土壤淹水的麻風樹（JatrophacurcasL.)生理參數估算數學模型的建立

中圖分類號：S181、Q945.78 文獻標志碼：A

【Research significance】 Jatropha curcas L plantation has been recommended by government agenciesand non-government organizationsas apossible source of biodiesel on waste and unattended lands.Wastelandsareeitherofpoorfertility，saltaccumulation in root zone or seasonal or perennial flooding.However，it is common that the jatropha production is affected by soil flooding.Cumulative photosynthetic CO₂ assimilation rate （A） ，transpiration rate （E） and stomatal conductance （g_s） under normalandsoilfloodedconditionscanbetakenasan indexofproductivity.Calculationsofcumulativevaluesof physiological responses requiredetermination ofphysiological responses/functions of individual leaves.A functional relationship between physiologi calresponsesand leaf position isthe basis forcalculating the cumulative physiological responses of all leavespositioned overatwig.Mathematicalmodeling of physiological responses may prove useful in avoidingtimeconsuming field experiments，andalsoinsitu measurements of physiological responses. The pro cessof developing mathematical models for physiologicalparametersassociateddifferentsteps，suchas （a），parameterization，assigning values to model parameters，eitherbasedondirectmeasurementsorestimation，optimization；and（b），determiningvalues for parameters related to solving the governing equations，and validation，comparing the model output to anexperimental data set that wasnot used in the parameterization process.【Current research progress】 Determination of physiological response pattern of J curcasleavesispossiblebymeasuringapex，middle， and bottom leaf responses.Physiological responses of leaves over a twig follow the normal distribution curves（VERMA et al.，2012；JAN et al.，2024）. However，models were developed to predict physiological responses from interrelated physiological responses.For example，if net CO₂ assimilation response pattern of leaves over a twig is known，the pattern of transpiration rate or stomatal conductance can be also calculated or vice versa. Soil flooding is a significant factor limiting plant productivity， leading to downregulate carbon assimilation due to stomatal and mesophyll conductance limitations （FLEXAS et al.，2006； ZHOU et al.，2013；FAHAD et al.，2017） and changes in carboxylation rates and electron transport（ZHOU et al.，2013，2014）.【Breakthrough of the study】J. curcas，a deciduous drought resistant oil tree species widely distributed in tropical and subtropical areas，grows in Central and South America，Africa，and South East Asia （SCHMOOK et al.，1997; KHALID et al.，2021）. Different parts of J. curcas have been used for various purposes，such as animal feed，medicinal product and ecorestoration plantation in disturbed areas（HELLER，1996；OPENSHAW， 2000；TANG et al.，2007）. Oil extracted from seeds of J. curcas can be used for making soap，cosmetics， and as a diesel （kerosene or extender）（OPENSHAW，2000；LIANG et al.，2007）as well. The vast patches of waterlogged land remain fallow along the large canals or in flood prone area of river basins of India. Soil flooding affects 10% of the global land area（SETTER and WATERS，20O3），and is one of the most important constraints in agricultural crop production （SHABALA，2011）. The yield penalty resulting from soil flooding may vary between 15% and 80% depending on species，soil type and duration of the stress（ZHOU，2010；JAN et al.，2024）. India has about 16 Mha of land underwater logging and about 8.2 Mha under salinity（ABROL，1994）.The main reason for selecting J. curcas is its recommendations for large scale plantation on unattended lands （VERMA et al.， 2014）.Studies found a strong， nonlinear correlation between physiological parameters and the duration of soil flooding，and that stagnant soil flooding significantly affected the growth，development，and performanceof J. curcas （ZHOU， 2010；JANetal.，2024）.However，thereare few reportson the physiological response modelsof J_? curcasunder soil flooding conditions.【Key issues to be resolved】The present study isdevoted to the development of a model and procedure for predicting any physiological response if at least one response pattern isknown. The developed models were tested for physiological responses of J. curcasunder normal and soil flooding conditions for wider applications.

1 Materialsand Methods

1.1 Plantmaterial and growthconditions

The forty-five-day old jatropha seedlings were raised from stem cuttings of 18～20cm in length grown in pots of 30cm in diameter and 30cm in depth filled with fertile soil. The pots were watered regularly to the field capacity. The uniform seedlings were subjected to two water regimes，i.e.，up to field capacity and continuous soil flooding. The soil flooding in the pots were maintained by retaining water level 5cm above the soil surface for a period of 4 weeks，and photosynthetic performance of the plant leavesat differentpositionsweremeasured thereafter. At the end of soil flooding stress，the average relative soil moisture was 65% and 36% for soil flooding and control，respectively. The soil texture was silty clay loam， pH7.1 ，with organic carbon，nitrogen，phosphorus and potassium of 0.86% ， ，35.5 and 172kghm^-2 ，respectively. The experiment was carried outatDepartmentofBotany，UniversityofLucknow，Lucknow，India.

