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30Journal of Siberian Federal University. Biology 4 (2009 2) 466-480
УДК 574.45 + 504.7
Mathematical and Computer Simulation of the Biological Life Support System Module 1/2. Description of the Model
a, Yury V. Barkhatova*, Nikolai S. Manukovskya, Alexander A. Tikhomirovab, Andrey G. Degermendzhyab, Jean-Bernard B. Grossc,
Christophe Lasseurd
a Institute of Biophysics of Siberian Branch of Russian Academy of Sciences,
Akademgorodok, Krasnoyarsk, 660036 Russia b Siberian Federal University, 79 Svobodny, Krasnoyarsk, 660041 Russia c LGCB, Universiter B.Pascal, CUST, BP206, 63174 Aubie're cedexFrance d Environmental Control and Life Support section, ESA-Estec, Postbus 299, 2200 AG, Noordwijk, Netherlands 1
Received 7.12.2009, received in revised form 14.12.2009, accepted 21.12.2009
The mathematical model based on kinetic coefficients and dependencies obtained during the experiments was constructed to estimate the character of functioning of the experimental module of biological life support system (BLSS) and the possibilities of its controlling. The mathematical model consists of two compartments - the 'phytotron' model (with wheat and radish) and the 'mycotron' model (for mushrooms). The following components are included into the model: edible mushrooms (mushroom fruit bodies and mycelium); wheat; radish; straw (processed by mycelium); dead organic matter in the phytotron (separately for the wheat unit and for the radish unit); worms; worms' coprolites; vermicompost used as a soil-like substrate (SLS); bacterial microflora; mineral nitrogen, phosphorus and iron; products of the system intended for humans (wheat grains, radish roots and mushroom fruit bodies); oxygen and carbon dioxide. At continuous gas exchange, the mass exchange between the compartments occurs at the harvesting time. The conveyor character of the closed ecosystem functioning has been taken into account - the number of culture age groups can be controlled (in experiments and in the model - 4 and 8 age groups). The conveyor cycle duration can be regulated as well. The module was designed for the food and gas exchange requirements of 1\30 of a virtually present human. The model estimates the values of all dynamic components of the system under various conditions and modes of functioning, especially those, which are difficult to be realized in the experiment. The model allows dynamic calculation of biotic turnover closedness coefficient for main considered elements. The coefficient of matter biotic cycle closure for systems based on matter supplies has been formalized.
Keywords: Mathematical modeling, biological life support system, closedness coefficient
Vladimir G. Gubanov
* Corresponding author E-mail address: barkh@ibp.ru
1 © Siberian Federal University. All rights reserved
Introduction
The mathematical model based upon kinetic coefficients and dependencies obtained during the experiments was constructed to estimate the functioning pattern of the biological life support system (BLSS) experimental module (Tikhomirov et al., 2003a, 2003b) and its controllability. The mathematical model consists of three compartments - two 'phytotron' models (with wheat and radish) and the 'mycotron' model (for mushrooms). The following components are included into the model: edible mushrooms (mushroom fruit bodies and mycelium); wheat; radish; straw (processed by mycelium); dead organic matter in the phytotron (separately for the wheat unit and for the radish unit); worms; worms' coprolites; vermicompost used as a soil-like substrate (SLS); bacterial microflora; mineral nitrogen, phosphorus and iron; products of the system intended for humans (wheat grains, radish roots and mushroom fruit bodies); oxygen and carbon dioxide. At continuous gas exchange, the mass exchange between the compartments occurs at the harvesting time. The conveyor character of the closed ecosystem functioning has been taken into account - the number of culture age groups can be controlled (in experiments and in the model - 4 and 8 age groups). The conveyor cycle duration can be regulated as well. The module was designed for the food and gas exchange requirements of 1\30 of a virtually present human.
The model also allows for the following processes: photosynthesis of wheat and radish in relation to the crop age, irradiance, the amount of biogenic elements; respiration of mushrooms, worms, bacteria and a human; consumption of grain and radish roots by a human and a return of biogenic elements in the mineral form; utilization of dead phytomass by worms and bacteria; processing ofwheat straw by mushroom mycelium; conversion of worms' coprolites into the mineral
form by bacteria. Continuous dynamic processes going in each of three system compartments are described by differential equations written in terms of mass using subsidiary conditions for parameters and discrete relations.
