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A revised and unified pressure-clamp/relaxation theory for studying plant cell water relations with pressure probes: In-situ determination of cell volume for calculation of volumetric elastic modulus and hydraulic conductivity

A revised and unified pressure-clamp/relaxation theory for studying plant cell water relations with pressure probes: In-situ determination of cell volume for calculation of volumetric elastic modulus and hydraulic conductivity
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  A revised and uni fi ed pressure-clamp/relaxation theory for studyingplant cell water relations with pressure probes: In-situ determinationof cell volume for calculation of volumetric elastic modulusand hydraulic conductivity T. Knipfer a, n , J. Fei a , G.A. Gambetta a,1 , K.A. Shackel b , M.A. Matthews a a Department of Viticulture and Enology, University of California, Davis, USA b Department of Plant Sciences/Pomology, University of California, Davis, USA H I G H L I G H T S   Accurate estimates of plant cell volume ( ν o ) can be determined in-situ using cell-pressure-probes.   A revised  ν o -theory was developed that is valid for the pressure-clamp and pressure-relaxation methods.   For the same cell, the pressure-clamp method gave a systematically lower (21%)  ν o  as compared to the pressure-relaxation method.   Effects of solute mixing could only explain a potential error in calculated  ν o  of   o 3%.   The results suggest that both methods are differentially affected by cell osmotic behavior (i.e. solute re fl ection coef  fi cient,  s ) in response to turgorchanges. a r t i c l e i n f o  Article history: Received 28 May 2013Received in revised form19 May 2014Accepted 22 May 2014Available online 5 June 2014 Keywords: ConvectionMicromanipulationSolute diffusionSolute re fl ection coef  fi cient Tradescantia virginiana a b s t r a c t The cell-pressure-probe is a unique tool to study plant water relations in-situ. Inaccuracy in theestimation of cell volume ( ν o ) is the major source of error in the calculation of both cell volumetricelastic modulus ( ε ) and cell hydraulic conductivity ( Lp ). Estimates of   ν o  and  Lp  can be obtained with thepressure-clamp (PC) and pressure-relaxation (PR) methods. In theory, both methods should result incomparable  ν o  and  Lp  estimates, but this has not been the case. In this study, the existing  ν o -theories forPC and PR methods were reviewed and clari fi ed. A revised  ν o -theory was developed that is equally validfor the PC and PR methods. The revised theory was used to determine  ν o  for two extreme scenarios of solute mixing between the experimental cell and sap in the pressure probe microcapillary. Using a fullyautomated cell-pressure-probe (ACPP) on leaf epidermal cells of   Tradescantia virginiana , the validity of the revised theory was tested with experimental data. Calculated  ν o  values from both methods were inthe range of optically determined  ν o  ( ¼ 1.1 – 5.0 nL) for  T. virginiana . However, the PC method produced asystematically lower (21%) calculated  ν o  compared to the PR method. Effects of solute mixing could onlyexplain a potential error in calculated  ν o  of   o 3%. For both methods, this discrepancy in  ν o  was almostidentical to the discrepancy in the measured ratio of   Δ V  / Δ P   (total change in microcapillary sap volume versus  corresponding change in cell turgor) of 19%, which is a fundamental parameter in calculating  ν o . Itfollowed from the revised theory that the ratio of   Δ V  / Δ P   was inversely related to the solute re fl ectioncoef  fi cient. This highlighted that treating the experimental cell as an ideal osmometer in both methods ispotentially not correct. Effects of non-ideal osmotic behavior by transmembrane solute movement maybe minimized in the PR as compared to the PC method. &  2014 Elsevier Ltd. All rights reserved. 