1.2 Measurementof photosynthetic characteristics

Photosynthetic CO₂ assimilation，transpiration and stomatal conductance were measured using an open system CIRAS-1，IRGA portable photosynthesissystem（PPSystem，England）undernatural sunlight at 9： 00-10： 00 am at photosynthetic photon flux density of .All measurements weretaken from the firsttotwelfthleaves（toptobottom）of jatropha plants during soil flooding stress （VERMA etal.，2014）.

1.2.1 Hypothesis

Response functions with similar trendsofvariations could form a relationship between two corresponding paired responses known as characteristic constants.Ifcharacteristic constantsbetween anytwo paired physiological response functions are known， the variation of one response function can easily be calculated froma set of another known response function. Net CO₂ assimilation rate，transpiration rate and stomatal conductanceofjatrophaleaveslocated over astem/twighavesimilartrendofvariation（VERMA et al.，2o12）and follow a Gaussian distribution as writtenbelow.

where， physiological response function with respect to leaf position， x= leaf position， p_m= physiological response of middle leaf， b= translation distance of peak， c= stretching factorof standard normaldistribution.

Since A ， E and g_s ofleaves overa stem/twig followsimilar trend[equation （1）] ，acharacteristic relationship existsbetween physiological response functions.Various characteristic constants between possible pairs of physiological response functions can be Writtenasbelow.

Thevaluesof physiological response characteristic constants are constantat given time for a plant.

1.2.2 CalculationProcedure

formation constants for converting E and g_s to A were calculatedbydividing A response of leaves by corresponding mean g_s of jatropha leaves.Characteristic constants for transformation of A and g_s to E response，andthemean E responsesofleaves were divided by corresponding A and g_s ，respectively. Similarly，characteristic transformation constants forconverting A and E responses to g_s were calculated by dividing mean of g_s by corresponding mean of A or E of jatropha leaves.Once these characteristic constants are worked out， the corresponding transformation maybe done as：

1，Net CO₂ assimilation （A ）from transpiration （E） （20

Net CO₂ assimilation response function in relation to transpiration response function can be written asbelow.

A_Ex=λ_AE×E_x

where，

A_Ex= calculatedvaluesof A from E forleaf position， x

characteristic constant to transform E to A E_x= observed values of E for leaf position， x ·λ_AE canbe calculated from equation（8）asunder.

Above equation can be rewritten asunder：

where，

Thevaluesof A ， E and g_s as functions of leaf positions were observed in three replications for control andsoil flooding conditions.The characteristic trans

Equation which can be used for transforming transpiration to photosynthetic rate can be finally writtenas：

2，Net CO₂ assimilation （A） from stomatal conductance （g_s ）

Net CO₂ assimilation response function in relation to stomatal response function can be written as below.

A_gsx=λ_Ags×g_sx

where，

A_gsx= calculatedvaluesof A from g_s for leaf position， x

λ_Ags= characteristic constant to transform g_s to A

g_sx= observed values of g_s asa function of leaf position， x

λ_Ags can be obtained from equation（16）in the following form.

where，

Thus equation which can be used for transforming g_s to A can be finally written as：

3，Transpiration （E） from net CO₂ assimilation 二

Transpiration response functions in relation to net CO₂ assimilation response function can be written asbelow.

E_Ax=λ_EA×A_x

where，

E_Ax= calculatedvalues of E from A for leaf posi-tion， x ：

λ_EA= characteristic constant to transform A to E

A_x= observed values of ?_A as a function of leaf po sition， x

λ_EA can be obtained from equation（16）and can bewritteninthe following form.

λ_EA=η_EAmη_bAE²η_bEA^2x

where，

The equation which can be used for transforming A into E can be finally written as ：

4，Transpiration （E） from stomatal conductance （g_s）

Transpiration response functions in relation to stomatal conductance response function can be written asbelow.

