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Feedback Control of Biological Rhythm in Crassulacean Acid Metabolism by CO2-Uptake Signal*
4th IFAC Conference on Analysis and Control of Chaotic Systems 4th Conference on August 2015. Tokyo, Japan and 4th IFAC IFAC26-28, Conference on Analysis Analysis and Control Control of of Chaotic Chaotic Systems Systems 4th IFAC Conference on Analysis and Control of Chaotic Systems August 26-28, 2015. Tokyo, Japan Available online at www.sciencedirect.com August 26-28, 2015. Tokyo, Japan August 26-28, 2015. Tokyo, Japan
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Feedback Control of Biological Rhythm Feedback Control of Biological Rhythm Feedback Control of Biological Rhythm Crassulacean Acid Metabolism by Crassulacean Acid Metabolism by Crassulacean Acid Metabolism by CO2-Uptake Signal CO2-Uptake CO2-Uptake Signal Signal
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Chisato Matoba ∗∗ Haruo Suemitsu ∗∗ Chisato Suemitsu ∗∗ Chisato Matoba Haruo TakamiMatoba Matsuo∗∗∗∗ Haruo Chisato Matoba Haruo Suemitsu Suemitsu ∗ Takami Matsuo ∗ Takami Matsuo Matsuo Takami ∗ ∗ Department of Architecture and Mechatronics, Oita University,Oita, ∗ of Architecture and Mechatronics, Oita University,Oita, ∗ Department Department of and Japan (e-mail: v13e6013,v14e6021,suemitu,matsuo@ oita-u.ac.jp). Department of Architecture Architecture and Mechatronics, Mechatronics, Oita Oita University,Oita, University,Oita, Japan (e-mail: v13e6013,v14e6021,suemitu,matsuo@ oita-u.ac.jp). Japan (e-mail: v13e6013,v14e6021,suemitu,matsuo@ Japan (e-mail: v13e6013,v14e6021,suemitu,matsuo@ oita-u.ac.jp). oita-u.ac.jp). Abstract: The mechanism of endogenous circadian photosynthesis oscillations of plants Abstract: The mechanism endogenous circadian photosynthesis oscillations plants Abstract: The of endogenous circadian photosynthesis oscillations of plants performing crassulacean acid of metabolism (CAM) is investigated in terms of a of nonlinear Abstract: The mechanism mechanism ofmetabolism endogenous(CAM) circadian photosynthesis oscillations of plants performing crassulacean acid is investigated in terms of a nonlinear performing crassulacean acid metabolism (CAM) is investigated investigated in terms terms equations of aa nonlinear nonlinear theoretical model. Blasiusacid et al.metabolism used throughout continuous time differential which performing crassulacean (CAM) is in of theoretical et al. used throughout continuous differential equations theoretical model. Blasius et al. used throughout continuous time differential equations which adequately model. reflect Blasius the CAM dynamics. The model shows time regular endogenous limit which cycle theoretical model. Blasius et al. used throughout continuous time differential equations which adequately reflect CAM The shows regular endogenous limit cycle adequately reflect the CAM dynamics. The model shows regular endogenous limit cycle oscillations that arethe stable for dynamics. a wide range of model temperatures in a manner that complies well adequately reflect the CAM dynamics. The model shows regular endogenous limit cycle oscillations that are stable for a wide range of temperatures in aa the manner that complies well oscillations that are stable for a wide range of temperatures in manner that complies well with experimental data. We have recently presented four types of feedback controllers for oscillations that aredata. stable for a wide range of temperatures in a the manner that controllers complies well with experimental We have presented four types for with experimental data. Wemodel have recently recently presented fourthe types ofintensity the feedback feedback controllers for the phase shift of the CAM of the single cell using lightof as ancontrollers input. In this with experimental data. We have recently presented four types of the feedback for the phase the CAM model of the single using the light intensity as an input. In this the phase shift of the CAM model cell using the intensity as input. this paper, we shift deal of with the synchronization issues cell in the coupled cells with light intensity input. the phase shift of thethe CAM model of of the the single single cell using the light light intensity as an an input. In In this paper, we deal with synchronization issues in the coupled cells with light intensity input. paper, we deal with the synchronization issues in the coupled cells with light intensity input. We show that the strong coupling causes the unsynchronization of the cells. paper, we deal with the synchronization issues in the coupled cells with light intensity input. We show that the strong coupling causes unsynchronization of cells. We show that the causes the the of the the We2015, show that(International the strong strong coupling coupling the unsynchronization unsynchronization the cells. cells. © IFAC Federationcauses of Automatic Control) Hosting byofElsevier Ltd. All rights reserved. Keywords: Photosynthesis oscillation, limit cycle, nullcline, pulse control, synchronization. Keywords: Photosynthesis oscillation, limit cycle, nullcline, pulse control, Keywords: Photosynthesis Photosynthesis oscillation, oscillation, limit limit cycle, cycle, nullcline, nullcline, pulse pulse control, control, synchronization. synchronization. Keywords: synchronization. 1. INTRODUCTION model showed regular endogenous limit cycle oscillations 1. model showed regular cycle oscillations 1. INTRODUCTION INTRODUCTION model showed regular endogenous limit cycle that were stable for a endogenous wide range limit of temperatures, in a 1. INTRODUCTION model showed regular endogenous limit cycle oscillations oscillations that were stable for a wide range of temperatures, in that were stable for a wide range of temperatures, manner that complies well with experimental data. Theaaa In plant, the circadian rhythms play important roles in that were stable for a wide range of temperatures, in in manner that complies well with experimental data. In plant, the circadian rhythms play important roles in manner that complies well with experimental The dynamical model CAM is discusseddata. fromThe the In plant, the circadian circadian rhythms growth, play important important roles in nonlinear gene expressions, photosynthesis, and many other manner that complies well of with experimental data. The In plant, the rhythms play roles in nonlinear dynamical model of CAM is discussed from the gene expressions, photosynthesis, growth, and many other nonlinear dynamical model of CAM is discussed from control theoretical viewpoint. The state-variables of the gene expressions, photosynthesis, growth, and many other physiological processes. The plant circadian rhythm is nonlinear dynamical model of CAM is discussed from the gene expressions, photosynthesis, growth, and many other control theoretical viewpoint. The state-variables of the physiological processes. The plant circadian rhythm is control theoretical viewpoint. The state-variables of the nonlinear dynamic equations are an internal CO conphysiological processes. Theof plant plant circadian rhythm is control theoretical viewpoint. The state-variables 2of the composed of a processes. large number self-sustained cellular oscilphysiological The circadian rhythm is nonlinear are anininternal CO con-a composed aa large self-sustained cellular oscildynamic equations are CO centration,dynamic a malateequations concentration the cytoplasm, composed ofsynchronize large number number ofother. self-sustained cellular oscil- nonlinear lations thatof eachof Precise and ecological nonlinear dynamic equations are an anininternal internal CO222 conconcomposed of a large number of self-sustained cellular oscilcentration, a malate concentration the cytoplasm, lations that synchronize each other. Precise and ecological centration, aa malate concentration the concentration in the vacuole, in and ancytoplasm, order of theaaa lations that synchronize each other. other. Precise andtechnology ecological malate control that of the circadian rhythm provides a key centration, malate concentration in the cytoplasm, lations synchronize each Precise and ecological malate in vacuole, and of control of aa key malate concentration concentration in the the vacuole, and an an order of the the tonoplast membrane. The input variables are order an external control of the the circadian circadian rhythm provides key technology technology for enhancing the plantrhythm growthprovides in a closed cultivation malate concentration in the vacuole, and an order of the control of the circadian rhythm provides a key technology tonoplast membrane. The input variables are an external for enhancing the plant growth in a closed cultivation tonoplast membrane. The input variables are an external CO concentration, a light intensity and a temperature. for enhancing the plant plantcycles growth in from a closed closed cultivation 2 system where light-dark differ the 24-h period tonoplast membrane. The input variables are an external for enhancing the growth in a cultivation CO2 concentration, aa light intensity and aa temperature. system where light-dark cycles differ the period concentration, and system where et light-dark cyclesClimatic differ from from the 24-h 24-h period CO (see Fukuda al. (2013)). extremes threaten CO concentration, a light light intensity intensity and estimator a temperature. temperature. system where light-dark cycles differ from the 24-h period We 22have recently presented a dynamic of the (see Fukuda et al. (2013)). Climatic extremes threaten (see Fukuda et al. (2013)). Climatic extremes threaten agricultural sustainability worldwide. Crassulacean acid We have recently presented a dynamic estimator of the (see Fukuda et al. (2013)). Climatic extremes threaten We have recently presented a dynamic estimator of tonoplast order and a fuzzy identifier of the nonlinear We have recently presented a dynamic estimator of the the agricultural sustainability worldwide. Crassulacean acid agricultural sustainability worldwide. Crassulacean acid metabolism (CAM) plants show a remarkable metabolic tonoplast order and a fuzzy identifier of the nonlinear agricultural sustainability worldwide. Crassulacean acid function tonoplast aa fuzzy of the nonlinear inorder the and dynamics of identifier the tonoplast order (see tonoplast order and fuzzy identifier of the nonlinear metabolism (CAM) plants show a remarkable metabolic metabolism (CAM) plantsnocturnal show aa remarkable remarkable metabolic plasticity for(CAM) modulating and diurnalmetabolic CO2 up- function dynamics of tonoplast order (see metabolism plants show functionetin inal. the the dynamics of the the tonoplastthe order (see Matsuo (2013)). Moreover, we proposed feedback in the dynamics of the tonoplast order (see plasticity for nocturnal and CO2 upplasticity for modulating modulating nocturnal and diurnal diurnal CO take and have been identified as competitive biomass ac- function Matsuo et al. (2013)). Moreover, we proposed the feedback 2 upplasticity for modulating nocturnal and diurnal CO upMatsuo et al. (2013)). Moreover, we proposed the feedback 2 controller by controlling the external CO2 concentration Matsuo et al. (2013)). Moreover, we proposed the feedback take and have been identified as competitive biomass actake and identified as competitive accumulators in been comparison with C3 andbiomass C4 crops. controller by controlling CO concentration take and have have been identified as many competitive biomass ac- and controller by the external CO concentration the light intensity the (seeexternal Sakamoto al. (2013)). controller by controlling controlling the external CO222et concentration cumulators in comparison with many C and C 3 efficiency 4 crops. cumulators in comparison with many C and C crops. One approach to increase plant water-use is to and the light intensity (see Sakamoto et al. the (2013)). 