BioSystems, 19 (1986) 111-122 111 Elsevier Scientific Publishers Ireland Ltd.

S O M E C Y B E R N E T I C A S P E C T S OF E A R L Y LEAF G R O W T H

JERZY PI~rKA Institute of Biocybernetics and Biomedical Engineering, Polish Academy of Sciences, KRN 55, 00-818 Warsaw (Poland)

(Received August 28th, 1985)

An analysis of photosynthesis for a leaf treated as a control system was carried out. Some similarities between selected functions of the leaf and certain mechanical and hydraulic processes were discussed, arid on this basis a leaf analog and its mathematical model were elaborated. Properties of the model, which can be treated as a mathematical model for the growth of a young leaf or a young plant in comfortable conditions, were worked out with the help of automatic control theory. The theoretical results obtained were compared with experimental results for lettuce cultivated in a greenhouse.

Keywords: Cybernetics; Control system; Leaf analog; Photosynthesis; Mathematical model; Leaf growth.

I n t r o d u c t i o n

Sztencel a n d Ze l aws k i (1981) desc r ibed the wel l -known effect of a pos i t i ve f e e d b a c k o c c u r r i n g du r ing p h o t o s y n t h e s i s . L igh t fa l l ing on a leaf s u r f a c e c a u s e s the s y n t h e s i s of g r o w t h - p r o m o t e r s , wh ich leads to g r o w t h of the leaf a rea . The b igger leaf a r e a i n c r e a s e s the to ta l p r o d u c t i o n of a s s i fn i l a t e s , c a u s i n g f u r t h e r g r o w t h of the leaf a rea . An e x p l a n a t i o n of such a n effect a n d i ts q u a n t i t a t i v e a n a l y s i s is the p u r p o s e of th i s paper .

The m e t h o d of a n a l o g y is one o f t en u sed in t echno logy . W h e n a n a n a l y s e d p roce s s is too compl i ca t ed a s imi la r , b e t t e r - k n o w n p roces s is ana ly sed . For example , i n s t e ad of fluid flow in a tube , we can cons ide r a c u r r e n t flow in an e lec t r ic c i rcui t . Such a c i rcu i t is called a p h y s i c a l ana log of the s y s t e m . I t is neces- s a r y to choose a p p r o p r i a t e c r i t e r i a of anal - ogy. T h e n d i f fe rent ia l e q u a t i o n s desc r ib ing p h e n o m e n a in the ana log can be appl ied to the s y s t e m itself .

The a n a l o g y m e t h o d was appl ied to a p h o t o s y n t h e s i s a n a l y s i s . The leaf was t r e a t e d as a cont ro l s y s t e m . : q ~ e s e were descr ibed:

(1) the leaf ana log ; (2) the m a t h e m a t i c a l model of the ana log ; (3) the cybe rne t i c a l f o r m a l i s a t i o n of the m a t h e m a t i c a l model ; (4) the d e t e r m i n a t i o n of s y s t e m p r o p e r t i e s ba sed on t h o s e of the m a t h e m a t i c a l model .

General assumptions

Accord ing to M i t h o r p e a n d M o o r b y (1979), only 85-90% of p lan t d ry m a s s is o rgan ic c o m p o u n d s p roduced as a r e su l t of p h o t o s y n t h e s i s . In order to s impl i fy the a n a l y s i s , i t is a s s u m e d t h a t an i nc r ea se in b i o m a s s depends on the o rgan ic c o m p o u n d (CH20) , . T h i s c o m p o u n d is p roduced du r ing p h o t o s y n t h e s i s only , in a c c o r d a n c e w i th the e q u a t i o n

C02 + 2HeO -* ( C H 2 0 ) . + H 2 0 + 02

One p a r t of th i s s u b s t a n c e m a i n t a i n s p lant - l iving p r o c e s s e s and the o the r i n c r e a s e s the b i o m a s s . I t is a s s u m e d t h a t

(1) the a b o v e - m e n t i o n e d s u b s t a n c e , wh ich we shal l r e fe r to as a suga r , is a l iquid w i th a dens i t y p,;

0303-2647/861503.50 O 1986 Elsevier Scientific Publishers Ireland Ltd. Published and Prinl~ed in Ireland

112

(2) one part of the sugar is stored in the leaf, and the other is sent to stem and roots;

(3) the sugar mass is converted into the plant dry mass;

(4) a sufficient amount of water is always available for the plant. Thus, water does not inhibit an increase of the biomass.

