Improved precision of hydraulic conductance measurements using a Cochard rotor in two different centrifuges

An improved way of calculating hydraulic conductance (K) in a Cochard cavitron is described. Usually K is determined by measuring how fast water levels equilibrate between two reservoirs while a stem spins in a centrifuge. A regression of log meniscus position versus time was used to calculate K and this regression method was compared to the old technique that takes the average of discrete values. Results of a hybrid Populus 84K shows that the relative error of the new approach is significantly lower than the old technique by 4~5 times. The new computational method results in a relative error less than 0.5% or 0.3% from 8 or 12 points of measurement, respectively. The improved precision of K measurement also requires accurate assessment of stem temperature because temperature changes K by 2.4% o C -1 . A computational algorithm for estimating stem temperature stability in a cavitron rotor was derived. This algorithm provides information on how long it takes stem temperature to be known to within an error of ±0.1 o C. Introduction The cavitron technique has been used for many years to measure vulnerability curves (VCs) of woody stems (Cochard 2002, Cochard, Damour et al. 2005). VCs have been widely viewed as a good measure of the drought resistance of woody plants (Cochard et al. 2013). Increasing drought will induce cavitation of water held in stem conduits (vessels or tracheids). A cavitation event occurs when water columns break under tension (=minus the pressure of water in xylem). A cavitated vessel first fills with water vapor then eventually fills with air-bubbles at atmospheric pressure because of Henry’s law of gas solubility at water/air interfaces. The time required for equilibrium of air-pressure depends on the rate of air penetration into the recently-cavitated vessel lumen via diffusion in the liquid phase (Fick’s Law). We have been conducting experiments that address the tempo of air bubble formation in recently cavitated vessels. The equilibrium for air bubble formation is defined by Henry’s law, from which we can predict that eventually the air pressure inside a cavitated vessel must equal ambient atmospheric pressure. While doing initial experiments we realized we needed to improve the precision of hydraulic conductance (K) measurement in stems spinning in a Cochard cavitron in order to study how long it takes air bubbles to form. Current measurements are reproducible to about 2% (see for example Cai et al 2014) over short periods of time (minutes). Over longer periods of time (1-2 h) K can also be influenced by the change of temperature in the centrifuge while the rotor is spinning due to nonstable temperature control in some centrifuges. If a spinning stem experiences a temperature change of ± 1 o C then this will cause a change in measured K of ±2.4% because of the affect of temperature on the viscosity of water near room temperature. Materials and Methods Theory Hydraulic conductance, K, can be measured in a cavitron at both high negative pressure and low (subatmospheric) pressure. A multi-point measurement of 1: e0007 The Journal of Plant Hydraulics 7: e-number 2 K can be completed in 20 to 60 seconds and hence we can assume K is constant during the measurement. In the traditional method, K is measured several times over small time and volume increments by observing two menisci: one that is stationary and the other that moves towards the stationary meniscus a distance x away relative to the stationary meniscus. During the measurement the meniscus moves a distance x in a cuvette in a time interval t and the flow rate is evaluated from F = Aw x/ t (kg s -1 ) where is the density of water, and Aw is the cross sectional area of water in the cuvette. At the same time the pressure drop causing flow is computed from P = 0.5 2 (2R�̅-�̅2) where �̅ is the value of x at the midpoint of x. In this equation is the angular velocity of the rotor spinning, and R = the maximum radius from the axis of rotation to the lower stationary meniscus. In Cochard’s original equations a lower case ‘r’ is used in place of x. The value of K is calculated from � = = ̅ ̅ (1) The problem with Eq. (1) is that the time and distance intervals are rather small and hence the standard deviation on repeated measurements of x/ t is rather large (around 10%). One of us (YW) noticed during experiments that x (the distance of the moving meniscus from stationary meniscus) declines exponentially with time towards 0 at constant , i.e., a plot of ln(x) declines linearly with t. This behavior follows from Eq. (1) written in differential format (dx/dt = x/ t) when we neglect �̅2, which is small compared to 2R�̅ (typically �̅ < 3 versus 2R = 254 mm). So we tried measuring the slope, m, of a plot of ln(x) versus t and computed K from the solution of differential Eq. (1) ignoring �̅ . The solution of Eq(1) ignorning �̅2 is ln = − �. (2A) Hence a plot of the ln(x/xo) versus t will yield a slope, m, from which we can compute K � = − (2B) The linear regression provides only one value for the slope, but a standard regression analysis can be used to compute the standard error of the slope, m. Preliminary experiments demonstrated to us that the computation of K from Equation (2) had a much smaller SE than from repeated measurements of Eq. (1). So we started looking for the exact solution to differential equation resulting from Eq. (1): � = = (3). The solution provided by (RB) is: �� � �� = �� − � �� �� � �� = �� − � �� −�� � � t − � = ln � � + �� � − � � − � , where � is equated to zero, i.e., the time when we begin to record the movement of the upper meniscus (at � ), and t is the time when we record the passage of the meniscus at position x. The second natural log term in Eq.