How do you read a xylem vulnerability curve? The central task is to connect declining xylem water potential with loss of the tissue’s capacity to transport water. In many studies, a vulnerability curve shows percent loss of hydraulic conductivity, or PLC, against negative xylem pressure or water potential. The much-searched P50 is the fitted potential at which 50 percent conductivity loss is estimated under that measurement method. This guide explains xylem vulnerability curves, P12, P50, P88, embolism, hydraulic safety margins and the methodological traps behind their numbers.
Someone asking how to interpret a plant P50 value may want to compare trees, diagnose water-transport limits, or understand why different instruments produce different curves. Those purposes should not be collapsed. P50 is not the water potential at which exactly half the leaves wilt, half the vessels visibly collapse, or half the plants die. It is a particular hydraulic threshold defined by a measurement and fitting procedure.
A vulnerability plot is also not identical to the pressure–volume curve used to estimate leaf turgor loss point. Pressure–volume analysis focuses on living tissue hydration, osmotic potential and turgor. Xylem vulnerability analysis focuses on the conducting system’s susceptibility to embolism-related loss of transport function. The axes may share MPa, but their biological questions differ.
The safest reading follows a chain: identify the xylem material and method; determine whether the vertical axis measures PLC, relative conductivity or an optical embolism signal; read the sign convention; inspect curve-fitting decisions; and only then interpret thresholds and ecological significance. The rest of the explanation shows how to preserve each link.
Why water transport depends on xylem
Xylem conducts water along a pathway from roots toward transpiring tissues. Its conduits and surrounding structures enable efficient movement under negative water pressure. The same water columns that allow transport can be vulnerable to cavitation and gas-filled embolism, particularly under sufficiently strong tension.
A plant is therefore not merely a container of water. It is a transport system whose hydraulic function depends on both driving potential differences and the continuity of conducting pathways. Losing conductivity can change the relationship between transpiration and potential.
The term ‘vulnerability’ names susceptibility to loss of transport function under a specified test. It does not automatically describe the amount of water present in a leaf or the probability of plant death under all environmental conditions.
To understand a plot, define the tissue: root, stem, branch, leaf vein or whole plant. A curve from a cut stem segment cannot be substituted without qualification for a whole-tree hydraulic response.
Cavitation and embolism are linked but distinct events
Cavitation is the formation of a gas phase within the water-transport pathway. Embolism describes gas that obstructs the liquid conducting system. These processes are related, but the number of cavitation events need not equal the amount of hydraulic conductivity lost.
A single large conduit and many narrow conduits may contribute different amounts to water flow. That is why counting visible air-filled conduits is not automatically equivalent to measuring the fraction of conductivity lost.
Some modern techniques use optical methods or acoustic emissions to observe or infer embolism-related events. Others measure the flow capacity of excised wood directly. Both can be informative, but their output variables must retain their original names.
When translating a figure, say ‘percent loss of conductivity’ only if conductivity was actually measured or calculated through a justified calibration. If the method measures optical embolism events, describe that signal instead.
What percent loss of conductivity means
Percent loss of conductivity, commonly PLC, compares the conductivity of a measured sample with an estimate of its maximum reference conductivity. A familiar expression is PLC = 100 × (1 − Kinitial/Kmax). It is a relative measurement whose validity depends on the reference and protocol.
Suppose Kinitial is 0.4 in the appropriate units and Kmax is 1.0 under the method. The calculated PLC is 60 percent. That does not mean 60 percent of conduits are blocked by count, or that 60 percent of tissue water has disappeared.
The meaning of Kmax is crucial. It may be obtained after a flushing or other procedure intended to remove embolism. But laboratory interventions can alter the sample, and not every tissue responds cleanly. Compare methods before comparing their PLC values.
Always record the units of conductivity and the way flow or pressure gradient was calculated. The ratio may be dimensionless, but questionable underlying measurements still produce a questionable percentage.
How to read the horizontal axis
Many xylem vulnerability curves plot xylem water potential in MPa, with progressively more negative values indicating stronger tension. Others plot the magnitude of negative pressure or a positive applied pressure. A reversed axis convention can make the same physiological deterioration appear to proceed in opposite directions.
On a conventional signed Ψ axis, −3 MPa is more negative than −1 MPa. If PLC increases as water potential becomes more negative, the visual line may rise while moving leftward or rightward depending on the plotting direction.