The leaf as a sys tem fulfils three functions:

(a) it represents a surface on which light energy falls;

(b) it is a specific t ransducer from light energy into an output product-- the sugar:

(c) it is a sugar tank, which has well-defined dynamics.

Thus, the leaf analog is composed of three subanalogs.

Leaf analog in photosynthesis

Leaf surface subanalog,

Figure la presents the general idea of the analogy. The light with a flux density @l falls on a surface 1. This surface is covered with a mobile plate 2. The leaf area is represented only by the non-covered part of the surface 1.

In other words, the analog represents a full- grown leaf in which a certain number of chloroplasts are inactive under the plate 2. The leaf growth is a process of chloroplast activation by increasing the leaf length It until a maximum value lira is reached. The leaf width s~ is constant. It is assumed that: (1) light falls on the leaf perpendicularly; (2) photosynthet ic activity of the surface subanalog is equal to the sum of activities of both leaf surfaces.

The two-dimensional leaf surface sub- analog has been simplified into a one- dimensional subanalog in Fig. lb. The leaf is represented by a beam 1' covered with a mobile plate 2'. The uncovered part of the beam is equivalent to a leaf area. The light flux density distribution is represented by a continuous distribution of generalized forces. These forces correspond to light flux vectors integrated on the leaf width sv When a leaf area increases, then the plate 2' uncovers the beam 1' and generalized forces exert pressure on it.

Leaf subanalog as an energy transducer

A linear dependence between a light flux density clh and a net photosynthet ic rate often occurs in the early-stage leaf growth

a) '= sL b) ¢#t'sL

u L~of ~t i I

I

dir~ct[on di re.cfion

Fig. 1. Leaf surface subanalog.

for ~z which is small enough. I f we assume a constant leaf thickness, then sugar pro- duction depends only on the leaf area and light flux density.

Figure 2a presents the general idea of biohydraulic analogy. A leaf as energy transducer was transformed into a hydraulic valve 4 in the pipeline 1. The pipeline has a rectangular shape with a width /~m and non- limited depth. The sugar flows laminarly by the pipeline. The, density ¢P,o of this flow is energetically equivalent to the. light flux density ~z. There: are no flow disturbances or pressure changes caused by a change of the valve position. The valve mask 3 is equivalent to the plate 2 in Fig. la. The return spring 5 (Fig. 2a) shuts the valve when CPz--cb,o = 0. A sugar mass flow Q~o in the output part 2, ,depends only on the value hr-- the opening valve 4 because other parameters of the subanalog, such as the sugar jet width i!l and the flow density of the

113

sugar dP~o, correspond to the same parameters from Fig. 1, as Iz and ~z.

Figure 2b presents a beam balance as a result of the biomechanical analogy. Forces P1 and P2 represent, respectively, generalized force proportional to energy absorbed by a leaf, and generalized force proportional to a deflection of the spring 5. This deflection is equal to the value hp of the valve opening. In this way hp corresponds to the light flux density.

Leaf subanalog as a sugar tank

The sugar is a product of photosynthesis. One part of it is stocked in the leaf and the other is sent to stem and roots. Applying a biohydraulic analogy, the leaf can be compared to a tank (Fig. 3) which stocks the sugar. The maximal length of the tank 10 is equal to /~, whereas the active length /~ = lz is limited by the plate 6. The width of the

plate r~o e m e n t di~c~:ion

b) q

Fig. 2. Leaf sul~na]log as an energy transducer.

Qso

114

plate is equal to the tank width and to the leaf width (s~ =st). In order to limit the analysis of the problem to the suitable subanalog, it is necessary to use both indices (s) and (l) referring to the leaf area.

The sugar mass flow Q~ from the tube 7 (Fig. 3) can fill the tank up to the height h~m. In order to counterbalance forces acting on the plate 6 in this subanalog, a certain amount of compensating fluid with the mass flow rate Qbo > Q~o flows through channel 5 into the tank. This fluid has the same physical and chemical parameters as the sugar. On the bottom of the tank there are two pressure sensors 3, placed on both sides

of the plate 6. The difference between the fluid pressure and the sugar pressure is amplified by the amplifier 1, which actuates the valve placed on the discharge channel 2 in such a way tha t both levels of liquids from both sides of the plate 6 are equal, at every moment in a range h~d~<h~<h~m. Thus, forces acting on the plate are counter- balanced.

