Page 397 - konishipaper

1. MATHEMATICALMODEL A. MinimalCAMModel
The C A M model that we wil l use has been studied by BlasiusandBeck(se[4，] [5])andhereweonlyoutlinethe minima1 C A M model. The model can be characterized by the major reactant po ols of C A M that generate the carbon flowduringthecircadiancycleasshowninFig.l.Thepol concentrationsarethefol1owing:
• internalCO2concentration，ω;
• malateconcentrationinthecytoplasm，x;
• malateconcentrationinthevacuole，y;and
• zisavariablethatdescribestheorderingofthelipid moleculesinthetonoplastmembrane.
一...--cω(t)externalCO2 cytoplasm
T(t)temperature vacuole
1.6 1.4 1.2
1~. 〉司
0.8 0.6 0.4 0._2
~
Ul Q malatecytopla幻n
tonoplast
Fig. l . 日 ow diagram of the C A M model showing as dynamic variables (encircIed)threreactantpols(internalC02concentration，ω;malatecon- centrationinthecytoplasm，x;andinthevacuole，y) andtheorderofthe tonoplast，z，withinaCAMcell.
These are the dynamic variables of the cyclic proces， whichareconectedbytheflows，Ul，U2，U3，duringthegain andlostermsofthemetabolites.Themodeldependsonthre externalcontrolparameters:temperature，T，lightintensity，L， andexternalCO2concentration，Cext'TheCAMmodelofa singlecelcanberewritenbythestate-spaceform:
Cco2 C O2 consumption by photosynthesi~ ， directlyproportional-tothe-externa1-controlp訂 ameter lightintensity，L(t)
Rco2 :CO2productionbyrespiration
Blasius etal. ca1culated the dynamic behavior using the dimensionlesvariableswiththep訂 ameters[4]:Cext 1， L(t) 1，T 0.238，0.242，0.246，0.2250，0.254， C = 5.5，C J = 1，C R = 1，ε = 0.0 01，T = 0.35，α = 1.5， 切 0.1，LK 0.5，R 0.1. The nonlinear function g(z，T)shownin2isaproximatedviathethird-orderspline interpolationfromthefigureshowninthereference(se[4]).
B.Calculation01ω usingJC02 TheCO2uptakeofasinglecelfromoutside，Jco2，isgiven
x=f(丸 L，T)+91(ω)(Cext(t)一切)+92L(t) ~(-U2+RC02) l
by
-~(-Ul+U2)
~(g(z，
WeproposedthereconstructionmethodoftheinternalCO2 concentration ，w(t) under the as sumption that Jco2，C，巴針。) areavailable[7].Moreover，wepresentedafedbackcontroler andafedforwardcontrolerwithapulsesignaltoshiftthe phaseoftheCAMmodelusingthelightintensityasaninput
I T)-y) J
Jm=CI(Cext(t)一切) 山 exp(αω)
(3)
Ul
，g2= I0 I I0 I
o 05
0• 1 O.15
0• 2
0.25 0.3 0.35
Fig.2. Thenonlinearfunctiony=g(z，T)whenT =0.238，0.24， 0.246，0.250，0.254.
U1=cz-f
U3=Jco2一 Cco2+Rco2
_ n_(Cext(t)-w) (2)
Rco2=CRポ告zdお wherethevariablesandtheparametersaredefinedasfolows:
切 ，x，y，z:statesvariables
T :temperature，controlparameter L:lightintensity，controlparameter Cext:extβrnalCO2concentration，controlparameter E :timeconstant
T :timeconstant
C ，c J ，c R ，L K ，W l ，α : constants
g(z，T) _thermodynamic_equilibrium value of malate concentrationinthevacuole
U1 the diference between malate infiux and efiux in-toandoutofthevacuole，modeledwiththedynamic hysteresis
U2 :thediferencebetweenmalateproductionfromCO ~ ~xa~ioI} byP~osp'hoe~olpy~u，:，ate carboxylase(PEPc)and itsdepletionbyaecarboxylation
U3 :thediferencebetweenCO2infiuxandefiux Jco2 : C O2 uptake from outside
U2=す-x
∞2ーしJ exp(αω) C∞2 =L(t)ω
色J
which is
、，‘，
1i
r 、.，、
=一叫
-rlu α
? CR000 1lil--」 一 位
ω Z U z - -E
「Ill1111111L
rpilli--Ill1L 一
一
司Eム ~中山 ~Qd
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