3 4 cumulators in comparison with many C and C crops. and the light intensity (see Sakamoto et (2013)). 3 4 Thought these estimator and controller require state the these light estimator intensity and (see controller Sakamotorequire et al. al. the (2013)). One increase plant water-use is One approach approach tointo increase plant water-use efficiency is to to and introduce CAMto C3 crops (see Borlandefficiency et al. (2014)). Thought state One approach to increase plant water-use efficiency is to Thought these estimator and controller require the state variables except for the order of the tonoplast membrane, Thought these estimator and controller require the state introduce CAM into C crops (see Borland et al. (2014)). 3 crops introduce CAM into C (see Borland et al. (2014)). Computational modeling of CAM accelerates the improvevariables except for the order of the tonoplast membrane, 3 crops introduce CAM into C (see Borland et al. (2014)). variables except for the order of the tonoplast membrane, 3 they are not measurable directly. variables except for the order of the tonoplast membrane, Computational modeling of CAM accelerates the improveComputational CAM accelerates improvement of CAM modeling crops in of biomass the productivity are not measurable directly. Computational modeling ofterms CAMof accelerates the improve- they they are measurable directly. they are not not measurable directly. ment of CAM crops in terms of biomass productivity To begin with, we propose the reconstruction method of ment of CAM crops in terms of biomass productivity and quality-related attributes (see Borland et al. (2013)). ment of CAM crops in terms of biomass productivity To begin with, we propose theusing reconstruction method of and quality-related attributes (see Borland et al. (2013)). To begin with, we propose reconstruction method of the internal CO concentration the CO2 uptake from and quality-related quality-related attributes (see Borland Borland et al. al. (2013)). (2013)). Blasius et al. (1999) investigated the mechanism of en- To begin with, 2we propose the theusing reconstruction method of and attributes (see et the internal CO the CO from Blasius et al. (1999) investigated the mechanism of en2 concentration 2 uptake the internal CO concentration using the CO uptake from outside, because the CO uptake can be determined the Blasius et al. (1999) investigated the mechanism of endogenous circadian photosynthesis oscillations of plants 2 concentration 2 uptake from 2 the internal CO using the CO Blasius et al. (1999) investigated the mechanism of en2 2 outside, because the CO be determined the dogenous circadian photosynthesis oscillations of plants 2 uptake outside, because the uptake can can be the is measured using CO2 analyzer dogenous circadian photosynthesis oscillations of plants performing CAM in terms of a nonlinear theoretical 2 2 exchange outside, becausewhich the CO CO be aadetermined determined the dogenous circadian photosynthesis oscillations of model. plants CO 2 uptake can CO exchange which is measured using CO analyzer performing CAM in terms of a nonlinear theoretical model. 2 2 CO exchange which is measured using a CO analyzer and an incubator (see Fukuda et al. (2004)). Next, we performing CAM in terms of a nonlinear theoretical model. They used throughout continuous time differential equa2 2 CO exchange which is measured using a CO analyzer performing CAM in terms of a nonlinear theoretical model. 2 an incubator (see Fukuda et al. (2004)).2 Next, we and They used continuous time differential equaand an Fukuda al. (2004)). Next, we aincubator feedback (see controller andet a feedforward controller They which used throughout throughout continuous timethe differential equa- present tions modes adequately reflect CAM dynamand an incubator (see Fukuda et al. (2004)). Next, we They used throughout continuous time differential equapresent aa feedback controller and aa feedforward controller tions which modes adequately reflect the CAM dynampresent feedback controller and feedforward controller with a pulse signal to shift the phase of the CAM model tions which modes adequately reflect the CAM dynamics. By incorporating results from both a complementary present a feedback controller and a feedforward controller tions which modes adequately reflect the CAM dynamaathe pulse signal to shiftasthe of the CAM model ics. By incorporating results from both a complementary with pulse signal to phase of the model using light intensity anphase input. controller ics. By incorporating results both a complementary and a continuous membrane model, descrip- with with athe pulse signal to shift shiftasthe the phase of These the CAM CAM model ics. By incorporating results from from botha adetailed complementary using light intensity an input. These controller and a continuous membrane model, a detailed descripusing the the light intensity intensity asand an process input. These These controller allows to control the timeas of budding and and of a continuous continuous membrane model, a a in detailed description the molecular malate transport and outdescripof the using light an input. controller and a membrane model, detailed allows to control the time and process of budding and tion molecular malate in of allows to control the time process of budding and of blossoms. For and healthy growth of a plant, tion of of the the molecular malate transport transport in and and out of the the unfolding vacuole through the tonoplast membrane was out achieved. allows to control the time and process of budding and tion of the molecular malate transport in and out of the unfolding of blossoms. For healthy growth of a plant, vacuole through the tonoplast membrane was achieved. unfolding of blossoms. For healthy growth of a plant, it is essential to maintain synchronized activities of the vacuole through the tonoplast membrane was achieved. Their analysis showed that the membrane effectively acts unfolding of blossoms. For healthy growth of a plant, vacuole through the tonoplast membrane was achieved. it is essential to maintain synchronized activities of the Their analysis showed that the membrane effectively acts it is essential to maintain synchronized activities of cellular oscillators so as to sustain the stable rhythm Their analysis showed that the membrane effectively acts as a hysteresis switch regulating the oscillations. It thus is essential to maintain synchronized activities of the the Their analysis showed that the membrane effectively acts it cellular oscillators so as to sustain the stable rhythm as a hysteresis switch regulating the oscillations. It thus cellular oscillators so as to to Finally, sustain the the stable rhythm rhythm (see Fukuda et al. so (2013)). we investigate the as aa hysteresis hysteresis switch basis regulating thecircadian oscillations. It thus thus provided a molecular for the clock. The cellular oscillators as sustain stable as switch regulating the oscillations. It (see Fukuda et al. (2013)). Finally, we investigate the provided aa molecular for the circadian clock. The synchronization Fukuda (2013)). Finally, we the phenomena two cells with different provided molecular basis basis (see Fukuda et et al. al. (2013)). of Finally, we investigate investigate the provided basis for for the the circadian circadian clock. clock. The The (see This worka ismolecular synchronization phenomena of two cells with different partially supported by the Grant-in-Aid for Scientific synchronization phenomena of two cells with different initial values by using the feedforward pulse signal of synchronization phenomena of two cells with different This work is partially supported by the Grant-in-Aid for Scientific This work initial values by using the feedforward pulse signal of Research (25420445), Japan Societyby for the Promotion for of Science. is partially supported the Grant-in-Aid Scientific initial values by using the feedforward pulse signal of This work is partially supported by the Grant-in-Aid for Scientific initial values by using the feedforward pulse signal of Research (25420445), Japan Society for the Promotion of Science. Ryo Ryo Ryo Ryo
Sakamoto ∗∗ ∗ Sakamoto Sakamoto Sakamoto ∗
Research Research (25420445), (25420445), Japan Japan Society Society for for the the Promotion Promotion of of Science. Science. Copyright © 2015 IFAC 59 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 59 Copyright © 2015 IFAC 59 Peer review under responsibility of International Federation of Automatic Copyright © 2015 IFAC 59 Control. 10.1016/j.ifacol.2015.11.011
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the light intensity. The simulation results are given to examine the performance of the proposed controllers using MATLAB/Simulink.
1.6 1.4
T=0.2238
1.2
2. THE MINIMAL CAM MODEL
y
1 0.8
The CAM model that we will use has been studied by Blasius and Beck (see Blasius et al. (1999); Beck et al. (2001)) and here we only outline the minimal CAM model. The model can be characterized by the major reactant pools of CAM that generate the carbon flow during the circadian cycle as shown in Fig.1. The pool concentrations are the following: • • • •
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0.2 0.05
cytoplasm L(t) light
vacuole
u2
malate vacuole x
u1 malate cytoplasm
y
z
0.2
z
0.25
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y u1 = cx − z w u2 = − x x u3 = Jco2 − Cco2 + Rco2 (C (t) − w) Jco2 = cJ ext exp(αw) Cco2 = L(t)w LK w1 Rco2 = cR L(t) + LK w + w1
T (t) temperature
CO2
0.15
τ , is the time-constant for relaxation into thermal equilibrium. The smallest parameter 1 reflects the volume ratio of cytoplasm to vacuole, which is typically of the order of 1/100 in CAM plants (see Blasius et al. (1999)). The flows ui , i = 1, 2, 3 involve modeling of the metabolic reactions and comprise the whole structure of the carbon circulation in CAM. They are described by the following equations:
Cext (t) external CO2
w
0.1
Fig. 2. The nonlinear function y = g(z, T ) when T = 0.2238, 0.2244, 0.2246, 0.2250, 0.2254.
internal CO2 concentration, w; malate concentration in the cytoplasm, x; malate concentration in the vacuole, y; and z is a variable that describes the ordering of the lipid molecules in the tonoplast membrane.
u3
T=0.2254
0.4
tonoplast
(2)
The meaning of each variable is the following: Fig. 1. Flow diagram of the CAM model showing as dynamic variables (encircled) three reactant pools (internal CO2 concentration, w; malate concentration in the cytoplasm, x; and in the vacuole, y) and the order of the tonoplast, z, within a CAM cell.