Do not interpret graph slope before checking which way the x-axis runs. The word ‘higher’ is especially dangerous here: a ‘higher absolute tension’ often corresponds to a lower signed potential.
A reliable description says ‘conductivity loss increased as xylem water potential became more negative’. This remains meaningful across common plotting conventions and avoids an unnecessary sign reversal.
The general shape of a vulnerability curve
A vulnerability curve often has a low-loss region, a transition where loss increases rapidly, and a high-loss region. Many stem datasets resemble a sigmoid, but other shapes appear and can reflect anatomy, measurement technique or problems in the data.
A sigmoid model does not prove that every sample belongs to the same biological class. When a curve has a long tail or reaches only a partial loss range, the location of the fitted midpoint may depend strongly on the chosen model.
The steepness of the transition is a separate characteristic from the midpoint. Two samples can share the same P50 but differ in how quickly conductivity is lost before and after it.
When reading two curves, compare not only their apparent horizontal positions but also their confidence intervals, data coverage, fitted shape and measurement method. These define how confidently a difference can be interpreted.
Defining P50 precisely
P50 commonly denotes the xylem water potential or pressure associated with an estimated 50 percent loss of hydraulic conductivity. If a fitted vulnerability curve crosses PLC = 50 percent at −3.2 MPa, then P50 is reported as approximately −3.2 MPa under that convention.
More negative P50 values are frequently interpreted as indicating greater resistance to embolism-related loss, because the tested tissue can sustain more negative potential before reaching the same relative conductivity loss. This interpretation assumes comparable methods and tissues.
P50 is not necessarily observed as one raw measurement exactly at 50 percent. It is often obtained by fitting a model to observations on either side of that loss level and solving for the midpoint.
If the measurements never approach 50 percent loss, P50 may be mostly extrapolated. Such an estimate requires explicit uncertainty and should not be presented with a false sense of precision.
What P12 and P88 add
P12 and P88 commonly refer to potentials associated with 12 percent and 88 percent loss of hydraulic conductivity, respectively. They characterize different parts of the vulnerability curve, often informing discussions about onset and severe loss.
Because xylem potential is negative, P88 is typically more negative than P50 for a usual monotonic curve, and P50 more negative than P12. For example, −1.5, −2.7 and −4.0 MPa could represent P12, P50 and P88 in a hypothetical dataset.
Those percentages are conventional summary points, not universal biological switches. P88 is not a guaranteed mortality threshold, and P12 does not necessarily mark the first gas-filled conduit.
Some publications report P20 or other levels rather than P12. Translate the exact percentage and method used instead of normalizing everything to familiar symbols without explanation.
Reading a numerical example
Imagine two illustrative woody samples. Sample A has fitted P50 = −2.5 MPa; sample B has fitted P50 = −4.0 MPa. Under otherwise comparable measurement methods and similar tissues, sample B’s tested xylem shows greater resistance to reaching 50 percent conductivity loss.
Now suppose both samples experience a minimum xylem potential of −2.0 MPa in an experiment. On a simple comparison, both operate at potentials less negative than their P50 values, but their margins differ.
This is not evidence that sample B necessarily grows faster or uses less water. Greater resistance to embolism can coexist with trade-offs in hydraulic efficiency, anatomy or ecology, and those relationships must be measured rather than assumed.
The example is hypothetical. Actual interpretation needs measured curves, confidence intervals and attention to whether field water potential and laboratory vulnerability were determined for comparable compartments.
Hydraulic safety margin and sign conventions
A hydraulic safety margin often compares an observed minimum water potential, Ψmin, with a threshold such as P50. One common signed formulation is Ψmin − P50, which yields a positive number when Ψmin remains less negative than P50.
For example, if Ψmin = −1.5 MPa and P50 = −3.0 MPa, the difference is +1.5 MPa. This indicates the observed minimum was 1.5 MPa on the less-negative side of the P50 threshold in that convention.
Other publications define margins with different ordering or refer to P12 instead of P50. A displayed negative ‘margin’ is not automatically erroneous; the author may have selected a different sign convention.
Always reproduce the exact formula, reference threshold and compartment. A safety margin is a derived comparison of potential values; it is not a direct instrument reading or a guarantee that a plant will avoid failure.
