship between air bubbles and embolism formation [63,64].Nevertheless, harmonizing macroscopic observations with microstructural findings using formulas has posed a considerable challenge.In response, we devised a novel formulation of the Bubble Radius-Sap Flow Rate-Water Potential Model for maize, offering insight into flow rate, water pressure, and bubble dynamics.Initial experimental measures substantiated various bubble alterations within the model's range, emphasizing the utility of the Bubble Radius-Sap Flow Rate-Water Potential Model in detecting xylem cavitation in maize leaves.Moreover, the model offers a platform for tracing potential bubble trends through water-potential and sap-flow-rate investigations, thereby shedding light on the cavitation phenomenon.It is noteworthy that our current formulation did not account for a multitude of underlying mechanisms such as the root pressure, ion transport from xylem parenchyma cells to the xylem vessels, and subsequent water fluxes.These critical components warrant further exploration and integration into the model in future research endeavors.Concurrently, we have also identified a device designed for evaluating plant stem sap flow under stable pressure conditions [65].While this apparatus currently lacks the capacity to reflect the bubble dynamics under varied pressure conditions encountered in our experiments, its scientific design holds substantial promise for future inclusion in our research.This holds the potential to yield data with even greater precision on plant sap flow.
Our model incorporated cavitation parameters to encapsulate the relationship with the liquid flow rate and used the PLC as a comparative water indicator.This approach harmonizes the equation with experimental data.Built on maize plants, our study holds significant implications for understanding the relationships between water uptake and transport and photosynthesis.As such, our findings will play a crucial role in shaping models of maize growth, yield, and crop production mechanisms.They will also help anticipate future species distributions and plant responses to climate change.By aligning the flow rate of maize with the bubble radius-sap flow rate-water potential model, supported by available experimental data, we can better synchronize it to reveal correlations between the maize's physiological state and subsequent yield.In the preliminary comparative analysis using PLC, our model appears to demonstrate a degree of feasibility.However, it is imperative to recognize the prevailing gap in flow velocity scales between hydrodynamic systems studied in existing research on cavitation numbers and those observed in plant physiology.This discrepancy necessitates further investigations to refine the model's parameters.Notably, research specifically targeting the behavior of maize xylem vessels is scant, yet within the broader scope of fluid mechanics, variations in flow velocity and cavitation occurrence manifest differently depending on environmental conditions and the state of the fluid [49].Leveraging the dimensionless nature of the cavitation number enhances the model's general applicability, a feature that undeniably confers significant advantages when addressing problems across diverse fields.It is noteworthy to highlight that studies, such as the one presented by [66], have delved into the intricacies of multi-vessel scenarios within the xylem vessel network.Such insights offer a broader conceptual framework for our subsequent investigations into the expansive water transport mechanisms within the plant xylem.
The symbiotic relationship between plant water uptake and transport and photosynthesis has immense implications for maize growth, yield, crop production modeling, and climate change predictions [28,67,68].Our experimental studies, based on environmental conditions, demonstrated the efficacy of the bubble radius-sap flow rate-water potential model in explaining the generation and expansion of air nuclei, the sap flow rate, and leaf water potential within corn leaves under various treatments.This model aids in comprehending the conditions necessary for cavitation occurrence in maize leaves, laying the groundwork for understanding the formation process of cavitation at the onset of drought-induced embolism.This understanding could pave the way for stabilizing corn yield under such conditions.

Figure 1 .
Figure 1.Bubbles within the xylem vessels of maize leaves.V denotes the flow rate of sap within the xylem, Ψw signifies the pressure of the surrounding water potential, and r is the radius of the resulting bubble.Pg represents the gas pressure inside the air pie, Pw indicates the water surface tension, and P1 is the pressure of the xylem sap, separated by the phase boundary layer.

Figure 2 .
Figure 2. Investigative findings from trial maize plots in Mengrun Township.(a) Tally of leaf quantities in exemplar maize plots; (b) examination of the principal lateral venation in maize foliage; and (c) cross-section study of the vascular bundles within maize leaf veins.

Figure 2 .
Figure 2. Investigative findings from trial maize plots in Mengrun Township.(a) Tally of leaf quantities in exemplar maize plots; (b) examination of the principal lateral venation in maize foliage; and (c) cross-section study of the vascular bundles within maize leaf veins.

Figure 3 .
Figure 3. Model demonstrating the interrelations between the bubble radius, sap flow rate, and water potential in vascular xylem conduits within maize leaves.(a) A three-dimensional depiction of the bubble radius, sap flow rate, and water potential under various simulated cavitation conditions.(b) The correlation between sap flow rate within the conduit and bubble radius under diverse cavitation parameters.(c) Depiction of varying sap-flow-rate intervals in relation to water potential under assorted cavitation states.(d) Demarcation of regions representing different cavitation states, based on bubble radius and water potential under distinct cavitation parameters.

Figure 3 .
Figure 3. Model demonstrating the interrelations between the bubble radius, sap flow rate, and water potential in vascular xylem conduits within maize leaves.(a) A three-dimensional depiction of the bubble radius, sap flow rate, and water potential under various simulated cavitation conditions.(b) The correlation between sap flow rate within the conduit and bubble radius under diverse cavitation parameters.(c) Depiction of varying sap-flow-rate intervals in relation to water potential under assorted cavitation states.(d) Demarcation of regions representing different cavitation states, based on bubble radius and water potential under distinct cavitation parameters.

Figure 4 .
Figure 4. (a) Correlation between the Percentage Loss of Conductivity (PLC) and the water potential within maize leaves.(b) Mapping of the relationship between the derived model for maize leaves and the corresponding PLC, utilizing specific parameters.

Figure 4 .
Figure 4. (a) Correlation between the Percentage Loss of Conductivity (PLC) and the water potential within maize leaves.(b) Mapping of the relationship between the derived model for maize leaves and the corresponding PLC, utilizing specific parameters.

Figure 5 .
Figure 5.Comparison of water potential in maize leaves under physiological conditions (CG), before embolism (BE), and after embolism (AE).

Figure 5 .
Figure 5.Comparison of water potential in maize leaves under physiological conditions (CG), before embolism (BE), and after embolism (AE).

Table 1 .
Climate conditions in the study area.

Table 2 .
An overview of growth characteristics, of a randomly selected sample of 21 maize plants observed from 2021 to 2023.