Jco2 : CO2 uptake from outside Cco2 : CO2 consumption by photosynthesis, which is directly proportional to the external control parameter light intensity, L(t) Rco2 : CO2 production by respiration.
These are the dynamic variables of the cyclic process. They are connected by the flows, u1 , u2 , u3 , during the gain and loss terms of the metabolites. The model depends on three external control parameters: temperature, T , light intensity, L, and external CO2 concentration, Cext . The dynamics are characterized by a set of four coupled, nonlinear differential equations of first order in time: w˙ = −u2 + u3 x˙ = −u1 + u2 (1) y˙ = u1 τ z˙ = g(z, T ) − y
Blasius et al. calculated the dynamic behavior using the dimensionless variables with the parameters (see Blasius et al. (1999)): Cext = 1, L(t) = 1, T = 0.2238, 0.2242, 0.2246, 0.2250, 0.2254, c = 5.5, cJ = 1, cR = 1, ε = 0.001, τ = 0.35, α = 1.5, w1 = 0.1, LK = 0.5, R = 0.1. Using these parameters, we perform the computer simulation by MATLAB/Simulink. The initial conditions of the states are given as w(0) = 0.4, x(0) = 0.62, y(0) = 0.56, z(0) = 0.2. The nonlinear function g(z, T ) is approximated via the third-order spline interpolation from the figure shown in the reference(see Blasius et al. (1999)). Figure 3 shows the dynamic responses of the state variables when T = 0.2242. Blasius et al. have shown the influence of temperature as an external control parameter (see Blasius et al. (1999)). With increasing temperature the steady state malate level decreases, until at the critical temperature, the system undergoes supercritical Hopf bifurcation, and the stable fixed point changes into unstable one surrounded by a small-amplitude limit cycle (see Blasius et al. (1999)).
where the function g(z, T ) is the thermodynamic equilibrium value of malate concentration in the vacuole ,y, and is a third-order nonlinear function depends on the temperature, T (see Blasius et al. (1999)). The temperaturedependent z-null cline, y = g(z, T ), corresponds to the hysteretic behavior of the phase diagram in the membrane model shown in Fig.2( seeNeff et al. (1998)). The constant, 60
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1.4
w x y z
1.2
manifold, we adopt Cext (t) and L(t) as the inputs of the control system. Using the control variables Cext (t) and L(t), we can reshape w-nullcline explicitly.
(state) (state) (state) (state)
We consider the following two types of the control input:
0.8 0.6
L(t) = g(m(t) − s0 ) (7) exp(αw) {g(m(t) − s0 )} (8) Cext = CJ where m(t) is a measurement signal and we select the measurement signal as w(t), The bias term, s0 , keeps the input variable positive. The input allows us to change wnullcline given by (5).
0.4 0.2 0
5
10 time
15
20
Fig. 3. Time responses of sustained endogenous rhythms in continuous light when T = 0.2246.
Figure 4 shows w and x nullclines with the feedback controller (7) for the feedback gain g = −0.1, −1. Figure 5 shows w and x nullclines with the feedback controller (8) for the feedback gain g = −0.1, −1. Thus, we can change the equilibrium point and the critical manifold by using feedback controllers.
3. FEEDBACK CONTROL OF BIOLOGICAL RHYTHM In this chapter, we review how to control the frequency of the CAM rhythm (see Sakamoto et al. (2013)) and propose the reconstruction method of the internal CO2 concentration, w by the CO2 uptake from outside. 3.1 State-Space Model
1.2
x
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1 x−nullcline
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w−nullcline
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1
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Fig. 4. w-nullcline (solid line) and x-nullcline (dotted line) with the feedback controller (7), where g = −0.1(lef t), −1(right) and s0 = 2. 1.6
1.6 w−nullcline
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x
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1
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1
Fig. 5. w-nullcline and x-nullcline with the feedback controller (8), where g = −0.1(lef t), −1(right) and s0 = 2.
(4)
3.3 Feedback Control with L(t)
because it can be measured easily than the other states x(t), y(t) and z(t).
We try to control the biological rhythm by using the light intensity L(t) as the control input and by setting both of the temperature and the external CO2 concentration are given constants. In this case, the input variable, L(t), is nonlinear, because the nonlinear function , f˜ (˜ x, L, T ), contains Rco2 . We design the following controller:
3.2 Reshaping of Nullcline by Output Feedback The CAM system consists of the fast modes w, y and the slow modes y, z, because the time constant is small. The feedback control allows us to reshape the nullclines of the fast modes w and x. From (1), w-nullcline and x-nullcline are given by w + x + Jco2 − Cco2 + Rco2 = 0 x y w −cx + + − x = 0 z x
1.4
1.2
0.6
The output m(t) is given by
−
1.6
1.4
1
The CAM model can be rewritten by the state-space form: x ˜˙ = f˜(˜ x, L, T ) + g˜1 (w)(Cext (t) − w) + g˜2 L(t) 1 (−u + R ) 2 co 2 w 1 x ˜ (−u + u ) 1 2 x ˜ = ,f = y u1 1 (3) z (g(z, T ) − y) τ 1 cJ 1 exp(αw) 0 0 g˜1 = , g˜2 = 0 0 0 0 m(t) = w(t),
1.6
x
0
x
w,x,y,z
1
61
L(t) = g(w(t) − s0 ) (9) where g is a feedback gain and s0 = 0.2 is a constant bias. The constant bias s0 is added to maintain the control signal positive.
(5) (6)
Figure 6 shows the responses of the CO2 uptake Jco2 for each feedback gain g = 0, 1.0, 1.5, 5.0, 20, 30. The other control variables Cext (t) and T are fixed as Cext = 1, T = 0.2246. These figures indicate that the feedback gain within the appropriate range can change the frequency of the rhythm.
The control variables Cext (t) and L(t) are included in Jco2 and in Cco2 and Rco2 , respectively. On the other hand, the control variable T affects the z-dynamics. Since T cannot be used to reshape w and x nullclines as the critical 61
1 0.8
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5
time
(a) g = 0
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(e) g = 20
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(c) g = −0.5
(d) g = −0.6
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0
2
4
6
8
10
0
12
0
time
2
4
6
8
10
12
time
(e) g = −0.75
Fig. 6. Responses of Jco2 for each feedback gain g = 0, 1.0, 1.5, 5.0, 20, 30.
10
0.1 0
time
(f) g = 30
8
(b) :g = −0.3
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(d) g = 5.0
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(a) g = −0.1
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(c) g = 1.5
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0
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(b) g = 1.0
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0
15
time
1
0
10
Jco2 value
5
Jco2 value
0
1 0.8
(f) g = −1.0
Fig. 7. Responses of Jco2 for each feedback gain g = −0.1, −0.3, −0.5, −0.6, −0.75, −1.0.
3.4 Feedback Control with Cext (t)
is not suitable for the real-time feedback control. Since 0.1 ≤ w ≤ 1, we approximate the exponential function as exp(αw) ≈ 1 + αw + 12 α2 w2 . In this case, we obtain w as a solution of the 2nd order polynomial equation:
We design a controller of the biological rhythm by using the external CO2 concentration, Cext , as the control input and by setting both of the temperature and the light intensity are given constants. In this case, the nonlinear state-space equation is rewritten by the affine system:
α2 Jco2 w2 + (cJ + αJco2 )w + (Jco2 − cJ Cext ) = 0(13) 2 Fig.8 shows the state variable w and its approximated value using (13). The error between two variable is small.
˜˙ = {f˜ (˜ ˜ 1 (w)w + g ˜ 2 L(t)} + g ˜ 1 (w)Cext (t). x x, L, T ) − g We design the following controller: exp(αw) {g(w(t) − s0 )} (10) CJ where g is a feedback gain and s0 = 0.15 is a constant bias. The constant bias s0 is added to maintain the control signal positive.
w values
Cext (t) =
1
1
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w values
0 -0.1
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Jco2 value
Jco2 value
IFAC CHAOS 2015 62 August 26-28, 2015. Tokyo, Japan
0.6 0.5 0.4
Figure 7 shows the responses of the CO2 uptake Jco2 for each feedback gain g = −0.1, −0.2, −0.3, −0.5, −0.6, −0.75. The other control variables L(t) and T are fixed as L(t) = 1, T = 0.2246. In the low-gain case, the equilibrium is stable and the limit cycle does not exist. In the high-gain case, the limit cycle also disappears. Therefore, there is a limit to make the rhythm to fast. However, the feedback controller with Cext (t) can change the frequency of the rhythm. 3.5 Calculation of w using Jco2
0.5 0.4
0.3 0.2 0
0.6
0.3 5
10 time
15
20
0.2 0
5
10 time
15
20
Fig. 8. Dynamic responses of w (left) and the approximated value using the solution the second order equation (right) ,where figure of top is g = 1. 4. PHASE CONTROL OF SINGLE CELL BY LIGHT INTENSITY INPUT We have recently proposed the linear and the nonlinear feedback controllers for the phase shift of the CAM model using the light intensity as an input (see Sakamoto et al. (2013)). The linear controller requires the external periodical signal as the reference input. The feedback gain of the linear controller effects the phase of the rhythm. On the other hand, the nonlinear controller does not need external signals.