Why the choice of tissue matters
Roots, stems and leaves may differ in vulnerability because their anatomy, conduits and functional roles differ. Some plants also show variation along a branch, with different ages or positions exposing tissues to different hydraulic histories.
A vulnerability curve measured from young twigs may not describe trunk behaviour exactly. Likewise, leaf hydraulic conductance decline may involve non-xylem pathways as well as leaf xylem.
Method descriptions should identify tissue diameter, length, age, species and position. Without those details, apparent differences among datasets may be partly sampling differences.
A world-facing interpretation should resist treating an organism as one uniform pipe. Plants operate through connected organs, each with its own hydraulic properties and possible constraints.
Bench dehydration methods
In bench dehydration experiments, plant material is allowed to lose water in a controlled manner while conductivity and water potential are assessed across samples or stages. This approach can generate curves that relate induced tension to loss of transport function.
The method depends on the water potential being representative of the conducting tissue and on adequate equilibration before hydraulic measurement. The exact drying and sampling sequence matters because different parts of the specimen may not respond uniformly.
Bench dehydration is often used as a comparator for evaluating alternative techniques, but it is still an experimental procedure with potential artefacts from cutting, handling, gas entry and measurement systems.
When comparing publications, identify whether their reference curves actually used bench dehydration with conductivity measurement or a related but different protocol. Naming the method clarifies how the observed loss was generated.
Centrifugation and Cavitron approaches
Some laboratories apply controlled centrifugal force to induce negative pressures in xylem while measuring changes in conductivity. Such systems can be efficient, but the reliability of the result depends on sample geometry, vessel length and apparatus assumptions.
In species with very long conduits relative to the segment dimensions, some centrifuge or pressure methods may produce open-vessel artefacts. These can yield unusually steep or exponential-looking vulnerability patterns and potentially exaggerate vulnerability.
Methodological reviews have documented that curves from different techniques are not always directly interchangeable, especially for ring-porous material with long vessels. The issue is not that every centrifuge result is wrong, but that anatomy and apparatus need validation.
A responsible reading mentions the method when summarizing P50. ‘The centrifuge-derived P50 was…’ conveys information that a bare species-level number hides.
Air injection and pressure-sleeve measurements
Air injection and pressure-sleeve techniques use applied gas pressure to investigate air entry and hydraulic vulnerability under specific experimental conditions. The method can be practical, but its pressure geometry and xylem conduit configuration influence interpretation.
A positive applied air pressure must not be confused with signed xylem water potential without a clear account of the relation used by the method. These values can share MPa units while representing different directly measured quantities.
Long-vessel pathways and sample preparation can affect apparent thresholds in some protocols. Researchers may use controls or independent methods to evaluate whether the observed curve reflects biological vulnerability rather than an artefact.
When reporting cross-study results, separate apparatus-specific evidence from broader biological claims. The curve should be identified by its induced-stress method and its measured response.
Optical vulnerability methods
Optical methods can track visible changes associated with embolism formation through images or image sequences. They often provide spatial or temporal information that bulk flow measurements do not show. However, optical event counts are not automatically identical to conductivity-based PLC.
An optical vulnerability curve may report cumulative embolism events, affected area or a normalized signal against potential. The label on the vertical axis determines what its P-type threshold means. Some studies discuss optical P50, but the precise measured quantity must be retained.
A large conduit can carry disproportionate flow, so an area or count fraction can differ from a flow-loss fraction. Calibration and anatomical context determine whether the quantities can be related meaningfully.
Use the phrase ‘optically detected embolism fraction’ where appropriate rather than silently translating it into ’50 percent loss of hydraulic conductivity’. The latter is a stronger claim requiring additional justification.
Acoustic signals and limitations
Acoustic emission techniques detect signals associated with processes such as cavitation, but interpretation can be complicated by background noise and other mechanical events. An acoustic count is not a universal direct measure of the hydraulic impact of each event.
Timing of emissions can help identify when processes accelerate, yet assigning a P50 from acoustic counts needs a defined normalization and supporting methodology. Not every emission corresponds to an equal share of lost conductance.
Acoustic monitoring can complement direct conductivity or imaging measurements, especially when multiple indicators agree. But an agreement in broad trend is not a license to equate their scales numerically.
Read the legend for the actual signal being plotted. A chart titled ‘vulnerability’ might contain emission counts rather than direct flow loss, and that difference must survive any summary.