1. Introduction The cell-pressure-probe has been used for more than 30 yearsto study water relations in-situ at the individual cell level (Hüskenet al., 1978; Steudle, 1993; Tomos and Leigh, 1999), mostly inhigher plants (Tomos et al.,1981; Tyerman and Steudle,1982; Zhuand Steudle, 1991; Moore and Cosgrove, 1991) but also in algalContents lists available at ScienceDirect journal homepage:  Journal of Theoretical Biology &  2014 Elsevier Ltd. All rights reserved.  Abbreviations:  ACPP, automated cell pressure probe; PC, pressure-clamp; PR,pressure-relaxation n Corresponding author. Tel.:  þ 1 530 752 7185. E-mail address: (T. Knipfer). 1 Present address: Institut des Sciences de la Vigne et du Vin, Bordeaux, France Journal of Theoretical Biology 359 (2014) 80 – 91  (Steudle and Tyerman, 1983; Henzler et al., 2004) and fungal cells(Cosgrove et al., 1987). All of these systems have a large centralvacuole, a tough cell wall, and can develop a relatively highhydrostatic turgor pressure ( P  ). Using the cell-pressure-probe thetip of a glass microcapillary is inserted into a cell, which causes cellsap volume entering the microcapillary ( V  ). Prior to measuring  P  ,  V  external to the cell is minimized so that it ideally equals thevolume of glass introduced into the cell (i.e. restoration of cellvolume). After the cell has reached its equilibrium state,  P   and  V  are monitored. Although the microcapillary tip is only in hydrauliccontact with the vacuole, both vacuolar and cytoplasmic compart-ments must exhibit the same  P   (the unsupported vacuolar mem-brane cannot support a pressure difference), and hence at waterpotential ( Ψ  )  equilibrium the same osmotic pressure ( π  ). Thus, thecell is regarded as an osmotic system having a volume ( ν )equivalent to that of the outermost cell membrane. Beside thedirect measurement of   P  , the cell-pressure-probe can be used toinduce small short-term changes in both  ν  and  P   by small changesin  V  , which allows the determination of two fundamental cellproperties, the cell volumetric elastic modulus ( ε ) and cellhydraulic conductivity ( Lp ) (Steudle, 1993). Both parameters depend on srcinal cell volume ( ν o ) ε ¼ ν o dP dV   ð 1 Þ and Lp ¼  ln 2 ν o  AT  1 = 2 b  ð 2 Þ where in practice, very small and rapid changes in  V   and thecorresponding measured changes in  P   are considered equivalent to dV   and  dP  , respectively (Eq. (1)). When  dV  / dP   is measured usingsuch changes, water  fl ows across the cell membrane are negligible,and therefore  dP   is proportional to the elastic change in cellvolume of   d ν  ( ¼ dV  ) (i.e., Eq. (1) is analogous to Hooke's law).These rapid changes in cell volume and pressure ( dV   and  dP  )should not be confused with  ∆ V   and  ∆ P   which represent changesas measured between the original and  fi nal equilibrium stateof the cell when subjected to the pressure-clamp (PC) andpressure-relaxation (PR) method (Steudle, 1993; Wendler andZimmermann, 1982) (Fig. 1). These two different hydraulic meth- ods for manipulating  P   and  V   can be used to induce water  fl owsacross the cell membrane and to determine  Lp  of Eq. (2). Mainlydue to the difference in cell elastic behavior between bothmethods,  b  of Eq. (2) is equivalent to  s π  o  when using the PCmethod and ( ε þ s π  o ) when using the PR method (Wendler andZimmermann, 1982). Nevertheless, both methods should giveequivalent estimates of   Lp  for the same cell, but this has not beenthe case. The  Lp  of parenchyma cells of sugar cane differed by 50 – 59% (Moore and Cosgrove,1991), and the  Lp  of wheat root corticalcells by 16 – 21% (Zhang and Tyerman, 1991), between bothmethods. Recent technical developments have included a compu-ter vision-based automated-cell-pressure-probe (ACPP) systemthat enables a  fi ner control and higher resolution of   P   and  V  changes with limited manual input from the operator (Wong et al.,2009). However, even after incorporating those improvements,  Lp of leaf epidermal cells of   Tradescantia virginiana  still differed by17% to 31% between both methods (Wada et al., 2014). It is notclear what causes this discrepancy in calculated  Lp  between the PCand PR methods for the same cell.The biggest source of error in determination of   ε  and  Lp  is theestimation of   ν o . Inaccuracy in optical  ν o  can be  4 50% due toirregularities in cell shape and cell to cell variations (Tomos et al.,1981; Malone and Tomos, 1990). Therefore, theories have beendeveloped independently for the PC (Wendler and Zimmermann,1982) and PR (Malone and Tomos, 1990) method to derive  ν o directly and in-situ for the punctured cell. This biophysicalapproach has the advantage that it eliminates errors in estimated ν o  caused by destructive tissue sampling. However,  ν o  calculatedusing these methods and existing theories differs by around 17%for the same cell (Zhang and Tyerman, 1991; Wada et al., 2014).Murphy and Smith (1998) reported that solute mixing between Nomenclature  A  cell membrane surface area (m 2 ) b  ( s π  o ) when using the PC method and ( ε þ s π  o ) whenusing the PR method (MPa)  J  v  water  fl ow across the cell membrane (m s  1 )  J  s  solutes  fl ow across the cell membrane (mol s  1 ) Lp  cell hydraulic conductivity (m s  1 MPa  1 ) N   amount of cell solutes (mol) Δ N   change in solutes of the cell compartment via themicrocapillary between srcinal and  fi nal equilibriumstate of the cell in the PC and PR method (mol) P   cell hydrostatic pressure, measured as gaugepressure (MPa) Δ P   change in cell hydrostatic pressure between srcinaland  fi nal equilibrium state of the cell in the PC and PR method (MPa) R  ideal gas constant (m 3 MPa K  1 mol  1 ) t   time (s) Δ t   time period of application of the PR and PC method (s) T   temperature (K) T  1/2  halftime of relaxation processes in  Δ P  ( t  ) or  Δ V  ( t  ) (s) V   cell sap volume in microcapillary (m 3 ) Δ V   change in microcapillary sap volume between srcinaland  fi nal equilibrium state of the cell in the PC and PR method (m 3 ) dP  / dV   cell elastic modulus (MPa m  3 ) ε  cell volumetric elastic modulus ( ¼ ν o  ( dP/dV  ))(m 3 MPa m  3 ¼ MPa) π   cell osmotic pressure (MPa) ν  cell volume, de fi ned as the volume bounded by thecell membrane, and calculated in (m 3 ) but expressedin (nL) s  solute re fl ection coef  fi cient (1) Ψ   cell water potential (MPa) Subscripts f   fi nal equilibrium in  J  v i  initial change in microcapillary sap volume corre-sponding to the induced step change from  P  o  to  P  f   inthe PC method m  change in microcapillary sap volume corresponding tothe  fl ow of water across the membrane when  P   isclamped at  P  f   in the PC method o  srcinal equilibrium in  J  v t   microcapillary tipex external T. Knipfer et al. / Journal of Theoretical Biology 359 (2014) 80 – 91  81  cell and microcapillary sap could be a potential source of errorwhen cell water relation parameters, such as  ν o , are calculatedwith the PC method. Those effects of solute mixing between celland microcapillary sap are not considered in existing  ν o  theories(Wendler and Zimmermann, 1982; Malone and Tomos, 1990).In this paper existing  ν o -theories (Wendler and Zimmermann,1982; Malone and Tomos, 1990) were critically reviewed. Weidenti fi ed some potential errors and discrepancies in the  ν o derivation for the PC and PR method, regarding the cell elasticresponse and osmotic behavior, which have remained unrecog-nized (see Section 2.2). Therefore, a revised  ν o -theory was devel-oped, which is the  fi rst uni fi ed theory for both methods. It allowsthe determination of an upper and lower physical estimate of   ν o ,based on two extreme scenarios of solute mixing between Fig.1.  