The CO2 uptake from outside, Jco2 , is measurable and is given by (Cext (t) − w) . (11) exp(αw) where the external CO2 concentration, Cext is known. When Jco2 = 0, the above equation is regarded as the nonlinear algebraic equation with respect to w as follows: Jco2 = cJ
In this chapter, we propose another controller using pulse signals, which is suitable for practical use. We select the pulse signals Pi (t)’s (i = 0, 1, 2, 3, 4) as follows:
Jco2 exp(αw) + cJ w = cJ Cext . (12) The iterative numerical calculation allows to obtain w using the known signals Jco2 , Cext (t). However, this solution
P0 (t) = { 1(0 ≤ t ≤ 20) 62
P2 (t) = P3 (t) = P4 (t) =
{ { {
0.9(0 ≤ t ≤ 2) 1(2 ≤ t ≤ 20) 0.9(0 ≤ t ≤ 5) 1(5 ≤ t ≤ 20)
J co2 value
P1 (t) =
{
Ryo Sakamoto et al. / IFAC-PapersOnLine 48-18 (2015) 059–064
0.2(5 ≤ t ≤ 10) 1(0 ≤ t ≤ 5, 10 ≤ t ≤ 20)
0.5
0.3 0.2
0.5
0.3 0.2 0.1
5
10 time
15
20
0 0
5
10 time
15
20
(b) P4 (t)
Fig. 10. Responses of Jco2 ’s using Lf f i ’s (solid line : feedforward control, dotted line : step input L(t) = 1). The state variable and the input of each cell are defined as follows: (wi , xi , yi , zi ), Li (t); i = 1, 2
(14)
All parameters of two cells are same, but the initial conditions are different. 5.1 Synchronization of Two Cells without Coupling When there is no coupling between two cells, the state space equations are given by
0.6
0.45
0.4
(a) P3 (t)
1.3(5 ≤ t ≤ 10) 1(0 ≤ t ≤ 5, 10 ≤ t ≤ 20).
where w ˆ is the recovered value using the measurable signal Jco2 as in Section 3.5. Fig.9 shows the response of Jco2 for each input Lf bi (t). From these responses, we can see that the feedback signal changes not only the phase but also the frequency and the wave form. 0.5
0.4
0.4 J co2 value
0.35 J co2 value
0.6
0.4
−0.1 0
ˆ Lf bi (t) = w(t)P i (t)
0.3 0.25 0.2 0.15
x1 , L1 , T ) + g˜1 (w1 )(Cext1 (t) − w1 ) + g˜2 L1 (t)(15) x ˜˙ 1 = f˜(˜ ˜ ˙x x2 , L2 , T ) + g˜1 (w2 )(Cext2 (t) − w2 ) + g˜2 L2 (t)(16) ˜2 = f (˜
0.3 0.2 0.1
0.1
0
0.05 5
10 time
15
−0.1 0
20
(a) P0 (t) 0.6
0.5
0.5
0.4
0.4
0.3 0.2
15
The external CO2 concentrations are same as follows:
20
0.1 0 15
20
Cext1 (t) = Cext2 (t) = 1. The feedforward inputs of the light intensities are also same as follows:
0.2
0 10 time
10 time
0.3
0.1
5
5
(b) P1 (t)
0.6
J co2 value
J co2 value
0.7
0.5
0
We select the feedback controller as
−0.1 0
0.6
0.1
4.1 Feedback Control with Pulse Signal
0 0
63
J co2 value
IFAC CHAOS 2015 August 26-28, 2015. Tokyo, Japan
−0.1 0
(c) P2 (t)
L1 (t) = L2 (t) = Pi (t), 5
10 time
15
i = 5, 6, 7
where Pi (t) are selected as:
20
(d) P3 (t)
P5 =
Fig. 9. Responses of Jco2 ’s using Lf bi ’s (solid line : feedback control, dotted line : step input L(t) = 1).
P6 =
{ {
0(1 ≤ t ≤ 2) 1(0 ≤ t ≤ 1, 2 ≤ t ≤ 200) 0(2 ≤ t ≤ 3) 1(0 ≤ t ≤ 2, 3 ≤ t ≤ 200)
Fig.11 shows the CO2 uptakes of two cells and the error of the CO2 uptakes for each input. We can see that the synchronization is dependent on the pulse input.
4.2 Feedforward Control with Pulse Signal We select the feedforward controller as
5.2 Synchronization of Two Cells with Coupling
Lf f i (t) = Pi (t). Fig.10 shows the response of Jco2 for each feedforward input Lf f i (t). This figure indicates that the feedforward input via the pulse signals can shift only the phase of rhythm.
Fukuda et al. (2004) proposed the coupling model of two or more cells of the CAM plants. Using this model, the coupling model of two cells is written by the state space equations as: x1 , L1 , T ) + g˜1 (w1 )(Cext1 (t) − w1 ) x ˜˙1 = f˜(˜ +˜ g2 L1 (t) + g˜31 w1 Dw (w1 − w2 ) (17) x Dx (w1 − w2 ) x ˜1 = 1 , g˜31 = 0 y1 0 z1
5. PHASE CONTROL OF TWO CELLS BY LIGHT INTENSITY INPUT From the simulation results in the previous chapter, it is enough to shift the phase of rhythm by the feedforward pulse input. In this chapter, we investigate the synchronization phenomena of two cells with different initial values by using the feedforward pulse signal of the light intensity. 63
Ryo Sakamoto et al. / IFAC-PapersOnLine 48-18 (2015) 059–064
x ˜˙ 2 = f˜(x˜2 , L2 , T ) + g˜1 (w2 )(Cext2 (t) − w2 ) +˜ g2 L2 (t) + g˜32 (18) w2 Dw (w2 − w1 ) x2 Dx (w2 − w1 ) x ˜2 = , g˜ = . 0 y2 32 0 z2
0.6
0.4
0.5
0.3 0.2
Jco2 error
Jco2
0.4 0.3 0.2
0.1 0 −0.1
0.1
−0.2
0
The connection term is given by
−0.1 0
−0.3 50
100 time
150
−0.4 0
200
(a) Jco2 ’s via P5 (t)
[ Dw D x ] = h [ 1 1 ] . where h is the strength of the connection. The feedforward inputs of the light intensities are given as follows:
0.4
0.5
0.3
100 time
150
200
0.2
0.4
Jco2
50
(b) Error of Jco2 ’s
0.6
Jco2 error
IFAC CHAOS 2015 64 August 26-28, 2015. Tokyo, Japan
0.3 0.2
0.1 0 −0.1
0.1
L1 (t) = L2 (t) = Pi (t), i = 5, 6 Fig.12 shows the CO2 uptakes of two cells and the error of the CO2 uptakes for each input when the connection strength is h = 10. We can see that the strong connection can cause large deviation of the phase with the passage of time. 0.6
0.3
0.5
0.2
0.3 0.2
50
100 time
150
(a) Jco2 ’s via P5 (t)
100 time
150
200
(b) Error of Jco2 ’s
0.6
0.3
0.5
0.2
0.4
0.1 Jco2 error
Jco2
50
0.3 0.2
0 −0.1
0.1 −0.2
0 −0.1 0
50
100 time
150
200
(c) Jco2 ’s via P6 (t)
−0.3 0
50
100 time
150
150
200
−0.4 0
50
100 time
150
200
(d) Error of Jco2 ’s
model biological clock and produces specific dynamic behaviour. Proc. R. Soc. Lond. B, 268:1307–1313. Blasius,B., Neff,R., Beck,F. and L¨ uttge,U. (1999). Oscillatory model of crassulacean acid metabolism with a dynamic hysteresis switch. Proc. R. Soc. Lond. B, 266: 93–101. Borland,A.M. and Yang,X. (2013). Informing the improvement and biodesign of crassulacean acid metabolism via system dynamics modelling. New phytologist, 200:946– 949. Borland,A.M., Hartwell,J., Weston,D.J., Schlauch,K.A., Tschaplinski, T.J., Tuskan,G.A., Yang,X., and Cushman, J.C. (2014). Engineering crassulacean acid metabolism to improve water-use efficiency. Trends in Plant Science, 19-5, 327–338. Fukuda,H., Kodama,J., and Kai,S. (2004). Circadian rhythm formation in plant seedling: global synchronization and bifurcation as a coupled nonlinear oscillator system. BioSystems, 77, 41–46. Fukuda,H., Murase,H., and Tokuda, I.T. (2013). Controlling Circadian Rhythms by Dark-Pulse Perturbations in Arabidopsis thaliana. Sci. Rep., 3, 1533; DOI:10.1038/srep01533. Matsuo,T., Totoki,Y., and Suemitsu,H. (2013). Adaptive Estimation of Biological Rhythm in Crassulacean Acid Metabolism with Critical Manifold. ISRN Applied Mathematics, Article ID 856404, 9 pages, 2013. doi:10.1155/2013/856404. Neff,R. Blasius,B., Beck,F., and L¨ uttge,U. (1998). Thermodynamics and energetics of the tonoplast membrane operating as a hysteresis switch in an oscillatory model of crassulacean acid metabolism. J. Memb. Biol., 165: 37–43. Owen,N.A and Griffiths,H. (2013). A system dynamics model integrating physiology and biochemical regulation predicts extent of crassulacean acid metabolism (CAM) phases. New phytologist, 200:1116–1131. Sakamoto,R.,Goto,A., Suemitsu,H., and Matsuo,T. (2013). Phase Shift of Biological Rhythm in Crassulacean Acid Metabolism by Controlling Light Intensity. Proc. of SICE2013, Nagoya,783–786.