Kmax and flushing are not trivial details
Conductivity assays may attempt to measure a maximum reference conductivity after removing embolism, often by flushing or another treatment. The assumption is that the reference represents a meaningful fully functional state for that sample.
Strong flushing can affect tissues or create artefacts in some conditions. The extent of conductivity recovery and any changes to the sample deserve evaluation. Kmax should not be assumed exact simply because a machine provides a precise number.
If Kmax is biased upward, the calculated PLC for a given Kinitial also increases. If Kmax is underestimated, PLC can appear smaller. The reference measurement can therefore shift the whole vulnerability curve.
A complete method names how reference conductivity was established and whether controls were used. This is an example of why ratios still depend critically on their denominator.
Open-vessel artefacts and long conduits
An open vessel can connect a cut surface with internal portions of a sample, potentially enabling air entry or altered flow behaviour under some measurement setups. Whether that is problematic depends on conduit lengths relative to sample dimensions and on the technique.
Ring-porous species often contain large, relatively long vessels; some have shown atypical vulnerability shapes under certain experimental methods. A curve that looks dramatically more vulnerable than field observations should prompt anatomical and procedural scrutiny.
Researchers may estimate vessel-length distributions, select longer segments or compare techniques to test for these artefacts. This is not mere equipment housekeeping; it protects the biological conclusion.
When a methods paper identifies possible bias, do not turn its warning into a universal condemnation of all data produced by that apparatus. The correct claim is conditional and should name the anatomical and methodological circumstances.
Sigmoid versus exponential curve shapes
Many bench-dehydration-based conductivity curves are sigmoidal, while some data collected with other methods show more exponential-looking patterns. The difference may reflect actual hydraulic behaviour, methodological bias or model fit constraints.
A visibly exponential curve should not immediately be interpreted as a new biological adaptation. First examine how tension was induced, what sample length was used and whether extreme values were adequately measured.
Likewise, forcing a sigmoid through data with limited coverage can mask unusual behaviour. A model is a tool for summarizing observations, not a substitute for checking them.
Compare residuals, endpoint coverage and independent validation when evaluating curve form. The scientific question is whether the selected function represents credible hydraulic loss under the tested conditions.
How fitting creates P50 estimates
Fitted P50 is often an inverse problem: given observed conductivity losses at several potentials, find the potential where the model predicts 50 percent loss. Different functional forms may produce different answers if the data do not constrain the midpoint.
Report the fitting equation or cite a method that defines it, and look for uncertainty intervals around P50. The position of the central estimate can be sensitive to a few observations near the steep transition.
If a curve has only low-loss observations but reports a precise P88, that threshold likely depends heavily on extrapolation. Precision should not be confused with empirical support.
In your own technical description, distinguish ‘observed points’ from ‘fitted curve’ and ‘derived P50’. This separation helps readers understand why an estimate is useful but not immutable.
Replication and independence
Multiple conductivity measurements from one stem segment are not necessarily independent biological replicates. If the study compares species or treatments, replication must reflect the experimental units rather than just the number of points on one curve.
Water potential uncertainty, segment-to-segment variation and fitting uncertainty can each contribute to variability. A published standard error around P50 may summarize only one of these sources.
Treat curves from different individuals as distinct evidence where the design permits. Report species and population sampling honestly instead of presenting all measurements as though they represented the global distribution of a trait.
Good plots show individual sample curves or uncertainty bands when practical. Means alone can hide substantial variation or a minority of unusually vulnerable samples.
Comparing two species fairly
Suppose study A reports a P50 of −2.2 MPa for one species and study B reports −4.1 MPa for another. A reader might immediately say the second is more resistant to conductivity loss. That may be a useful starting hypothesis but is not yet a controlled comparison.
Ask whether studies used the same organ, measurement method, preparation, water-potential convention and environmental history. Species grown in different conditions may differ in acclimation, and apparatus differences can shift estimates.
If the experiments are directly comparable, a more negative P50 is evidence of greater resistance to reaching the specified conductivity-loss threshold. It does not alone quantify drought survival probability.
State precisely what the comparison supports and which uncertainties remain. That is more valuable than ranking all species on a single number without reporting how the numbers were generated.