Schematic overviews of (a,b) the pressure-clamp and (c,d) pressure-relaxation methods. Both methods share four stages in cell behavior as indicated by numbers: 1,srcinal equilibrium prior to the experiment; 2, initial elastic deformation in volume when the experiment is initiated; 3, non-equilibrium during the experiment; 4,  fi nalequilibrium after suf  fi cient waiting time. (a,c) Recordings of cell hydrostatic pressure ( P  ) and corresponding changes in microcapillary sap volume ( V  ) over time as measuredwith the automated cell-pressure-probe together with (c,d) corresponding changes in cell volume ( ν ) and osmotic pressure ( π  ) in stages 1 – 4 for an exosmotic experiment.Changes in  ν  are indicated by arrows, and dashed lines indicate  ν  at the previous stage. Changes in  V   that result in the movement of microcapillary sap into the cell arede fi ned as positive, and out of the cell as negative (equivalent to the de fi nition of the direction of transmembrane water  fl ow across the cell membrane,  J  v , of Eq. (5)). In anexosmotic experiment, an increase in ( þ ) P   results in the movement of the oil/sap meniscus towards the cell, and of cell sap ( þ ) Δ V   into the cell. This induces atransmembrane water  fl ow of (  )  J  v  out of the cell, and  vice versa  in an endosmotic experiment. In (a – d)  J  s ¼ solute  fl ow across the cell membrane,  T  1/2 ¼ halftime,  ν ¼ cellvolume,  π  ¼ osmotic pressure; subscripts  “  f  ” ¼ fi nal equilibrium for  J  v ¼ 0,  “ i ” ¼ initial,  “ m ” ¼ across cell membrane,  “ o ” ¼ srcinal equilibrium for  J  v ¼ 0. T. Knipfer et al. / Journal of Theoretical Biology 359 (2014) 80 – 91 82  microcapillary and cell sap (Murphy and Smith, 1998). A possiblerole of transmembrane solute movement (solute re fl ection coef  fi -cient,  s ) on  ν o  calculations from both methods was investigatedand is discussed. Cell water relations methodology would bene fi tfrom the ability to determine precise estimates of   ν o  from bothmethods in order to minimize the error in cell  Lp  and  ε . 2. Theory   2.1. Cell biophysical behavior  In order to evaluate  Lp  and  ν o  with cell-pressure-probes, cellwater status is temporarily perturbed. The PC and PR methodinduce a small  Δ Ψ   across the cell membrane, which results in atransient transmembrane water  fl ow (  J  v ) until equilibrium isreestablished. In the PC method, a step-change in  P   is imposedwhich causes a water in fl ow (endosmotic) or out fl ow (exosmotic)while changes in  V  ( t  ) in the microcapillary are followed (Fig. 1aand b). In the PR method, a set volume of   V   is injected (exosmotic)into or withdrawn (endosmotic) from the cell, and changes in  P  ( t  )are recorded (Fig. 1c and d). For both methods, the imposedchanges in cell volume are relatively small ( o 5%). Based on therelaxation kinetics in  P  ( t  ) or  V  ( t  ), the halftime ( T  1/2 ) of waterexchange is determined for calculation of   Lp  (Eq. (2)), and  fi nalvalues of   P   and  V   are determined by extrapolation to  t  - 1  forcalculation of   ν o  (Zhang and Tyerman, 1991). The revised theorythat we develop here takes advantage of the fact that bothmethods share four comparable stages in biophysical behavior(Fig. 1).  2.1.1. (Stage 1) Original equilibrium Prior to the application of both methods,  J  v ¼ 0, and  P  o ,  π  o , and ν o  are constants (subscript  “ o ” , srcinal).  2.1.2. (Stage 2) Elastic deformation in cell volume The initial change in  P   results in an elastic change in cellvolume observed as a change in  V   in the microcapillary, while  J  v remains zero. In the PC method the initial change in micocapillarysap volume is de fi ned as  Δ V  i  ( ¼ ν f   ν o ) and induces a correspond-ing change in  P   of ( P  f   P  o ). In contrast, in the PC method the initialchange in microcapillary sap volume is equivalent to the totalchange in microcapillary sap volume of   Δ V   ( ¼ ν max  ν o ) andinduces a corresponding change in  P   of ( P  max  P  o ). When trans-membrane solute  fl ow (  J  s ) is negligible,  π  o  remains constant.  