The sugar flows out of the tank through three resistances Rd, R~, R, shown sym- bolically in Fig. 3a. The mass flow rate Qd through the resistance Rd represents sugar losses in the tank for maintaining metabolism in the plant during the dark. The

iV• q b o ~ ,

'l Q° 1 qbo c ~ > .,.,

4

a) Qso

~-~ 6 '7 8

9

t0

4t

42

S

Fig. 3. Leaf subanalog as a sugar tank.

qd qa qr

115

mass flow rate Qa through the resis tance R~ represents additional sugar losses caused by photorespirat ion. This resis tance is con- trolled by the raass flow sensor 8 and the amplifier 9, because greater photorespirat ion means greater sugar losses. Thus, an output signal of the sensor 8 is proportional to the supply mass flow rate Q~o, and as a consequence, to the rate of metabolism. The mass flow rate Qr through the resis tance Rr represents sugar sent to s tem and roots.

The above-mentioned res is tances shown in Fig. 3a are presented in a different way in Fig. 3b. The mass flow rates Qd, Qa, Qr increase in accordance with an increase of the leaf area. :Every uni t of the leaf area needs a certain amount of sugar in order to maintain its life processes, and every uni t produces a sugar excess which is sent to other parts of the plant. Thus, increase of the leaf area is equal to the increase of the parameter /8. Figure 3b presents channels 11 and 12 from Fig. 3a in the form of slits along the length of the tank. The sugar flows over the edges of the slits and the mass flow rates Qd, Qa, Qr depend on the posit ion /8 of plate 6. Only Qa depends additionally on Q~o. This influence is simulated by the variable slit height, which is shown symbolically by the valve resis tance Ra.

The possible changes of the sugar level may be classified into 3 types:

(A) The sugar level h~ rises up to the maximum value h~,~. This increase appears j u s t af ter the light has fallen down on a leaf. The level of the compensat ing fluid follows the sugar level.

(B) l ~ e sugar level h~ = h~,~ is constant all the time, and the sugar is sent by channel 11 and resis tance Rr to the other plant organs. The presence of the r e s i s t a n c e R~ c a u s e s a m i n i m a l i n c r e a s e of the s u g a r level A h ~ ~-0, and as a consequence a sugar p r e s s u r e o n plate 6. The compensat ing fluid level cannot

(c)

exceed the maximum value h.m, because of channel 4, through which a fluid excess flows out. Thus, plate 6 affected by this sugar p r e s s u r e m o v e s to the left s ide and i n c r e a s e s the distance/8. Light, and as a consequence the sugar inflow Q~o, is cut off. The plate movement is stopped in a new position, and the resis tance R~ is closed. The sugar flows out of the tank only through resis tance Rd in order to maintain life processes. The sugar level decreases f rom maximum value b~,~ to minimum value h~d. Further drop of the level is impossible, because Qd =0. As a result the plant dies. The level of the compensat ing fluid follows the sugar level.

Leaf analog,

The analog presented in Fig. 4 is composed of 3 subanalogs described above. The beam balance of the analog is under pressure of the generalized forces (~zsl) on one side and on the other there are the hydraulic valve and closing spring. The valve controls the sugar quant i ty Q~o flowing by the pipeline into the tank. The sugar affects the plate, which moves to the left side of the tank. This movement is t ransmit ted through a mech- anical gear into the cross beam. The mechanical gear is composed of the rack and the rack wheel fixed on the threaded roller with the nut. The general t ransmiss ion ratio of the gear is 1:1. The cross beam moves two mask plates: one masks the field of generalized forces and the other one masks an equivalent of these forces, the sugar flow. Arrows in Fig. 4 show the movement directions of the analog elements.

In the described analog there are: one input quant i ty , the light flux densi ty ¢Pl, and two output quanti t ies , the sugar level height h~ and the tank area fs = / ~ (equal to the leaf area fz =/lSl). Both output quanti t ies vary alternately depending on successive light and dark periods.

116

5l a b

S

@ ~ ts

Qd Qo Qr

Fig. 4. Leaf analog composed of three subanalogs.