The importance of minimum observed water potential
A P50 value is a potential failure reference, but its functional meaning depends on the water potentials the plant actually experiences. Field monitoring may estimate a minimum seasonal Ψ; that exposure can be compared with the laboratory curve.
If Ψmin is far less negative than P50, the observed period may leave a substantial nominal margin relative to the midpoint of conductivity loss. If Ψmin approaches P50, the hydraulic risk interpretation changes.
However, field measurements may represent leaf or stem compartments that are not identical to the sample in the vulnerability test. Temporal duration, recovery and accumulated damage complicate the simple margin.
Use safety margins as comparisons, not guarantees. A plant may face other limits before P50, and a single measured Ψmin may miss short events or heterogeneous conditions.
How drought recovery complicates the picture
Water transport may recover after stress through several pathways depending on plant anatomy and conditions, while some embolism or associated damage persists. The vulnerability curve itself often describes induction of loss, not the full recovery trajectory.
Rewatering can restore leaf water potential rapidly while hydraulic conductivity remains impaired. These variables need not recover at the same speed. Concluding ‘the xylem healed’ from a less negative leaf Ψ is therefore unwarranted.
Studies examining recovery should report measurements of conductivity, embolism state and tissue performance over time. Separating potential from transport capacity avoids an important physiological error.
Read an experiment’s temporal design carefully: a curve measured during increasing tension may not describe the behaviour of the same sample during rehydration. Hysteresis, damage and measurement disturbance may matter.
Water potential, turgor loss and xylem failure form different thresholds
Leaf water potential is an instantaneous state variable. The turgor loss point is an inferred tissue threshold related to cellular pressure. Xylem P50 is a hydraulic threshold from a vulnerability measurement. Their shared negative units can tempt readers to treat them as synonyms.
Consider a hypothetical species with leaf Ψtlp = −1.8 MPa and stem P50 = −3.4 MPa. At −2.2 MPa, the leaf may have crossed its fitted turgor-loss threshold while the tested stem remains on the less-negative side of P50.
That situation is physiologically possible; it does not imply that the plant is safe from all impairment or that exactly a specified fraction of xylem remains functional. Each threshold addresses a different compartment and mechanism.
The pressure–volume and turgor-loss guide explains the living-tissue part of this comparison. Keeping the two curves separate is essential for meaningful integration.
When water use and hydraulic safety interact
High sap flow requires conductive pathways and potential differences, but a low observed flow does not automatically demonstrate xylem damage. Stomatal closure, cloud cover or reduced atmospheric demand can reduce flow in otherwise functional tissue.
Similarly, a more negative potential may arise under strong transpiration before substantial embolism develops. You need both state and function measurements to distinguish a high-demand operating point from transport impairment.
A careful plant-water narrative therefore combines leaf or stem Ψ, stomatal behaviour, actual sap flow and hydraulic vulnerability where available. Each contributes different evidence.
The stomatal conductance and VPD guide helps clarify demand-side regulation without confusing those measurements with conductivity-loss thresholds.
Why P88 does not universally mean death
A loss of 88 percent hydraulic conductivity sounds severe, and some literature has explored strong relationships between hydraulic failure and mortality. But the exact relationship depends on plant type, organ, exposure, recovery processes and the study’s outcome definition.
No universal equation says every plant dies the instant a curve reaches P88. Mortality mechanisms can involve carbon starvation, tissue damage, pathogens and other constraints alongside hydraulic failure.
When writing for learners, avoid turning a useful proxy into a deterministic diagnosis. ‘The fitted curve predicts severe conductivity loss at this potential’ is more accurate than ‘the tree dies here’.
If a study claims a mortality threshold, check whether survival was directly measured and under what conditions. Do not import the threshold from a different population or method without evidence.
A measurement artefact can create a convincing biological story
Suppose a sample with long vessels is tested using a technique susceptible to open-vessel effects. The resulting curve might suggest early conductivity loss and a relatively less negative P50. An unwary reader could interpret this as exceptional drought sensitivity.
Before proposing a mechanism, examine vessel lengths, segment lengths and independent controls. If a reference method produces a different shape, the discrepancy becomes a methodological question rather than proof of unusual plant biology.
Methodology is not a footnote when the measured quantity is difficult. It defines which claims can reasonably be made from the figure.
A strong educational interpretation offers more than a number: it states what the apparatus measured, how the curve was built and which validation steps support its use.