(4) contributes 1 to 2% to the slope depending on the range of x. Hence our initial use of just ln(xo/x) = constant –ln(x) was approximately correct but the exact solution is preferable. Theoretically � should be proportional to y = �� + �� with a slope, m =− . From the recorded x vs t, we compute y vs t and from the regression we got the slope, m, to calculate K as shown in Eq. (2B) above. Experiments for computation of K Test experiments were conducted on clonal hybrid Populus 84K shoots (Populus alba × Populus glandulosa). Branches were cut from the trees growing on the campus of Northwest A&F University. Segments of 0.274 m long and 6-7 mm in basal diameter were cut from harvested branches (> 0.5 m long) under water, and were fully flushed with 10 mM KCl under 200 kPa pressure for 30 min. A Cochard cavitron was used to test our improved method, in which we used a 20X or 40X microscope to observe the water level changes. The difference between two holes on the two reservoirs was 6.5 mm or 3.8 mm depending on the microscope used. Some measurements were performed on a cavitron rotor in a Beckmann Coulter model Allegra X-22R centrifuge and the temperature-dependent experiments were done in a Xiang Yi model H2100R centrifuge because it had better temperature regulation. We measured vulnerability curves to see whether hydraulic conductance obtained by two methods (regression and traditional) were the same at different tensions. We collected typically 11 points of x versus t and obtained the means and SE of the slope by the regression method and the mean and SE by the traditional method using 10 values of x and t and equation (1) to compute K and then computed the traditional mean and SE. Since the two methods manipulate the same data set, it is essential to recognize that the methods being compared are differentiated by a purely computational difference except in one way. In the old method x and t (Eq. 1) are usually determined 6 to 11 times by 1 0007 The Journal of Plant Hydraulics 7: e-number 3 refilling the cuvette 2 or 3 times during the sequence of measurements and applying Eq. (1) 6 to 11 times. In this study the cuvette is filled only once and 8 to 11 measurements of x and t are made, which provides data sets that can be evaluated by use of Eq. (1) 8 to 11 times and this is compared to a regression using Eq. (4). Since the time required for the meniscus to move a fixed distance, x, increases as the experiment progresses the value of t also increases, so successive measurements of K using Eq. (1) become more accurate as the measurements progress. This should make the ratio of (standard error)/(mean K) smaller hence giving a more accurate mean K. Despite this trend we will demonstrate that the ratio decreases faster by the regression-computational method. Temperature tests: an algorithm to estimate Tstem Enhanced precision of measurement of K is of little value if the stem temperature, Tstem, is unknown and variable because K is proportional of 1/ where is the viscosity of water. The value of 1/ changes about 2.4% o C -1 near 20 o C. The temperature depedence of 1/ ranges from 3.3 to 1.95 % o C -1 for temperatures from 0 to 40 o C, respectively. Hence Tstem must be known or controlled to within ±0.1 o C to achieve ±0.24% accuracy in the computation of percent loss of conductance (PLC) which needs to be determined by repeated measurements of K at the same Tstem but differing tensions during the construction of a vulnerability curve. The approach we took was to make use of the fact that a stem is a defacto thermometer when repeated measurements of K are made at low tension, i.e., when only temperature affects K. Most centrifuges have refrigeration (or a heat pump that can both heat and cool) and a thermostatic way to set and control temperature. But thermostatic control of temperature is never completely precise. In order to assess the impact of fluctuations of temperature inside a centrifuge on Tstem, we programmed large changes in temperature and monitored changes in K vs time while measuring the air temperature inside the centrifuge with a temperature sensor near the rotor. Many types of temperature sensors are capable of ±0.1 o C resolution after calibration; these include thermistors, thermocouples and LM335 chips; we used the latter. But air temperature will not reflect the likely temperature of the spinning stem. Hence we tried to devise a computational algorithm to compute Tstem adequately to predict the observed changes in K. Then this algorithm was used to assess the ability of different centrifuges to control Tstem. We tested two kinds of computational algorithms: The first was a running mean of the air-temperature in the centrifuge. We monitored air-temperature every 5 seconds, and we computed the running mean air temperature for various length of time. The second algorithm was a first-order rate reaction equation where the change in stem temperature at any time interval is given by Tstem= α t(Tair,i-Tstem,i), where α = the ‘heat transfer’ rate constant and t is the time step. Hence if we know Tstem at time t=0 (Tstem,0) then the stem temperature at a later time (Tstem,t) after n measuring intervals is given by