E_gsx=λ_Egs×g_sx

where，

E_gsx= calculated values of E from g_s for leaf posi

tion， x

λ_?Egs= characteristicconstantto transform g_s to E

g_sx=g_s as a function of leaf position， x ：

λ_Egs canbe obtained from equation（22）in the following form.

where，

The equation which can be used for transforming g_s into E can be finally written as ：

E_gsx=η_Egsmη_bgsE²η_bEgs?gsx^2x

5，Stomatal conductance （g_s） from net CO₂ as-similation （A ）

Stomatal conductance response functionsin relation to net CO₂ assimilation rate response function canbewrittenasbelow.

g_s，Ax=λ_gs，A×A_x

where，

（204號 g_sAx= calculated values of g_s from A for leaf posi-tion， x ：

λ_gsA= characteristic constant to transform A to g_s

λ_gs，4 can be obtained from equation（34）in the following form.

where，

The equation finally used for transforming net CO₂ assimilation rate into g_s is expressed as below：

g_sAx=η_gs，dmη_bAgs²η_bgs，A^2xA_x

6，Stomatal conductance （g_s） from transpiration

Stomatal conductance response functions in relation to transpiration rate response function can be writtenasbelow.

g_sEx=λ_gsE×E_x

where，

（204號 g_sEx= calculated values of g_s from E for leaf position， x

λ_gs，E= characteristic constant to transform E to g_s

λ_gsE can be obtained from equation（34）in the following form.

where，

The final equation for transforming transpiration

ratetostomatal conductanceisexpressedasbelow：

g_sEx=η_gsEmη_bEgs²η_bgsE^2xE_x

The different transformation characteristic constants were first calculated for different pairs of physiological responses，and later multiplied with the values of physiological responses needed to be transformed.Tocalculate A of J. curcas leaves from corresponding E response，the characteristic constants for converting E to A 1 （λ_AE ）multipliedwith E responses and to calculate A from g_s ，characteristic constants for converting g_s to A（λ_Ags） were multiplied with values of g_s of jatropha leaves. Similar calculations were done for converting A and g_s responses to E and A and E to g_s . Calculation of characteristic constants was also made for converting A to ，and constants for converting g_s to . The calculation of characteristic constants for converting A to and constants for converting g_s to E （λ_g，E） were presented inTable2，and calculationofcharacteristicconstants for converting A to g_s （λ_Ag_s ）and constants for converting E to g_s （λ_Eg_s） ）were presented in Table3.The calculatedvaluesof A fromobservedvaluesof E and g_s ， E fromobserved values of and g_s and g_s from observed A and E for control as well as under soil floodingconditions against leafpositions were presented in Table1to Table3.Percentdeviationsofcalculated values of and g_s responses with respect to observedvalueswerecalculatedasbelow：

Deviation （%）=[ （Observed value-Calculated value）/Observed value] ×100

Root mean square errors （RMSE）were also usedasparameter to compare calculated values with observed values.The root mean squares error can be calculated as：

2 Results

The variations of calculated values of E and g_s under control and soil flooding conditions with respect to leaf positions （x） ）wereshown in Fig.1 to Fig. 3. The patterns of observed values of A ， E and g_s with respect to leaf positions were similarand followed the normal distribution.Therefore，itis possible to calculate A ， E and g_s responses with respect to leaf positionsat a given time with each other.Percent deviationsand RMSEofcalculated valuesof A ， E and g_s withobserved values under control and soil flooding conditions are presented in Table 1 to Table 3.

2.1 Physiological characteristicconstants

It can seen that the values of λ_AE rangedfrom 1.05 to 3.65 under soil flooding and from 1.23 to 2.80 without soil flooding（control），and the values of λ_Ags ranged from O.07 to O.22under soil floodingand from 0.06 to O.12 under control conditions.Similarly，the values λ_AE ranged from 0.27 to 0.95 under soil flooding and from 0.35 to 0.82 under control conditions， and the values of λ_Egs ranged from 0.03 to 0.07under soil flooding and from O.02 to O.06 under control conditions.Thevaluesof λ_AE ranged from 7.68to 15.00 with soil floodingand from 8.50 to 16.58without soil flooding. The values of λ_gsE ranged from 16.88 to 41.45without soil flooding and from 15.38 to 37.84 during soil flooding stress（Fig.1 to Fig.3）.