Mathematical model of the analog

Introduction

The analysis of the analog functioning can be divided into four problems: (a) force- equilibrium on the beam balance; (b) sugar flow through t h e valve; (c) sugar tank dynamics; (d) analog dynamics. The math- ematical model of the analog refers to an early-stage of the leaf growth. Not too great a value of ¢Pt is taken into account when a linear dependence between a light flux densi ty rate and a photosynthesis rate is considered.

Evaluation of the sugar quantity flowing into the tank

From an ins tant equilibrium of the forces on the beam balance (Fig. 2b) follows that

P~a = P2b (1)

where P1 is the generalized force proportional to the energy absorbed by the leaf in a unit t ime in [N]; P2 is the generalized force proportional to the va lue hp of the valve 4 in [N]; a and b are the distances of the forces P1 and P2 from the pivoting point in [m].

P, = k~b (2)

where k~ is the constant in [N/W]; Ii is the light energy absorbed by the leaf in [W].

b = q)~slll = (Ptf~ (3)

where q)l in [Wire's]; sz and It in [m]; ~ in [m 2]

Pe = kphp (4)

where kp is the ,~pring constant in IN/m]; hp in [m]. It was assumed tha t the sugar mass flow rate Q~o in [kg/s] is the following:

Q~ = k~h. (5)

where k~o is the constant in [kg/(s m)]. After evaluating hp from Eqns. 1-4, Q~o may be rewritten as

Q~ : K , , f z , ~ z (6)

where /~j = ~?k~j; ~ includes the efficiency of all biochemical processes (7 = alb); k~j is the coefficient showing how much of the sugar in [kg] can be obtained from 1 J of energy, when the metabolic processes occur without any losses (k~j = k~k1/kv). I~j is the sugar con- version coefficient of the value (bt, when the plant grows in comfortable conditions. Deviations from comfortable conditions ef- fect ,/.

Sugar tank dynamics

In order to evaluate dynamic properties of any system, it is necessary to evaluate its t ransfer function. As mentioned above, in the cases A and C of the process we have

,hsd. <~ hs, < hsm (7)

and plate 6 (Fig. 3a) remains in a fixed position at the distance/~ from the tank edge.

Case A In this case, h~ rises within the limits

shown in Eqn. 7. The flow continuity

117

equation of the sugar passing through the tank may be writ ten as

dh~ 1 c - ( Q ~ - Q d - Qo) (8)

dt ps

where C is the hydraulic capacitance of the tank in [m2]. (C= dVJdb~=f~); V, is the sugar volume in the tank in [m3]. (Vs = l~s~h~ = f~h~); f~ is the tank area in [m2]; p~ is the sugar density in [kg/ma]; Qd is the sugar loss for maintenance of metabolism in the plant during the dark in [kg/s]; Qa is the additional sugar loss caused by photo- respiration in [kg/s]. Substi tuting Eqns. A4 and A7 into Eqn. 8, we obtain

dh~ + psRodC-~ b~ = R ~ ( 1 - x)Q~o (9)

After applying the Laplace t ransform we get

h~(s) /¢~ Q~o(s) 7d~S + 1 (10)

where ' /~ is the time constant of the tank in [s], (Td~=psRoaC=Rjapsfsj); Rd~ is the equivalent resistance of the outflow quan- tities in [ms/kg], (Rd~ = Rod(l--A); X(s) are the indices showing the input and output quanti t ies in the domain of the complex variable function. The time constant 7~ does not depend on the current leaf area fs, because every uni t of the leaf area has the same properties. It depends only on these properties.

It follows from Eqn. 10 tha t the tank is the first-order inertial element. When the input signal Q~o of the tank is the uni t step function, the sugar level is found from the inverse Laplace t ransform of Eqn. 10 as

h~ = Rd, Q~o(1- e-tin',,,) (11)

In this analysis the delay ~ between the output and input signals was not taken into account. In the analog this delay is time for the sugar to flow from the valve into the

118

tank. It represents the time necessary for chemical reactions in the leaf. It is possible to include the delay by replacing (t) by ( t - ~'~).