2.1.3. (Stage 3) Non-equilibrium During the application of both methods,  J  v a 0. In the PCmethod, cell volume remains constant at  ν f   because  P   is clampedat  P  f   (neglecting the very small, 0.0001 MPa  Δ P  , associated with fl ow through the capillary tip (Wada et al., 2014)), and  T  1/2  of therelaxation in  Δ V  m ( t  ) ( ¼ Δ V  ( t  )  Δ V  i ¼  J  v  A Δ t  ), as measured in themicrocapilllary, exclusively re fl ects water movement across thecell membrane. In the PC method, the volume of water lost(exosmotic) or gained (endosmotic) across the cell membrane isconstantly replaced bycell sap from the microcapillary. In contrast,in the PR method cell volume changes because  P   relaxes while  Δ V  is clamped, and  T  1/2  of the relaxation in  Δ P  ( t  ) re fl ects watermovement across the cell membrane as well as elastic deforma-tion effects (Wendler and Zimmermann, 1982). In the PR method,water lost or gained across the cell membrane is not replaced bymicrocapillary sap.During both methods, cell  π   changes over time predominantlydue to solute transfer via the microcapillary, but also due to  J  s  if the cell membrane is not ideally semipermeable to solutes.  2.1.4. (Stage 4) Final equilibrium After a suf  fi cient time period,  J  v ¼ 0 again, and  P  f  ,  π  f  , and  ν f   areconstant again (subscript  “  f  ” ,  fi nal). In the PC method, the  fi nalequilibrium is obtained predominantly by changes in  π  ( t  ), while  P  is clamped at  P  f  . In the PR method,  fi nal equilibrium is reachedpredominantly by the rapid changes in  P  ( t  ) while changes in  π  ( t  )are rather small. The time to reach the  fi nal equilibrium state ismuch longer with the PC ( 4 120 s) as compared to the PR ( o 20 s)method.In both methods the change in original to  fi nal cell volume( Δ ν ) is equivalent to the total change in microcapillary sap volume( Δ V  ), minus (exosmotic) or plus (endosmotic) the volume lost orgained, respectively, by transmembrane water  fl ow during theapplication of the method ( ¼  J  v  A Δ t  ); i.e.  Δ ν ¼ [ Δ V  þ (  J  v  A Δ t  )].  2.2. Analysis of existing theories for extracting cell volume Existing  ν o -theories for the PC (Wendler and Zimmermann,1982) and PR (Malone and Tomos, 1990) methods were reviewed in accordance with the concept of cell biophysical behaviordescribed in Section 2.1.  2.2.1. Pressure clamp Based on the thermodynamics of irreversible processes (Kedemand Katchalsky, 1958), Wendler and Zimmermann (1982) devel- oped a pressure probe theory for the PC method, where ν o ¼ Δ V  Δ P  ð s π  o þ Δ P  Þ ð 3 Þ In Eq. (3),  Δ P   describes the imposed change to the new,clamped pressure ( ¼ P  f   P  o ) and  Δ V   was de fi ned as the changein original to  fi nal cell volume ( ¼ ν f   ν o ). Effects of non-idealsemipermeability of the membrane to solutes were accounted forby  s . Their derivation of Eq. (3) ( ¼ Eq. (A5)) is summarized indetail in Appendix A.1. Testing Eq. (3) on giant algae cells of   Characoralline  gave  ν o  values   7 – 20% smaller than optical  ν o  (Wendlerand Zimmermann, 1982). More importantly, we identi fi ed somepotential errors in their  ν o  derivation, which have remainedunrecognized:(1) In response to  Δ P  , cell volume changes by elastic deformation(Fig. 1a and b; stage 2). This results in an initial swelling(exosmotic) or shrinkage (endosmotic) of the cell while  J  v ¼ 0.This elastic change in cell volume was not considered in Eq.(3).