The model estimates the values of all dynamic components of the system under various conditions and modes of functioning, especially those, which are difficult to be realized in the experiment. The model allows dynamic calculation of biotic turnover closedness coefficient for main considered elements. The ratio of any ith biogenic element flow on the producer link to the sum of the same flow and the flow of element coming into the deadlock sediments is the closure measure of any i- element (coefficient Cli).
1. BLSS experimental module
We present brief description of the BLSS module and methods of working with it (more detailed description can be found elsewhere (Tikhomirov et al., 2003a, 2003b)). The experimental model of a BLSS consists of three interrelated components: autotrophic, heterotrophic, and physicochemical. The autotrophic component is represented by two plant species: the short-stemmed Triticum aestuvi L. spring wheat (cultivar 232, selected by G.M. Lisovsky) and Raphanus sativus L. the radish cultivar Virovskii White. These plant species have been extensively used in life support systems (Gitelson et al., 1975; Zamknutaya sistema... (Closed system.), 1979). In this BLSS model the plants were grown under continuous lighting at 150-170 W/m2 of photosynthetically active radiation (PAR). The plants were grown in rectangular chambers 0.155 m high, with the bottom area 0.032 m2 (0.22 x 0.145). Wheat was grown in 16 chambers, radish in 8. During the growth period, the chambers were placed very close to each other, so that wheat and radish formed their
cenoses. The growth period of the wheat was 64 days. The growth period of the radish was 32 days. To decrease the age dependence of plant photosynthesis, an uneven-aged conveyor was organized in the system (Gitelson, Lisovsky, Tikhomirov, 1997). In the conveyor, wheat was represented by 8 age groups and radish by 4.
The main representatives of the heterotrophic unit were mushrooms, worms, and bacterial microflora. The mushroom component was represented by the oyster mushroom Pleurotus ostreatus (Jacq.: Fr.) Kummer, a higher edible mushroom. The mushrooms P. ostreatus were grown in non-sterile conditions, on straw wheat, during 60-70 days; carposomes formed within two, sometimes three, generations and constituted 6-7 g per 100 g of the substrate (dry basis). When the growth was completed, the unutilized portion of lignocelluloses - the residual substrate - amounted to 40-45% of the initial mass. The residual substrate was used to prepare vermicompost or as an organic fertilizer for the plants.
The worm component was represented by the red California worm, a hybrid close to Eisenia foetida Savigny. The worms facilitated the utilization of the abovementioned residual substrate, radish tops, and other organic substrates. During 60 days, five worms normally introduced into 100 g of moist substrate formed 25-30 g of vermicompost -organic substance used as soil-like substrate (SLS) to grow wheat and radish plants in the system. SLS is a product of continuous conversion of gramineous culture straw (wheat, rice etc.) by wood-destroying fungus P. ostreatus and worms E. foetida. In its main characteristics SLS is similar to organic soils (histosols), but it differs from them by the absence of aluminosilicate matrix. Prior to the launch of the system, the starting SLS was made from wheat straw, using mushrooms and
worms, following the procedure described elsewhere (Manukovsky et al., 1996, 1997).
Structurally, the BLSS model consisted of two hermetically sealed chambers, 3 m3 in volume each (Fig. 1). The gas exchange between the chambers was continuous, through air pipes. The air exchange between the chambers was effected with a fan. In the chamber used for cultivation of wheat and radish on the SLS the air temperature was maintained at 24±1°C. In the other chamber, where mushrooms matured and bore fruit, the temperature was maintained at 19 °C, the humidity at 90%, and the irradiation at 15 W/m2 PAR. The CO2 concentration in the entire system was maintained at 0.11±0.03% and the oxygen concentration at 20,6-21,5%. Higher plant photosynthesis produced surplus oxygen, necessary for human respiration, i.e. O2/CO2 > 1 (the last ratio refers to 02 and C02consumption-production not to their concentrations). Every day, a human periodically breathed through a special mask to equalize this ratio and maintain it balanced. The system was designed for the food and gas exchange requirements of 1/30 of a virtually present human.