Case C In this case ¢Pz = 0 therefore Q~ = 0, and

the sugar level h~ decreases within the limits of Eqn. 7. The flow continui ty equation (Eqn. 8) when Qo = 0 can be rewrit ten as

d~ 1 C - Qd (12)

dt

From Eqns. A4 and 12 we obtain

h~ = h ~ e -"% (13)

where :/~ is the time constant of the tank in [s], (7~ =p~RodC=Rjdp~fsj). l~nus, the time constant Td = Td,,.

Case B In case B, the sugar level remains constant

during photosynthesis , and the sugar presses on plate 6 (Fig. 3) shift ing it to the left side of the tank. The growth conditions are the following:

0 < /s < /sin h s = ham = constant (14)

The flow continuity equation of the sugar passing through the tank may be written as

dV~ 1

dt m ( Qso- Qdm- Qa- Qr) (15)

where d V d d t = d ( l ~ h ~ m ) / d t = h ,mdfdd t ; Qr is the mass flow rate of the sugar sent into the stem and roots in [kg/s]. Substi tuting Eqns. A5, A7 and A9 into Eqn. 15 we obtain

Tr ~--~ft~ + f~ -(1- A) A Q~o (16)

where Tr is the time constant in [s], (Tr = psh~m/A); A is the constant in [kg/(sm2)]. (A = (h~mlRjd + h d R j D I f , fl.

Equation 16 is equivalent to Eqn. 9. Thus

f~(s) (1 - A)/A

Q~o(s) TrS + 1 (17)

In case B, the tank has the same properties as in cases A and C, i.e. the properties of the first-order inertial element. When the input signal Q~o is the unit step function, the output signal f~ is the following

1 - A f~ - Q~,(1 - e -#%) (18)

A

Replacing (t) by ( t - r r ) we can take into account the delay Tr necessary for chemical reactions.

Analog, d y n a m i c s

In the analog presented, there are three kinds of input signals:

(1) The sugar mass flow rate Q~o is the input signal for the tank. As presented above, the tank has properties of the first-order inertial element. The element has dif- ferent time constants suitable for cases A, B or C. According to Eqn. 18, Fig. 5 presents a dependence between the input and output signals, when Q~o is a unit step function. The step function response curve has a saturation.

(2) The light flux density ~z is the input signal for the leaf treated as a control system.

(3) The generalized force P, is the input signal for the leaf analog. It follows from Eqns. 2 and 3, when Cbz = constant , the input signal P, increases in a continuous way, because/~ also increases.

We can find the dependence between fs and ¢bz putting Eqn. 6 into Eqn. 16 for/~ = f~, and subst i tut ing fs = fJf~o. Solving this relation we obtain

fs = fsoe(B*'-l)~ (19)

119

20 - - T - - [ - F- I I ] /

q6

46

I 42 o - caLcuLa~d data

x - ex~xzrim~ntol data ~ I ~3 40 ao=ordir, a~ ~0 Dec [.t] / :

~3 ,

4

I I _ i I I [ " 400 £0C 5,30 .~00 ,500 600

J~Qve~'expo-%[tion CO Licjht [hours~

Fig. 5. T h e d ry m a s s of l e t tuce cu l t iva t ed in a g r e e n h o u s e a s a f u n c t i o n of t h e t i m e of e x p o s u r e of l eaves to l ight . T h e l igh t f lux d e n s i t y w a s t a k e n in to a c c o u n t a cco rd ing to Eqn . 21.

f

l 7C0

where m is the current leaf mass, and me is the initial leaf mass. The quant i ty Tr in Eqn. 18 is the time constant of the leaf treated as the sugar tank, but in Eqns. 19 and 21 it plays the role of a certain time constant characterist ic of a given species.

A typical exponential expression des- cribing the mass increment of an organ- ism is presented in the form

rn = mee ~t (22)

where a is the parameter chosen in such a way tha t Eqn. 22 could be fitted to an experimental curve. In this equation the growth depends only on the initial value me and the time t. But the time is not the cause of the growth. Only light initiates it. Therefore, Eqn. 21 gives more information because it relates cause and effect. This relationship between the input and output quanti t ies Ct and m includes parameters B and T~ characteristic of a given plant species. Equation 21 describes the growth of the young leaf in comfortable conditions. When a plant lacks a stem as in the lettuce, this equation also describes the growth of the young plant.