0
−0.3 0
200
100 time
Fig. 12. Jco2 ’s of two cells and their error (h = 10).
−0.2
0
−0.3 50
(c) Jco2 ’s via P6 (t)
−0.1
0.1
−0.1 0
−0.1 0
0.1 Jco2 error
Jco2
0.4
−0.2
0
200
(d) Error of Jco2 ’s
Fig. 11. Jco2 ’s of two cells and their error (h = 0). 6. CONCLUSION We proposed the reconstruction method of the internal CO2 concentration using the CO2 uptake from outside and presented the feedback controller and the feedforward controller. Using the computer simulations, we showed that the feedback controller allows us to control the frequency, the wave form, and the phase of the rhythm, but the feedforward controller with the pulse can change only the phase of the rhythm. Moreover, in the case of the coupling cells, we indicated that the timing of the pulse in the feedforward controller affects the synchronization of cells. Owen et al. have recently proposed a new physiological model of the CAM plants (see Borland et al. (2013); Owen et al. (2013)). The construction of the coupling model of the cells based on the physiology is a next future problem. REFERENCES Beck,F., Blasius,B., L¨ uttge,U., Neff,R., and Rascher,U. (2001). Stochastic noise interfers coherently with a 64
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Feedback Control of Biological Rhythm Feedback Control of Biological Rhythm Feedback Control of Biological Rhythm Crassulacean Acid Metabolism by Crassulacean Acid Metabolism by Crassulacean Acid Metabolism by CO2-Uptake Signal CO2-Uptake CO2-Uptake Signal Signal
in in in
Chisato Matoba ∗∗ Haruo Suemitsu ∗∗ Chisato Suemitsu ∗∗ Chisato Matoba Haruo TakamiMatoba Matsuo∗∗∗∗ Haruo Chisato Matoba Haruo Suemitsu Suemitsu ∗ Takami Matsuo ∗ Takami Matsuo Matsuo Takami ∗ ∗ Department of Architecture and Mechatronics, Oita University,Oita, ∗ of Architecture and Mechatronics, Oita University,Oita, ∗ Department Department of and Japan (e-mail: v13e6013,v14e6021,suemitu,matsuo@ oita-u.ac.jp). Department of Architecture Architecture and Mechatronics, Mechatronics, Oita Oita University,Oita, University,Oita, Japan (e-mail: v13e6013,v14e6021,suemitu,matsuo@ oita-u.ac.jp). Japan (e-mail: v13e6013,v14e6021,suemitu,matsuo@ Japan (e-mail: v13e6013,v14e6021,suemitu,matsuo@ oita-u.ac.jp). oita-u.ac.jp). Abstract: The mechanism of endogenous circadian photosynthesis oscillations of plants Abstract: The mechanism endogenous circadian photosynthesis oscillations plants Abstract: The of endogenous circadian photosynthesis oscillations of plants performing crassulacean acid of metabolism (CAM) is investigated in terms of a of nonlinear Abstract: The mechanism mechanism ofmetabolism endogenous(CAM) circadian photosynthesis oscillations of plants performing crassulacean acid is investigated in terms of a nonlinear performing crassulacean acid metabolism (CAM) is investigated investigated in terms terms equations of aa nonlinear nonlinear theoretical model. Blasiusacid et al.metabolism used throughout continuous time differential which performing crassulacean (CAM) is in of theoretical et al. used throughout continuous differential equations theoretical model. Blasius et al. used throughout continuous time differential equations which adequately model. reflect Blasius the CAM dynamics. The model shows time regular endogenous limit which cycle theoretical model. Blasius et al. used throughout continuous time differential equations which adequately reflect CAM The shows regular endogenous limit cycle adequately reflect the CAM dynamics. The model shows regular endogenous limit cycle oscillations that arethe stable for dynamics. a wide range of model temperatures in a manner that complies well adequately reflect the CAM dynamics. The model shows regular endogenous limit cycle oscillations that are stable for a wide range of temperatures in aa the manner that complies well oscillations that are stable for a wide range of temperatures in manner that complies well with experimental data. We have recently presented four types of feedback controllers for oscillations that aredata. stable for a wide range of temperatures in a the manner that controllers complies well with experimental We have presented four types for with experimental data. Wemodel have recently recently presented fourthe types ofintensity the feedback feedback controllers for the phase shift of the CAM of the single cell using lightof as ancontrollers input. In this with experimental data. We have recently presented four types of the feedback for the phase the CAM model of the single using the light intensity as an input. In this the phase shift of the CAM model cell using the intensity as input. this paper, we shift deal of with the synchronization issues cell in the coupled cells with light intensity input. the phase shift of thethe CAM model of of the the single single cell using the light light intensity as an an input. In In this paper, we deal with synchronization issues in the coupled cells with light intensity input. paper, we deal with the synchronization issues in the coupled cells with light intensity input. We show that the strong coupling causes the unsynchronization of the cells. paper, we deal with the synchronization issues in the coupled cells with light intensity input. We show that the strong coupling causes unsynchronization of cells. We show that the causes the the of the the We2015, show that(International the strong strong coupling coupling the unsynchronization unsynchronization the cells. cells. © IFAC Federationcauses of Automatic Control) Hosting byofElsevier Ltd. All rights reserved. Keywords: Photosynthesis oscillation, limit cycle, nullcline, pulse control, synchronization. Keywords: Photosynthesis oscillation, limit cycle, nullcline, pulse control, Keywords: Photosynthesis Photosynthesis oscillation, oscillation, limit limit cycle, cycle, nullcline, nullcline, pulse pulse control, control, synchronization. synchronization. Keywords: synchronization. 1. INTRODUCTION model showed regular endogenous limit cycle oscillations 1. model showed regular cycle oscillations 1. INTRODUCTION INTRODUCTION model showed regular endogenous limit cycle that were stable for a endogenous wide range limit of temperatures, in a 1. INTRODUCTION model showed regular endogenous limit cycle oscillations oscillations that were stable for a wide range of temperatures, in that were stable for a wide range of temperatures, manner that complies well with experimental data. Theaaa In plant, the circadian rhythms play important roles in that were stable for a wide range of temperatures, in in manner that complies well with experimental data. In plant, the circadian rhythms play important roles in manner that complies well with experimental The dynamical model CAM is discusseddata. fromThe the In plant, the circadian circadian rhythms growth, play important important roles in nonlinear gene expressions, photosynthesis, and many other manner that complies well of with experimental data. The In plant, the rhythms play roles in nonlinear dynamical model of CAM is discussed from the gene expressions, photosynthesis, growth, and many other nonlinear dynamical model of CAM is discussed from control theoretical viewpoint. The state-variables of the gene expressions, photosynthesis, growth, and many other physiological processes. The plant circadian rhythm is nonlinear dynamical model of CAM is discussed from the gene expressions, photosynthesis, growth, and many other control theoretical viewpoint. The state-variables of the physiological processes. The plant circadian rhythm is control theoretical viewpoint. The state-variables of the nonlinear dynamic equations are an internal CO conphysiological processes. Theof plant plant circadian rhythm is control theoretical viewpoint. The state-variables 2of the composed of a processes. large number self-sustained cellular oscilphysiological The circadian rhythm is nonlinear are anininternal CO con-a composed aa large self-sustained cellular oscildynamic equations are CO centration,dynamic a malateequations concentration the cytoplasm, composed ofsynchronize large number number ofother. self-sustained cellular oscil- nonlinear lations thatof eachof Precise and ecological nonlinear dynamic equations are an anininternal internal CO222 conconcomposed of a large number of self-sustained cellular oscilcentration, a malate concentration the cytoplasm, lations that synchronize each other. Precise and ecological centration, aa malate concentration the concentration in the vacuole, in and ancytoplasm, order of theaaa lations that synchronize each other. other. Precise andtechnology ecological malate control that of the circadian rhythm provides a key centration, malate concentration in the cytoplasm, lations synchronize each Precise and ecological malate in vacuole, and of control of aa key malate concentration concentration in the the vacuole, and an an order of the the tonoplast membrane. The input variables are order an external control of the the circadian circadian rhythm provides key technology technology for enhancing the plantrhythm growthprovides in a closed cultivation malate concentration in the vacuole, and an order of the control of the circadian rhythm provides a key technology tonoplast membrane. The input variables are an external for enhancing the plant growth in a closed cultivation tonoplast membrane. The input variables are an external CO concentration, a light intensity and a temperature. for enhancing the plant plantcycles growth in from a closed closed cultivation 2 system where light-dark differ the 24-h period tonoplast membrane. The input variables are an external for enhancing the growth in a cultivation CO2 concentration, aa light intensity and aa temperature. system where light-dark cycles differ the period concentration, and system where et light-dark cyclesClimatic differ from from the 24-h 24-h period CO (see Fukuda al. (2013)). extremes threaten CO concentration, a light light intensity intensity and estimator a temperature. temperature. system where light-dark cycles differ from the 24-h period We 22have recently presented a dynamic of the (see Fukuda et al. (2013)). Climatic extremes threaten (see Fukuda et al. (2013)). Climatic extremes threaten agricultural sustainability worldwide. Crassulacean acid We have recently presented a dynamic estimator of the (see Fukuda et al. (2013)). Climatic extremes threaten We have recently presented a dynamic estimator of tonoplast order and a fuzzy identifier of the nonlinear We have recently presented a dynamic estimator of the the agricultural sustainability worldwide. Crassulacean acid agricultural sustainability worldwide. Crassulacean