At the start, the experimental system was supplied with 120 L of distilled water, including 70 L of irrigation water and 50 L of the water as part of the SLS, plants, and other organic matter.
To collect and analyze the experimental data, the system was opened once in eight days for four hours. That time was spent on removing chambers with mature plants out of the chamber and on collecting samples of the phytomass: wheat grains, straw, and roots; radish tops and both edible and inedible roots. After these components of the phytomass were removed from the growth chambers, worms were counted in the remaining SLS, the SLS mass was determined, and samples were taken for analysis. The organic matter left after sampling was returned to the respective growth chambers, which were placed back
Chamber 1
Chamber 2
Fig. 1. A small experimental life support system (with a 1/30 fraction of virtually present human) with increased closure degree. Chamber 1 - phytotron - autotrophic unit; chamber 2 - mycotron - heterotrophic unit
into the plant growth chamber. The harvested wheat straw was used to make the substrate for mushrooms. The P. florida mycelium was grown in stainless-steel cylindrical containers in the plant growth chamber at a temperature of 25°C during 3 weeks. Then the containers were placed into the other chamber, where the mushrooms matured and fruited.
Edible biomass - grains, roots, mushrooms -were oxidized in the physicochemical reactor, to simulate their consumption by a human. The oxidation products - inorganic compounds readily assimilated by plants - were returned into the system. The physicochemical method of waste utilization is based on using hydrogen peroxide, which can be derived from water within the system. Unlike other physicochemical
methods based on this principle, this one does not require high temperatures or pressures and is energy efficient, environmentally friendly, and safe (Kudenko et al, 1997). To maintain the physicochemical process, no auxiliary chemical substances or other ingredients, which have to be taken as a store, are needed.
The output oxidation products were metabolite water, carbon dioxide, and mineral residue. The mineral residue consisted of the mineralized organic matter (NH4+, SO42-) and phytomass ash (P, K, Mg, Ca, Na, S, Si). The mineral residue was returned to the irrigation water. Thus, oxidation yielded a solution of oxides that did not contain any substances harmful for plants. It was added to the SLS as a nutrient for plants.
Fig. 2. Scheme of mass exchange in the experimental system системе (Tikhomirov et al., 2003a, 2003b). Material flows are given as g/day. Broad arrows show the water exchange circuit. Flows of metabolite water are given separately. The SLS mass in the system was 15 kg (dry basis)
After the oxidation of the edible biomass in the reactor, the carbon dioxide was returned into the system. The destruction rates of the substance were calculated based on the chemical composition of the phytomass. The experiment with the system, which was in a quasi-stationary state, lasted about 90 days. The general scheme of the material balance and water exchange in the experimental system can be seen in Fig. 2. This scheme displays the material balance in the experimental system where the processes of photosynthesis of wheat and radish biomass are counterbalanced by the processes of
biodestruction of inedible phytomass in the SLS and by oxidation of grains, edible roots, and mushroom carposomes in the physicochemical reactor.
In the scheme the photosynthesis of the plant biomass and the biosynthesis of mushrooms from the wheat straw are shown as parts of one unit. In this unit, the processes of synthesis are counterbalanced by the processes of oxidation of edible biomass (grains, edible roots, and mushrooms) in the physicochemical reactor and by the processes of biodestruction of inedible phytomass (tops, roots, residual substrate) in the
SLS and of extractive substances in the irrigation water.
2. Mathematical model of the BLSS module
The mathematical model (primary description of which is given in (Gubanov et al., 2007) based on kinetic coefficients and dependencies obtained during the experiments was constructed to estimate the character of functioning of the experimental module BLSS and the possibilities of its controlling.