where Comparison of the theoretical and experimental results

B = (1 - ~.)K,.i (20) A

The result given by Eqn. 19 is interesting. The exponential expression describes well many biological organisms at the early- growth stage. This fact was emphasised by Kowalik (1976), Hunt (1981), Sztencel and Zelawski (1981) . . . . etc. Multiplying both sides of Eqn. 19 by (pa), where p is the leaf density and a is the leaf thickness, we obtain the mass increment equation of the young leaf. Thus

r n : ?r loe(B¢,-1)~r (21)

The results given above were compared with experimental results for lettuce cul- t ivated in a greenhouse. The comparison was based on data given by Dec (1982). The parameters B and Tr were calculated from Eqn. 21 in which the light flux density as well as the dry matter of the lettuce were taken from the data obtained for the first three measurement points. Other points for dry mass of the lettuce were calculated from Eqn. 21. The result of one of the experimental series is presented in Fig. 5. It shows tha t calculated data correspond well with experimental data up to 500 h of the leaves' exposure to light. In this comparison it was

120

assumed that the change of sugar level during one night was sufficiently small to be neglected.

Conclusions

This analysis leads to the following con- clusions:

(1) In the analysis of plant physiology processes it is possible to use the method of biomechanical or biohydraulic analogy in order to work out an analog of an analysed process. Mathematical des- cription of the analog facilitates a time- quant i ta t ive analysis of the process.

(2) The mathematical model of the analog of photosynthes is relates effect to cause. The effect is the dry mass of the leaf or plant and the cause is the light. The cybernetic formalization of the description was helpful in the analysis. The preliminary comparison between experimental and theoretical data shows a good agreement for a small value of the light flux densi ty q~z, when the plant grows in a greenhouse. In this case the plant is not influenced by environmental conditions. The model needs an ad- aptat ion to every value of ~bt.

(3) Theoretical results obtained from the analysis are encouraging but it is necessary to include in the model some parameters involving the senescence of the plant and its environment.

Appendix

Calculation o f the sugar losses Qd

The resis tance R for the laminar flow is defined as follows

dh Ah R - d Q - A Q (A1)

where R is in [ms/kg]; Ah is the pressure drop on the resis tance R in [m]; AQ is the

mass flow rate through the resis tance R under the influence of the pressure drop Ah in [kg/s].

Such a general relation is not adequate for the calculation of sugar losses. When the leaf thickness is constant , the mass flow rate Qd depends on the leaf area, because the greater leaf volume needs more sugar to maintain life processes. So, Qd increases according to the increment of the active tank length /8, and to the slit length (Fig. 3b). In Eqn. A1 for h = 0 is Q = 0, so we can rewrite the equation as Q = h/R. Defining the uni tary mass flow rate Qj as: Q~ = h/Rj, where Rj is the uni tary resis tance in [ms/kg] for the slit length /~j, when /~j refers to the uni tary tank area fsj = / ~ = 1[m2], we can calculate any mass flow rate referring to any leaf area as

fs f ~ h _ h Q =-~j QJ=f~j " Rj RO (A2)

where RO is the current flow resis tance for any slit in [ms/kg]

Ro : ( f . j l f~)Rj (A3)

The resis tance Ro has the same features as the resis tance R, and additionally takes into account the leaf area, because f~=/~. Assuming that the plant is exposed to light every day, and even after one night the value h~ >> b~, we can apply Eqns. A2 and A3 to the calculation of the sugar losses Qd in [kg/s].

Qd ~ Rod Rid fsj (A4)

where Rid is the uni tary flow resis tance for the uni tary leaf area during the dark respiration.

Calculation o f the mass flow rate Qa

The mass flow rate Qa through the resis tance R~ depends on the value of output signal from sensor 8 and amplifier 9 (Fig. 3a). Thus, the resis tance R~ depends on Q.o and

~L

Fig. A1. The linear dependence of net photosynthesis P on the light flux density ¢bl, for the small value of dPt.

takes into account the dependence of chloroplasts acti~rity on the light flux densi ty ¢l. Figure A1 shows the linear dependence of net photosynthesis P on the light flux densi ty dpz, for small opt value, when Qd = Qd~. Assuming tha t P is proportional to the net quant i ty of sugar ( Q~o - Qd - Qa) remaining in the tank according to Eqn. 8, We derive the furtction ( Q~, - Qd - Qa) = f (¢z) as a result of overlapping of the following functions (Fig. A2):

(1) Q~o = f(cl)l), according to Eqn. 6;

Q~o

Q

//

~ , ~ ~ fLow otJi~ of the tonk

Fig. A2. The dependence of the sugar mass flow rates Q Onto and out of the tank) on the light flux density ¢l-

121

(2) Qd = Qdm -- f(dPl) = constant, the maxi- mum value Qd;

(3) Qa = f(q)z), the unknown quantity; (4) ( Q d m + Q ~ ) = f ( d p z ) , the result of over-

lapping of two curves; (5) ( Q ~ - Q d m - Q o ) = f ( d p l ) , the result of

overlapping of three curves.