acid metabolism (CAM) plants show a remarkable metabolic tonoplast order and a fuzzy identifier of the nonlinear agricultural sustainability worldwide. Crassulacean acid function tonoplast aa fuzzy of the nonlinear inorder the and dynamics of identifier the tonoplast order (see tonoplast order and fuzzy identifier of the nonlinear metabolism (CAM) plants show a remarkable metabolic metabolism (CAM) plantsnocturnal show aa remarkable remarkable metabolic plasticity for(CAM) modulating and diurnalmetabolic CO2 up- function dynamics of tonoplast order (see metabolism plants show functionetin inal. the the dynamics of the the tonoplastthe order (see Matsuo (2013)). Moreover, we proposed feedback in the dynamics of the tonoplast order (see plasticity for nocturnal and CO2 upplasticity for modulating modulating nocturnal and diurnal diurnal CO take and have been identified as competitive biomass ac- function Matsuo et al. (2013)). Moreover, we proposed the feedback 2 upplasticity for modulating nocturnal and diurnal CO upMatsuo et al. (2013)). Moreover, we proposed the feedback 2 controller by controlling the external CO2 concentration Matsuo et al. (2013)). Moreover, we proposed the feedback take and have been identified as competitive biomass actake and identified as competitive accumulators in been comparison with C3 andbiomass C4 crops. controller by controlling CO concentration take and have have been identified as many competitive biomass ac- and controller by the external CO concentration the light intensity the (seeexternal Sakamoto al. (2013)). controller by controlling controlling the external CO222et concentration cumulators in comparison with many C and C 3 efficiency 4 crops. cumulators in comparison with many C and C crops. One approach to increase plant water-use is to and the light intensity (see Sakamoto et al. the (2013)). 3 4 cumulators in comparison with many C and C crops. and the light intensity (see Sakamoto et (2013)). 3 4 Thought these estimator and controller require state the these light estimator intensity and (see controller Sakamotorequire et al. al. the (2013)). One increase plant water-use is One approach approach tointo increase plant water-use efficiency is to to and introduce CAMto C3 crops (see Borlandefficiency et al. (2014)). Thought state One approach to increase plant water-use efficiency is to Thought these estimator and controller require the state variables except for the order of the tonoplast membrane, Thought these estimator and controller require the state introduce CAM into C crops (see Borland et al. (2014)). 3 crops introduce CAM into C (see Borland et al. (2014)). Computational modeling of CAM accelerates the improvevariables except for the order of the tonoplast membrane, 3 crops introduce CAM into C (see Borland et al. (2014)). variables except for the order of the tonoplast membrane, 3 they are not measurable directly. variables except for the order of the tonoplast membrane, Computational modeling of CAM accelerates the improveComputational CAM accelerates improvement of CAM modeling crops in of biomass the productivity are not measurable directly. Computational modeling ofterms CAMof accelerates the improve- they they are measurable directly. they are not not measurable directly. ment of CAM crops in terms of biomass productivity To begin with, we propose the reconstruction method of ment of CAM crops in terms of biomass productivity and quality-related attributes (see Borland et al. (2013)). ment of CAM crops in terms of biomass productivity To begin with, we propose theusing reconstruction method of and quality-related attributes (see Borland et al. (2013)). To begin with, we propose reconstruction method of the internal CO concentration the CO2 uptake from and quality-related quality-related attributes (see Borland Borland et al. al. (2013)). (2013)). Blasius et al. (1999) investigated the mechanism of en- To begin with, 2we propose the theusing reconstruction method of and attributes (see et the internal CO the CO from Blasius et al. (1999) investigated the mechanism of en2 concentration 2 uptake the internal CO concentration using the CO uptake from outside, because the CO uptake can be determined the Blasius et al. (1999) investigated the mechanism of endogenous circadian photosynthesis oscillations of plants 2 concentration 2 uptake from 2 the internal CO using the CO Blasius et al. (1999) investigated the mechanism of en2 2 outside, because the CO be determined the dogenous circadian photosynthesis oscillations of plants 2 uptake outside, because the uptake can can be the is measured using CO2 analyzer dogenous circadian photosynthesis oscillations of plants performing CAM in terms of a nonlinear theoretical 2 2 exchange outside, becausewhich the CO CO be aadetermined determined the dogenous circadian photosynthesis oscillations of model. plants CO 2 uptake can CO exchange which is measured using CO analyzer performing CAM in terms of a nonlinear theoretical model. 2 2 CO exchange which is measured using a CO analyzer and an incubator (see Fukuda et al. (2004)). Next, we performing CAM in terms of a nonlinear theoretical model. They used throughout continuous time differential equa2 2 CO exchange which is measured using a CO analyzer performing CAM in terms of a nonlinear theoretical model. 2 an incubator (see Fukuda et al. (2004)).2 Next, we and They used continuous time differential equaand an Fukuda al. (2004)). Next, we aincubator feedback (see controller andet a feedforward controller They which used throughout throughout continuous timethe differential equa- present tions modes adequately reflect CAM dynamand an incubator (see Fukuda et al. (2004)). Next, we They used throughout continuous time differential equapresent aa feedback controller and aa feedforward controller tions which modes adequately reflect the CAM dynampresent feedback controller and feedforward controller with a pulse signal to shift the phase of the CAM model tions which modes adequately reflect the CAM dynamics. By incorporating results from both a complementary present a feedback controller and a feedforward controller tions which modes adequately reflect the CAM dynamaathe pulse signal to shiftasthe of the CAM model ics. By incorporating results from both a complementary with pulse signal to phase of the model using light intensity anphase input. controller ics. By incorporating results both a complementary and a continuous membrane model, descrip- with with athe pulse signal to shift shiftasthe the phase of These the CAM CAM model ics. By incorporating results from from botha adetailed complementary using light intensity an input. These controller and a continuous membrane model, a detailed descripusing the the light intensity intensity asand an process input. These These controller allows to control the timeas of budding and and of a continuous continuous membrane model, a a in detailed description the molecular malate transport and outdescripof the using light an input. controller and a membrane model, detailed allows to control the time and process of budding and tion molecular malate in of allows to control the time process of budding and of blossoms. For and healthy growth of a plant, tion of of the the molecular malate transport transport in and and out of the the unfolding vacuole through the tonoplast membrane was out achieved. allows to control the time and process of budding and tion of the molecular malate transport in and out of the unfolding of blossoms. For healthy growth of a plant, vacuole through the tonoplast membrane was achieved. unfolding of blossoms. For healthy growth of a plant, it is essential to maintain synchronized activities of the vacuole through the tonoplast membrane was achieved. Their analysis showed that the membrane effectively acts unfolding of blossoms. For healthy growth of a plant, vacuole through the tonoplast membrane was achieved. it is essential to maintain synchronized activities of the Their analysis showed that the membrane effectively acts it is essential to maintain synchronized activities of cellular oscillators so as to sustain the stable rhythm Their analysis showed that the membrane effectively acts as a hysteresis switch regulating the oscillations. It thus is essential to maintain synchronized activities of the the Their analysis showed that the membrane effectively acts it cellular oscillators so as to sustain the stable rhythm as a hysteresis switch regulating the oscillations. It thus cellular oscillators so as to to Finally, sustain the the stable rhythm rhythm (see Fukuda et al. so (2013)). we investigate the as aa hysteresis hysteresis switch basis regulating thecircadian oscillations. It thus thus provided a molecular for the clock. The cellular oscillators as sustain stable as switch regulating the oscillations. It (see Fukuda et al. (2013)). Finally, we investigate the provided aa molecular for the circadian clock. The synchronization Fukuda (2013)). Finally, we the phenomena two cells with different provided molecular basis basis (see Fukuda et et al. al. (2013)). of Finally, we investigate investigate the provided basis for for the the circadian circadian clock. clock. The The (see This worka ismolecular synchronization phenomena of two cells with different partially supported by the Grant-in-Aid for Scientific synchronization phenomena of two cells with different initial values by using the feedforward pulse signal of synchronization phenomena of two cells with different This work is partially supported by the Grant-in-Aid for Scientific This work initial values by using the feedforward pulse signal of Research (25420445), Japan Societyby for the Promotion for of Science. is partially supported the Grant-in-Aid Scientific initial values by using the feedforward pulse signal of This work is partially supported by the Grant-in-Aid for Scientific initial values by using the feedforward pulse signal of Research (25420445), Japan Society for the Promotion of Science. Ryo Ryo Ryo Ryo
Sakamoto ∗∗ ∗ Sakamoto Sakamoto Sakamoto ∗
Research Research (25420445), (25420445), Japan Japan Society Society for for the the Promotion Promotion of of Science. Science. Copyright © 2015 IFAC 59 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 59 Copyright © 2015 IFAC 59 Peer review under responsibility of International Federation of Automatic Copyright © 2015 IFAC 59 Control. 10.1016/j.ifacol.2015.11.011
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the light intensity. The simulation results are given to examine the performance of the proposed controllers using MATLAB/Simulink.