2.2 Calculation of photosynthetic responses

Thepercentdeviationsand RMSEofthecalculatedvaluesof A ， E and g_s from the observed values wereshowninTable1toTable3.Itcanbeseenfrom Fig.1 that the predicted values of A ， E and g_s were in closeagreementto the observed valuesundercontrol as well as waterlogged conditions. Average percent deviationsof thecalculatedvalues of A from E were 7.36% ， 1.69% and 5.14% forthree replications under control conditions and 10.91% ， 10.11% and 8.40% for the corresponding replications under waterlogged conditions. The corresponding root mean squares errors（RMSE）were found to be 0.3614，0.1091 and 0.2864 for controland 0.2392，0.1902and 0.2270 for waterloggedtreatment forall the three replications. Similarly，the average deviationsof the calculated valuesof A from g_s were found to be 4.725， 1.69% and 5.14% under control and 11.17% ， 10.31% and 6.33% underwaterlogged conditions forall thereplications.ThecorrespondingRMSEwerecalculatedas 0.4367，0.5075 and 0.3601 for control and 0.1611， 0.1660 and O.1445 for waterlogged treatment in three replications（Table1）.

Variationsoftheobservedandcalculatedvalues of E wereshownin Fig.2 and the calculated percent deviationsof E from the A and g_s were presented in Table2.It could be seen from Table2 that the percent deviationsofthe calculated E from the observed A were 5.80% ， 1.70% and 5.58% under control and 9.35% ， 9.96% and 8.21% under waterlogged conditions for all the three replications.The corresponding RMSEwere0.1716，0.0569and0.1350undercontrol and 0.0977，0.0812 and 0.0996 under waterlogged conditions for all the three replications.

Averagedeviationsofthecalculatedvaluesof E from g_s were 6.71% ， 4.51% and 5.58% under control and 11.61% ， 5.48% and 8.46% under waterlogged conditions for the corresponding replications. The correspondingRMSE were foundto be 0.2039，0.2474 and 0.1954 for control and 0.0881， 0.0548 and 0.0814 for waterlogged conditions，respectively.The variationsof observed and calculated response functions of g_s and calculated percent deviations of g_s werepresentedinTable3.

Fig.2 showed a close agreement between the observed and predicted values of g_s Thepercent deviationsofcalculatedvaluesof g_s from A with observed valueswere foundtobe 4.84% ， 4.33% and 3.87% withthecorrespondingRMSEas4.5987，5.4724and 3.9477 under control and average deviationsof 13.06% ， 9.65% and 5.88% with the corresponding RMSEas1.4677，1.7429 and1.3510underwaterlogged conditions，respectively.The averagedeviationsofcalculated g_s from E were 7.43% ， 4.27% and 5.83% forcontrol and 13.21% ， 5.35% and 7.99% underwaterloggedconditionsforallthethreereplications.ThecorrespondingRMSEwere4.6322，5.6754 and4.0070undercontrol and2.2123，1.5386，2.2547 underwaterlogged conditions，respectively，for all thereplication.Thedeviationswereslightlyhighereither for apex leaves（leaf positions 1 and 2）or most bottom leaves when photosynthetic responses were verylow. The RMSE were comparatively higher for the calculated physiological responses under soil flooding conditions，however，the valueswere low. The calculated and observed values of A ， E and g_s wereincloseagreement.Hence，thehypothesiswas verified makingitpossibletocalculate physiological responses such as A ， E and g_s from each other with verylessdeviationsfromtheobservedvalues.

3 Discussion

The search for alternate source of fossil fuel overthe globeis on.The oilof J curcas seeds has beenusedasbiodiesel.Jatrophaplantationisrecommended asapossible sourceofbiodiesel onwasteand unattendedlands.Indiaishavingvastpatchofwaterlogged land along the large canals，where jatropha plantation is also being recommended. Physiological responses of J. curcas leaves located over twigvaries asthatof normallydistributedcurve.Thecumulative A ， E and g_s under control and waterlogged conditions canbe takenas an index forplantperformance and productivity（VERMA etal.，2014；KHALID et al.， 2021；JANetal.，2024）.

Cumulative values of photosynthetic responses requiredeterminationofphysiological observations of individual leavesundernormal aswellaswaterlogged conditions or to any other conditions.Mathematical modeling of physiological responses may prove useful in avoiding time consuming in situ measurements. Inthe present study，characteristic physiological responsetransformation constantsweredetermined fromthe observed data and prediction of the other physiological responsesweredone.Theaverage deviations of the predicted values of photosynthetic responsesover theobserved values were in acceptable range，hence，the hypothesis for existing a specific characteristic functional relationship with all possible pairsof ?_A ， E and g_s werevalidated.

4 Conclusion

The present study demonstrated a possibility of calculation for physiological responses following similarpattern from any other observed or measured physiological response pattern，which could reduce the fieldmeasurements soto save time，laborand cost.

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