We can find the dependence Q~ = f (¢ t ) f rom Fig. A2, when:

(a) a chloroplast photosynthet ic activation depends only on the light flux density cPl, and it increases linearly;

(b) this activation does not depend on the sugar level in the tank and it is a special feature of the species.

A value Q~ does not depend on the current mass flow rate Qd. Thus, the inclination angle a--constant, and we can calculate Q~ for Qd = Qdm, h , = h~n, dPt = dPlo, which means in the point where the absolute value of photerespirat ion is equal to the absolute value of dark respiration. From Eqn. A4 we obtain

Qdm = hsm = hsm f8 (A5) Rod Rjd fsj

From Fig. 2 it follows

tagc¢ : Qdm _ Qa (A6) q)~ cI)l

Taking into account Eqn. 6 we obtain

~t Qdm Qa = ~ - ~ Qdm = Q~oo Q~o = AQ,o (A7)

where

/z,m 1 1 (A8)

Thus, A is a constant characteristic of a given species.

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Calculation o f the mass flow rate Q~

Defin ing the m a s s flow r a t e Qr of the s u g a r p a s s i n g to the s t e m and roo t s s imi la r ly to Eqn. A4, we can wr i t e

h~ h~ f~ Qr - - Ro~ - Rj~ fsj (A9)

w h e r e h~ is the s u g a r level he i gh t in channe l 11 (Fig. 3). The fol lowing a s s u m p t i o n s a re t a k e n into a c c o u n t in th i s case:

(1) The s u g a r f lowing in to channe l 11 fills i t and c a u s e s a level to r i se a t the he igh t h~ b e c a u s e of the r e s i s t a n c e Rot (in Fig. 3, Rr). D i m e n s i o n s of o the r p l an t o r g a n s i nc r ea se in a c c o r d a n c e w i th the in- c r e m e n t of t a n k a r e a fs, i.e. to the leaf a rea . The d i m e n s i o n s of channe l 11 i n c r e a s e in s u c h a w a y , t h a t hr = c o n s t a n t . T h u s the c o n s t a n t p r e s s u r e on p lan t o r g a n s c a u s e s the i r g rowth .

(2) The re is no s u g a r t r a n s p o r t be tween p lan t o r g a n s du r ing d a r k n e s s ; howeve r , du r i ng

p h o t o s y n t h e s i s the s u g a r is t r a n s p o r t e d f r o m the leaves , t h r o u g h the s t e m down to the roo ts .

(3) All p lan t leaves a re suff ic ient ly y o u n g , so t he r e is no s u g a r t r a n s p o r t f r o m the y o u n g leaves in to old ones .

R e f e r e n c e s

Dec, E., 1982, Modelowanie procesu wzrostu ro~lin. Doktors Thesis, AR Krak6w, Wydzial Rolniczy, pp. 91-97.

Hunt, R., 1981, The fitted curve in plant growth studies, in: Mathematics and Plant Physiology, D. A. Rose and D. A. Charles-Edwards (ed.) (Academic Press, London) pp. 283-298.

Kowalik, P., 1976, Podstawy teoretyczne agrohydrologii Zulaw (Gda6skie Towarzystwo Naukowe, Wydzial IV Nauk Technicznych, Acta Technica Gedanensia Nr 11, Gdafisk) p. 18.

Milthrope, F.L. and Moorby, J., 1979, Wst~p do fizjologii plonowania ro~lin (PWRL, Warszawa) p. 94 (trans., 1974, An Introduction to Crop Physiology, Cambridge University Press).

Sztencel, J. and Zelawski, W., 1981, Akumulacja suchej masy w ro~linie w fazie tzw. wzrostu wykladniczego. Post~py Nauk Rolniczych 5,,61-68.