1.6 1.4
T=0.2238
1.2
2. THE MINIMAL CAM MODEL
y
1 0.8
The CAM model that we will use has been studied by Blasius and Beck (see Blasius et al. (1999); Beck et al. (2001)) and here we only outline the minimal CAM model. The model can be characterized by the major reactant pools of CAM that generate the carbon flow during the circadian cycle as shown in Fig.1. The pool concentrations are the following: • • • •
0.6
0.2 0.05
cytoplasm L(t) light
vacuole
u2
malate vacuole x
u1 malate cytoplasm
y
z
0.2
z
0.25
0.3
0.35
y u1 = cx − z w u2 = − x x u3 = Jco2 − Cco2 + Rco2 (C (t) − w) Jco2 = cJ ext exp(αw) Cco2 = L(t)w LK w1 Rco2 = cR L(t) + LK w + w1
T (t) temperature
CO2
0.15
τ , is the time-constant for relaxation into thermal equilibrium. The smallest parameter 1 reflects the volume ratio of cytoplasm to vacuole, which is typically of the order of 1/100 in CAM plants (see Blasius et al. (1999)). The flows ui , i = 1, 2, 3 involve modeling of the metabolic reactions and comprise the whole structure of the carbon circulation in CAM. They are described by the following equations:
Cext (t) external CO2
w
0.1
Fig. 2. The nonlinear function y = g(z, T ) when T = 0.2238, 0.2244, 0.2246, 0.2250, 0.2254.
internal CO2 concentration, w; malate concentration in the cytoplasm, x; malate concentration in the vacuole, y; and z is a variable that describes the ordering of the lipid molecules in the tonoplast membrane.
u3
T=0.2254
0.4
tonoplast
(2)
The meaning of each variable is the following: Fig. 1. Flow diagram of the CAM model showing as dynamic variables (encircled) three reactant pools (internal CO2 concentration, w; malate concentration in the cytoplasm, x; and in the vacuole, y) and the order of the tonoplast, z, within a CAM cell.
Jco2 : CO2 uptake from outside Cco2 : CO2 consumption by photosynthesis, which is directly proportional to the external control parameter light intensity, L(t) Rco2 : CO2 production by respiration.
These are the dynamic variables of the cyclic process. They are connected by the flows, u1 , u2 , u3 , during the gain and loss terms of the metabolites. The model depends on three external control parameters: temperature, T , light intensity, L, and external CO2 concentration, Cext . The dynamics are characterized by a set of four coupled, nonlinear differential equations of first order in time: w˙ = −u2 + u3 x˙ = −u1 + u2 (1) y˙ = u1 τ z˙ = g(z, T ) − y
Blasius et al. calculated the dynamic behavior using the dimensionless variables with the parameters (see Blasius et al. (1999)): Cext = 1, L(t) = 1, T = 0.2238, 0.2242, 0.2246, 0.2250, 0.2254, c = 5.5, cJ = 1, cR = 1, ε = 0.001, τ = 0.35, α = 1.5, w1 = 0.1, LK = 0.5, R = 0.1. Using these parameters, we perform the computer simulation by MATLAB/Simulink. The initial conditions of the states are given as w(0) = 0.4, x(0) = 0.62, y(0) = 0.56, z(0) = 0.2. The nonlinear function g(z, T ) is approximated via the third-order spline interpolation from the figure shown in the reference(see Blasius et al. (1999)). Figure 3 shows the dynamic responses of the state variables when T = 0.2242. Blasius et al. have shown the influence of temperature as an external control parameter (see Blasius et al. (1999)). With increasing temperature the steady state malate level decreases, until at the critical temperature, the system undergoes supercritical Hopf bifurcation, and the stable fixed point changes into unstable one surrounded by a small-amplitude limit cycle (see Blasius et al. (1999)).
where the function g(z, T ) is the thermodynamic equilibrium value of malate concentration in the vacuole ,y, and is a third-order nonlinear function depends on the temperature, T (see Blasius et al. (1999)). The temperaturedependent z-null cline, y = g(z, T ), corresponds to the hysteretic behavior of the phase diagram in the membrane model shown in Fig.2( seeNeff et al. (1998)). The constant, 60
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1.4
w x y z
1.2
manifold, we adopt Cext (t) and L(t) as the inputs of the control system. Using the control variables Cext (t) and L(t), we can reshape w-nullcline explicitly.
(state) (state) (state) (state)
We consider the following two types of the control input:
0.8 0.6
L(t) = g(m(t) − s0 ) (7) exp(αw) {g(m(t) − s0 )} (8) Cext = CJ where m(t) is a measurement signal and we select the measurement signal as w(t), The bias term, s0 , keeps the input variable positive. The input allows us to change wnullcline given by (5).
0.4 0.2 0
5
10 time
15
20
Fig. 3. Time responses of sustained endogenous rhythms in continuous light when T = 0.2246.
Figure 4 shows w and x nullclines with the feedback controller (7) for the feedback gain g = −0.1, −1. Figure 5 shows w and x nullclines with the feedback controller (8) for the feedback gain g = −0.1, −1. Thus, we can change the equilibrium point and the critical manifold by using feedback controllers.
3. FEEDBACK CONTROL OF BIOLOGICAL RHYTHM In this chapter, we review how to control the frequency of the CAM rhythm (see Sakamoto et al. (2013)) and propose the reconstruction method of the internal CO2 concentration, w by the CO2 uptake from outside. 3.1 State-Space Model
1.2
x
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1 x−nullcline
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w−nullcline
0.2
0.2 0.4
0.6 w
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1
0.2
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0.6 w
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Fig. 4. w-nullcline (solid line) and x-nullcline (dotted line) with the feedback controller (7), where g = −0.1(lef t), −1(right) and s0 = 2. 1.6
1.6 w−nullcline
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x
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0.6 w
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1
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1
Fig. 5. w-nullcline and x-nullcline with the feedback controller (8), where g = −0.1(lef t), −1(right) and s0 = 2.
(4)
3.3 Feedback Control with L(t)
because it can be measured easily than the other states x(t), y(t) and z(t).
We try to control the biological rhythm by using the light intensity L(t) as the control input and by setting both of the temperature and the external CO2 concentration are given constants. In this case, the input variable, L(t), is nonlinear, because the nonlinear function , f˜ (˜ x, L, T ), contains Rco2 . We design the following controller:
3.2 Reshaping of Nullcline by Output Feedback The CAM system consists of the fast modes w, y and the slow modes y, z, because the time constant is small. The feedback control allows us to reshape the nullclines of the fast modes w and x. From (1), w-nullcline and x-nullcline are given by w + x + Jco2 − Cco2 + Rco2 = 0 x y w −cx + + − x = 0 z x
1.4
1.2
0.6
The output m(t) is given by
−
1.6
1.4
1
The CAM model can be rewritten by the state-space form: x ˜˙ = f˜(˜ x, L, T ) + g˜1 (w)(Cext (t) − w) + g˜2 L(t) 1 (−u + R ) 2 co 2 w 1 x ˜ (−u + u ) 1 2 x ˜ = ,f = y u1 1 (3) z (g(z, T ) − y) τ 1 cJ 1 exp(αw) 0 0 g˜1 = , g˜2 = 0 0 0 0 m(t) = w(t),
1.6
x
0
x
w,x,y,z
1
61
L(t) = g(w(t) − s0 ) (9) where g is a feedback gain and s0 = 0.2 is a constant bias. The constant bias s0 is added to maintain the control signal positive.
(5) (6)
Figure 6 shows the responses of the CO2 uptake Jco2 for each feedback gain g = 0, 1.0, 1.5, 5.0, 20, 30. The other control variables Cext (t) and T are fixed as Cext = 1, T = 0.2246. These figures indicate that the feedback gain within the appropriate range can change the frequency of the rhythm.
The control variables Cext (t) and L(t) are included in Jco2 and in Cco2 and Rco2 , respectively. On the other hand, the control variable T affects the z-dynamics. Since T cannot be used to reshape w and x nullclines as the critical 61
1 0.8
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(e) g = −0.75
Fig. 6. Responses of Jco2 for each feedback gain g = 0, 1.0, 1.5, 5.0, 20, 30.
10
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(f) g = 30
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(b) :g = −0.3
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5
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0
1 0.8
(f) g = −1.0
Fig. 7. Responses of Jco2 for each feedback gain g = −0.1, −0.3, −0.5, −0.6, −0.75, −1.0.
3.4 Feedback Control with Cext (t)
is not suitable for the real-time feedback control. Since 0.1 ≤ w ≤ 1, we approximate the exponential function as exp(αw) ≈ 1 + αw + 12 α2 w2 . In this case, we obtain w as a solution of the 2nd order polynomial equation:
We design a controller of the biological rhythm by using the external CO2 concentration, Cext , as the control input and by setting both of the temperature and the light intensity are given constants. In this case, the nonlinear state-space equation is rewritten by the affine system:
α2 Jco2 w2 + (cJ + αJco2 )w + (Jco2 − cJ Cext ) = 0(13) 2 Fig.8 shows the state variable w and its approximated value using (13). The error between two variable is small.
˜˙ = {f˜ (˜ ˜ 1 (w)w + g ˜ 2 L(t)} + g ˜ 1 (w)Cext (t). x x, L, T ) − g We design the following controller: exp(αw) {g(w(t) − s0 )} (10) CJ where g is a feedback gain and s0 = 0.15 is a constant bias. The constant bias s0 is added to maintain the control signal positive.
w values
Cext (t) =
1
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0 -0.1
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0.6 0.5 0.4
Figure 7 shows the responses of the CO2 uptake Jco2 for each feedback gain g = −0.1, −0.2, −0.3, −0.5, −0.6, −0.75. The other control variables L(t) and T are fixed as L(t) = 1, T = 0.2246. In the low-gain case, the equilibrium is stable and the limit cycle does not exist. In the high-gain case, the limit cycle also disappears. Therefore, there is a limit to make the rhythm to fast. However, the feedback controller with Cext (t) can change the frequency of the rhythm. 3.5 Calculation of w using Jco2
0.5 0.4
0.3 0.2 0
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10 time
15
20
0.2 0
5
10 time
15
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Fig. 8. Dynamic responses of w (left) and the approximated value using the solution the second order equation (right) ,where figure of top is g = 1. 4. PHASE CONTROL OF SINGLE CELL BY LIGHT INTENSITY INPUT We have recently proposed the linear and the nonlinear feedback controllers for the phase shift of the CAM model using the light intensity as an input (see Sakamoto et al. (2013)). The linear controller requires the external periodical signal as the reference input. The feedback gain of the linear controller effects the phase of the rhythm. On the other hand, the nonlinear controller does not need external signals.
The CO2 uptake from outside, Jco2 , is measurable and is given by (Cext (t) − w) . (11) exp(αw) where the external CO2 concentration, Cext is known. When Jco2 = 0, the above equation is regarded as the nonlinear algebraic equation with respect to w as follows: Jco2 = cJ
In this chapter, we propose another controller using pulse signals, which is suitable for practical use. We select the pulse signals Pi (t)’s (i = 0, 1, 2, 3, 4) as follows:
Jco2 exp(αw) + cJ w = cJ Cext . (12) The iterative numerical calculation allows to obtain w using the known signals Jco2 , Cext (t). However, this solution
P0 (t) = { 1(0 ≤ t ≤ 20) 62
P2 (t) = P3 (t) = P4 (t) =
{ { {
0.9(0 ≤ t ≤ 2) 1(2 ≤ t ≤ 20) 0.9(0 ≤ t ≤ 5) 1(5 ≤ t ≤ 20)
J co2 value
P1 (t) =
{
Ryo Sakamoto et al. / IFAC-PapersOnLine 48-18 (2015) 059–064
0.2(5 ≤ t ≤ 10) 1(0 ≤ t ≤ 5, 10 ≤ t ≤ 20)
0.5
0.3 0.2
0.5
0.3 0.2 0.1
5
10 time
15
20
0 0
5
10 time
15
20
(b) P4 (t)
Fig. 10. Responses of Jco2 ’s using Lf f i ’s (solid line : feedforward control, dotted line : step input L(t) = 1). The state variable and the input of each cell are defined as follows: (wi , xi , yi , zi ), Li (t); i = 1, 2
(14)
All parameters of two cells are same, but the initial conditions are different. 5.1 Synchronization of Two Cells without Coupling When there is no coupling between two cells, the state space equations are given by
0.6
0.45
0.4
(a) P3 (t)
1.3(5 ≤ t ≤ 10) 1(0 ≤ t ≤ 5, 10 ≤ t ≤ 20).
where w ˆ is the recovered value using the measurable signal Jco2 as in Section 3.5. Fig.9 shows the response of Jco2 for each input Lf bi (t). From these responses, we can see that the feedback signal changes not only the phase but also the frequency and the wave form. 0.5
0.4
0.4 J co2 value
0.35 J co2 value
0.6
0.4
−0.1 0
ˆ Lf bi (t) = w(t)P i (t)
0.3 0.25 0.2 0.15
x1 , L1 , T ) + g˜1 (w1 )(Cext1 (t) − w1 ) + g˜2 L1 (t)(15) x ˜˙ 1 = f˜(˜ ˜ ˙x x2 , L2 , T ) + g˜1 (w2 )(Cext2 (t) − w2 ) + g˜2 L2 (t)(16) ˜2 = f (˜
0.3 0.2 0.1
0.1
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20
(a) P0 (t) 0.6
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The external CO2 concentrations are same as follows:
20
0.1 0 15
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Cext1 (t) = Cext2 (t) = 1. The feedforward inputs of the light intensities are also same as follows:
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10 time
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5
(b) P1 (t)
0.6
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−0.1 0
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4.1 Feedback Control with Pulse Signal
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−0.1 0
(c) P2 (t)
L1 (t) = L2 (t) = Pi (t), 5
10 time
15
i = 5, 6, 7
where Pi (t) are selected as:
20
(d) P3 (t)
P5 =
Fig. 9. Responses of Jco2 ’s using Lf bi ’s (solid line : feedback control, dotted line : step input L(t) = 1).
P6 =
{ {
0(1 ≤ t ≤ 2) 1(0 ≤ t ≤ 1, 2 ≤ t ≤ 200) 0(2 ≤ t ≤ 3) 1(0 ≤ t ≤ 2, 3 ≤ t ≤ 200)
Fig.11 shows the CO2 uptakes of two cells and the error of the CO2 uptakes for each input. We can see that the synchronization is dependent on the pulse input.
4.2 Feedforward Control with Pulse Signal We select the feedforward controller as
5.2 Synchronization of Two Cells with Coupling
Lf f i (t) = Pi (t). Fig.10 shows the response of Jco2 for each feedforward input Lf f i (t). This figure indicates that the feedforward input via the pulse signals can shift only the phase of rhythm.
Fukuda et al. (2004) proposed the coupling model of two or more cells of the CAM plants. Using this model, the coupling model of two cells is written by the state space equations as: x1 , L1 , T ) + g˜1 (w1 )(Cext1 (t) − w1 ) x ˜˙1 = f˜(˜ +˜ g2 L1 (t) + g˜31 w1 Dw (w1 − w2 ) (17) x Dx (w1 − w2 ) x ˜1 = 1 , g˜31 = 0 y1 0 z1
5. PHASE CONTROL OF TWO CELLS BY LIGHT INTENSITY INPUT From the simulation results in the previous chapter, it is enough to shift the phase of rhythm by the feedforward pulse input. In this chapter, we investigate the synchronization phenomena of two cells with different initial values by using the feedforward pulse signal of the light intensity. 63
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x ˜˙ 2 = f˜(x˜2 , L2 , T ) + g˜1 (w2 )(Cext2 (t) − w2 ) +˜ g2 L2 (t) + g˜32 (18) w2 Dw (w2 − w1 ) x2 Dx (w2 − w1 ) x ˜2 = , g˜ = . 0 y2 32 0 z2
0.6
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Jco2 error
Jco2
0.4 0.3 0.2
0.1 0 −0.1
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The connection term is given by
−0.1 0
−0.3 50
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(a) Jco2 ’s via P5 (t)
[ Dw D x ] = h [ 1 1 ] . where h is the strength of the connection. The feedforward inputs of the light intensities are given as follows:
0.4
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(b) Error of Jco2 ’s
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IFAC CHAOS 2015 64 August 26-28, 2015. Tokyo, Japan
0.3 0.2
0.1 0 −0.1
0.1
L1 (t) = L2 (t) = Pi (t), i = 5, 6 Fig.12 shows the CO2 uptakes of two cells and the error of the CO2 uptakes for each input when the connection strength is h = 10. We can see that the strong connection can cause large deviation of the phase with the passage of time. 0.6
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(a) Jco2 ’s via P5 (t)
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model biological clock and produces specific dynamic behaviour. Proc. R. Soc. Lond. B, 268:1307–1313. Blasius,B., Neff,R., Beck,F. and L¨ uttge,U. (1999). Oscillatory model of crassulacean acid metabolism with a dynamic hysteresis switch. Proc. R. Soc. Lond. B, 266: 93–101. Borland,A.M. and Yang,X. (2013). Informing the improvement and biodesign of crassulacean acid metabolism via system dynamics modelling. New phytologist, 200:946– 949. Borland,A.M., Hartwell,J., Weston,D.J., Schlauch,K.A., Tschaplinski, T.J., Tuskan,G.A., Yang,X., and Cushman, J.C. (2014). Engineering crassulacean acid metabolism to improve water-use efficiency. Trends in Plant Science, 19-5, 327–338. Fukuda,H., Kodama,J., and Kai,S. (2004). Circadian rhythm formation in plant seedling: global synchronization and bifurcation as a coupled nonlinear oscillator system. BioSystems, 77, 41–46. Fukuda,H., Murase,H., and Tokuda, I.T. (2013). Controlling Circadian Rhythms by Dark-Pulse Perturbations in Arabidopsis thaliana. Sci. Rep., 3, 1533; DOI:10.1038/srep01533. Matsuo,T., Totoki,Y., and Suemitsu,H. (2013). Adaptive Estimation of Biological Rhythm in Crassulacean Acid Metabolism with Critical Manifold. ISRN Applied Mathematics, Article ID 856404, 9 pages, 2013. doi:10.1155/2013/856404. Neff,R. Blasius,B., Beck,F., and L¨ uttge,U. (1998). Thermodynamics and energetics of the tonoplast membrane operating as a hysteresis switch in an oscillatory model of crassulacean acid metabolism. J. Memb. Biol., 165: 37–43. Owen,N.A and Griffiths,H. (2013). A system dynamics model integrating physiology and biochemical regulation predicts extent of crassulacean acid metabolism (CAM) phases. New phytologist, 200:1116–1131. Sakamoto,R.,Goto,A., Suemitsu,H., and Matsuo,T. (2013). Phase Shift of Biological Rhythm in Crassulacean Acid Metabolism by Controlling Light Intensity. Proc. of SICE2013, Nagoya,783–786.
0
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Fig. 12. Jco2 ’s of two cells and their error (h = 10).
−0.2
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(c) Jco2 ’s via P6 (t)
−0.1
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Jco2
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Fig. 11. Jco2 ’s of two cells and their error (h = 0). 6. CONCLUSION We proposed the reconstruction method of the internal CO2 concentration using the CO2 uptake from outside and presented the feedback controller and the feedforward controller. Using the computer simulations, we showed that the feedback controller allows us to control the frequency, the wave form, and the phase of the rhythm, but the feedforward controller with the pulse can change only the phase of the rhythm. Moreover, in the case of the coupling cells, we indicated that the timing of the pulse in the feedforward controller affects the synchronization of cells. Owen et al. have recently proposed a new physiological model of the CAM plants (see Borland et al. (2013); Owen et al. (2013)). The construction of the coupling model of the cells based on the physiology is a next future problem. REFERENCES Beck,F., Blasius,B., L¨ uttge,U., Neff,R., and Rascher,U. (2001). Stochastic noise interfers coherently with a 64