(2) Surprisingly,  Δ V   was de fi ned as  ν f   ν o . As stated by theauthors  “ the change in cell volume, is equivalent to thedisplacement of the oil/cell sap meniscus in the microcapil-lary ” . If this de fi nition is true,  J  v ¼ 0 during the entire clampingtime which violates Eq. (A1).(3) After insertion, the microcapillary is  fi lled with cell sapcontaining solutes (Malone and Tomos, 1990). When micro-capillary sap is injected into or withdrawn out of the cell,solutes also enter (exosmotic), or leave (endosmotic), and thissolute transfer affects the  fi nal amount of cell solutes ( N  f  ) andsubsequently  π  f   ( ¼ ( N  f  / v f  ) RT  )). The relationship of   π  o ν o ¼ π  f  ν f   of Eq. (A4) only holds, if the amount of cell solutes remainsconstant and this movement of solutes was not considered. Inother words,  π  f   is only affected by the change in cell volumefrom  ν o  to  ν f  .(4) Effects of solute mixing between cell and microcapillary sapwere not addressed.  2.2.2. Pressure relaxation Based in part on Wendler and Zimmermann (1982), Malone and Tomos (1990) developed a pressure probe theory for the PR  T. Knipfer et al. / Journal of Theoretical Biology 359 (2014) 80 – 91  83  method, where ν o ¼ π  o ðð P  o  P  f  Þð dV  = dP  Þ Δ V  Þ P  f   P  o þð P  o  P  f  Þ dV dP   ð 4 Þ In Eq. (4),  Δ V   is the induced step change in the microcapillary,and  dV  / dP   is the cell volumetric elastic modulus obtained empiri-cally. Their derivation of Eq. (4) ( ¼ Eq. (A10)) is summarized indetail in Appendix A.2. Eq. (4) was tested on leaf epidermal cells of  wheat, where calculated  ν o  was comparable to optically deter-mined mean cell volume. In this theory(1)  s was not included, because it was assumed that a cell acts likean ideal osmometer with  s ¼ 1 (i.e. perfect semipermeablemembrane).(2)  ν o  was only derived for an endosmotic experiment.(3)  Δ N   was de fi ned as  π  o ν o ( Δ V  / ν o ) which is mathematicallyincorrect (see Eq. (A7) for details).(4) Effects of solute mixing between cell and microcapillary sapwere not addressed.  2.3. The revised  ν o -theory Our objective is to correct the potential errors in existing  ν o -theories (Wendler and Zimmermann, 1982; Malone and Tomos,1990) and combine them into a single, revised  ν o -theory that (i) isequally valid for the PC and PR methods, and (ii) accounts for theeffects of solute mixing between microcapillary and cell sap. LikeWendler and Zimmermann (1982) and Malone and Tomos (1990), we begin by describing the water  fl ow across the cell membrane(  J  v ) with the phenomenological equation of the thermodynamicsof irreversible processes (see Eq. (A1))  J  v ¼ Lp Δ Ψ   ¼ Lp ð Ψ  ex  Ψ  Þ¼ Lp ½ð P  ex  P  Þ s ð π  ex  π  Þ ;  ð 5 Þ where  J  v  is driven by a gradient in water potential ( Δ Ψ  ) betweencell interior and exterior (subscript  “ ex ” ). By convention, Eq. (5)de fi nes transmembrane water  fl ows out of the cell (for  Ψ  ex o Ψ  ) asnegative and  fl ows into the cell (for  Ψ  ex 4 Ψ  ) as positive. Both,  Ψ  ex and  Ψ   are composed of a hydrostatic ( P  ) and an osmotic ( π  )pressure component. The solute re fl ection coef  fi cient ( s ) accountsfor non-ideal semipermeability of the membrane, which affects Δ π  ¼ π  ex  π  ,  and  s ¼ 1 if the cell behaves like an ideal osmometer.Based on Eq. (5), we will derive an expression for the calculation of cell volume by comparing the srcinal (subscript  “ o ” , stage 1)with the  fi nal (subscript  “  f  ” , stage 4) equilibrium in  J  v  (Fig. 2).For both equilibrium states (  J  v ¼ 0), Eq. (5) reduces to P  o  P  ex ¼ s ( π  o  π  ex ) and  P  f   P  ex ¼ s ( π  f   π  ex ). Combining thosetwo expressions yields P  f   P  o ¼ s ð π  f   π  o Þ ð 6 Þ According to the van't Hoff equation ( π  ¼ ( RTN  )/ ν ),  π  f   of Eq. (6)can be substituted for ( N  f  / ν f  ) RT   which gives P  f   P  o ¼ s N  f  v f  RT   π  o    ð 7 Þ whereas cell volume ( ν ) is de fi ned as the volume bounded by thecell membrane. In order to account for the change in  ν  betweensrcinal and  fi nal equilibrium byelastic deformation,  ν f   of Eq. (7) issubstituted for [( P  o  P  f  )( dV  / dP  ) þ ν o ] (Eq. (A9)) which yields P  f   P  o ¼ s N  f  ð dV  = dP  Þð P  f   P  o Þþ ν o RT   π  o    ð 8 Þ where the cell elastic modulus ( dV  / dP  ) was treated as a constantthat predominantly depends on the mechanical properties of thecell wall. Eq. (8) holds for non-growing cells where changes in cellvolume from  ν o  to  ν f   are relatively small and reversible (Proseuset al., 1999).  2.3.1. Accounting for solute movement between microcapillaryand cell sap Eq. (8) accounts for transmembrane solute  fl ow (  J  s ) by  s , butdoes not account for solute movement between experimental celland microcapillary, i.e. when either cell sap containing solutes isremoved (endosmotic) via the microcapillary and the amount of cell solutes is reduced or when microcapillary sap containingsolutes is introduced (exosmotic) into the cell and the amount of cell solutes is increased.To address this issue, microcapillary and cell were treated astwo compartments, with solutes in the cell compartment beingperfectly mixed (Fig. 2). At srcinal equilibrium (stage 1) it wasconsidered that both compartments are in diffusional equilibrium,which means that the osmotic pressure of cell sap in the micro-capillary ( π  t ) is equivalent to  π  o  of the cell (i.e. no difference insolute concentration ( c  , ¼ π  / RT  )). When not at equilibrium duringboth methods, it was considered that the change in concentrationof cell solutes potentiallyaffects the concentration of solutes in themicrocapillary (and subsequently  π  t ) due to solute mixing bydiffusion. The amount of solutes transferred between microcapil-lary and cell by diffusion depends on the cross-sectional area of the microcapillary tip, the change in concentration gradient overtime, and the diffusion coef  fi cient (see Fick's law); whereas thediffusion coef  fi cient depends on temperature and viscosity of cellsap (see Stokes – Einstein equation). Those parameters can varydepending on the experimental conditions and cell type. Here, weconsidered the two possible extreme scenarios of (i) no mixing  – when solute movement is purely by  “ convection ”  (solutes are onlycarried by  Δ V  ) and solute diffusion is negligible, or (ii) completemixing  –  when solute movement is by  “ super-diffusion ”  withmaximal solute diffusion and no concentration difference of solutes between cell and microcapillary sap. It follows that at  fi nal Fig. 2.  Schematic presentation of the experimental cell in (a) srcinal and (b)  fi nalequilibrium in the pressure-clamp and pressure-relaxation method. Cell volume ( ν )was de fi ned as the volume bounded by the cell membrane. Cell sap volume inmicrocapillary and cell were treated as two compartments. The revised  ν o -theorywas derived based on the parameters as shown here, for the two possible extremescenarios of solute mixing between the cell and the microcapillary (i.e. byconvection and super-diffusion). (a) In srcinal equilibrium,  π  t  is identical to  π  o .(b) In  fi nal equilibrium,  π  t  is either identical to  π  o  (convection, i.e. no mixing) or  π  f  (super-diffusion, i.e. complete mixing). In (a,b),  N  ¼ amount of solutes, P  ¼ hydrostatic pressure,  R ¼ ideal gas constant,  T  ¼ temperature,  V  ¼ microcapillarysap volume,  Δ V  ¼ change in microcapillary sap volume (less than 5% of   ν ), π  ¼ osmotic pressure,  ν ¼ cell volume,  Ψ  ¼ water potential; subscripts  “ ex ” ¼ cellexternal,  “  f  ” ¼ fi nal,  “ o ” ¼ srcinal, and  “ t  ” ¼ microcapillary tip. T. Knipfer et al. / Journal of Theoretical Biology 359 (2014) 80 – 91 84
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