How to write an accurate figure description
Begin with the axes: ‘Percent loss of stem hydraulic conductivity was plotted against xylem water potential in MPa.’ Add the method if known, such as bench dehydration or centrifugation, and identify the tissue and sampling population.
Next describe the observed direction: ‘Conductivity loss increased as potential became more negative.’ Then report the fitted threshold and uncertainty: ‘The estimated P50 was −3.1 MPa, with the interval reported by the study.’ Do not invent an interval where none appears.
Interpret with measured specificity: ‘The tested samples required a more negative xylem potential to reach 50 percent conductivity loss than the comparison samples under this method.’
Finally state scope: the value does not directly measure mortality, photosynthetic rate or total water content. This structure gives readers both an intelligible result and a boundary against overclaiming.
Typical problems in tables and translations
One common error is to erase the minus sign and call positive applied chamber pressure a signed xylem potential. Another is to label an embolism-count plot as PLC. Both change what the data mean even if the graph looks polished.
A third error is comparing P50 estimates from different organ types or apparatus without qualification. A fourth is confusing P12, P50 and P88 by assuming the numerical subscript describes a percentage of surviving plants.
A fifth error is computing hydraulic safety margins using the wrong subtraction order, then interpreting a sign reversal as a physiological difference. A sixth is reporting an extrapolated threshold as directly observed.
These are technical literacy problems as much as botany problems. Exact nouns, signs, percentages and model definitions matter in every language. Clear prose should preserve them rather than simplify them away.
A classroom exercise using a conceptual curve
Draw a horizontal axis from zero toward more negative potential and a vertical axis from 0 to 100 percent PLC. Mark hypothetical values P12 = −1.5 MPa, P50 = −2.8 MPa and P88 = −4.1 MPa. Ask learners to place the milestones in the correct order.
Next add a seasonal minimum potential of −2.0 MPa. Ask whether that observation is more or less negative than P50 and what the conventional signed safety margin would be. With Ψmin − P50, the example margin is +0.8 MPa.
Then change the vertical label to ‘cumulative optical embolism area’. Ask what can no longer be assumed about the reported P50. The exercise shows how a small caption change alters the parameter’s meaning.
Finally ask for two sentences: one strictly describing the plotted observation and one proposing a possible biological interpretation. That separation trains scientific reasoning beyond this particular topic.
Frequently asked questions
Is a more negative P50 always better? It indicates greater resistance to the defined conductivity-loss level under comparable methods, but whole-plant performance depends on many other traits and conditions.
Does P50 mean 50 percent of vessels embolized? Usually not. PLC concerns hydraulic conductivity, which weights conduits by their contribution to transport rather than merely counting them.
Can I use an optical curve to obtain hydraulic P50? Only if the method and its calibration justify that inference. An optical midpoint should otherwise be labelled as an optical endpoint.
Are P12 and P88 exact universal boundaries? No. They are useful conventional levels on a fitted response curve. Their significance and uncertainty depend on the experiment.
A disciplined reading route
First confirm the data type: measured conductivity loss, acoustic events or optical embolism indicators. Then record the tissue, pressure convention and method used to induce vulnerability.
Second examine raw observation coverage and fitting. Locate the part of the curve around 50 percent response and note whether P12 or P88 is mostly extrapolated. Look for reported uncertainty.
Third compare any claimed hydraulic safety margin with an appropriately defined field minimum potential. Note where the source mixes organs or measurement times, and do not conceal that mismatch.
Fourth connect the hydraulic result to whole-plant physiology through independent measurements. This is where stomatal control, leaf water status and eventual recovery become useful without being mislabeled as PLC.
Sources and connected educational routes
A critical methods review by Cochard and colleagues in the Journal of Experimental Botany discusses vulnerability techniques, curve shapes and open-vessel concerns. It provides a methodological reference for readers who want to evaluate rather than merely copy reported P50 values.
The leaf water potential interpretation guide explains signed MPa values and chamber readings. The separate pressure–volume guide addresses turgor and osmotic traits.
For people writing or translating laboratory explanations, the eduKateSG cross-linguistic transfer resource is relevant because a technical term must keep its measurement definition across languages.
Keep the core distinctions visible: potential is a state, conductivity is a transport property, percent loss is a normalized response, and P50 is a derived threshold. Once these are separate, vulnerability curves become far easier to read responsibly.