The mathematical model consists of two compartments - the 'phytotron' model (with wheat and radish) and the 'mycotron' model (for mushrooms). The following components are included into the model: edible mushrooms (mushroom fruit bodies and mycelium); wheat; radish; straw (processed by mycelium); dead organic matter in the phytotron (separately for the wheat unit and for the radish unit); worms; worms' coprolites; vermicompost used as a soil-like substrate (SLS); bacterial microflora; mineral nitrogen, phosphorus and iron; products of the system intended for humans (wheat grains, radish roots and mushroom fruit bodies); oxygen and carbon dioxide. Flow chart of the model is presented in Fig. 3.
At continuous gas exchange, the mass exchange between the compartments occurs at the harvesting time. The conveyor character of the closed ecosystem functioning has been taken into account - the number of culture age groups can be controlled (in experiments and in the model -4 and 8 age groups). The conveyor cycle duration can be regulated as well. The module was designed for the food and gas exchange requirements of 1\30 of a virtually present human.
The model also allows for the followeing processes: photosynthesis of wheat and radieh in relation to the crop age, irradiance, the amount of biogenic elements; respiration of mushrooms, worms, bacteria and a human; consumption of
grain and radish roots by a human and a return of biogenic elements in the mineral form; utilization of dead phytomass by worms and bacteria; processing of wheat straw by mushroom mycelium; conversion of worms' coprolites into the mineral form by bacteria.
2.1. Description of continuous dynamic processes
Let us examine the structure of the model in greater detail. First, the continuous dynamic processes going in each of three system compartments, described by differential equations written in terms of mass, should be conceived.
2.1.1. 'Mycotron' model
The growth rate of mushroom mycelium Fm is defined as:
dt
(M m /m ) Fm
Fm > 0
(Mm y m )Fm ~ F, ^m < 0
(1)
where /um is mycelium specific growth rate, ym is the specific rate of mycelium metabolism, f is the specific rate of fruit bodies growth (formation), F is the mass of mushroom fruit bodies.
H + Km
(2)
y m y m Fm
0 < k < 1.
where LO2 is the coefficient of heterotroph growth limitation by the oxygen content in the atmosphere, (!m is the maximum specific growth rate of mycelium, H is the straw biomass in mycotron, KH is the constant of mycelium halfsaturation on straw, y*m = const, k - exponent, any rational number from 0 to 1.
Lo, is defined as:
LOt =■
1 , Q >Qoi K„Q , Q<Qol
(3)
Growth period
Redistribution period
straw
o2 co2
1 4 1 r
mycelium
O2 CO2
+ t 1 r
fruit
bodies
PHYTOTRON
Fig. 3. Flow chart of the mathematical model of the «phytocenosis - soil-like substrate - gas exchange with a human» system. Here, bacteria-1 are responsible for consumption of dead organic matter, bacteria-2 are responsible for consumption of worms' coprolites
where Q is the mass of molecular oxygen in system atmosphere, Qop, is the eontent of oxygen in system atmosphere optimal for heterotrophs, KQ is the constant of heterotroph growth limitng by the oxygen content in atmosphere.
The growth rate of mushroom fruit bodies F is written in the form:
dF dt
JF, dF->0 dt
0 , Fm < 0
when the ratio of fruif bodies' mass and mushroom mycelium amounts to its maximal pobsible value p and at this moment the value F i s «zeroing» -fruit bodies are gathered in the harvest.
The rate of straw mass reduction in mycotron H can be written as:
H = _ ymFm dt YH '
(5)
(4)
with subsidiary condition describing the harvesting of mushroom fruit bodies:
F
F = 0 3 —<m,
F '
m
where p is the maximum possible proportion of mushroom fruit bodies and mycelium mass, 0< p <1, p = const; the condition means that harvesting of mushroom fruit bodies takes place
where YH is the coefficient of mycelium yield on straw.
The rate of «dead end» (the organic matter, which does not take part in matter cycling) formation for mycotron (BF) is defined as:
dt
(6)
2.1.2. 'Wheatphytotron' model
The growth rate of the total wheat mass X1 is defined as:
X dt
~ MxiJi ,
(7)
where fiXi is wheat specific growth rate, J1 is the surface area of wheat growth chamber, at that tliat,
Mxi = kiLco2LExi-Lsici(ai(A _pii) bi_hù ) , (8)
where ki is tlie coefficient of photosynthesis dependencis of wheat growth rate, L0i is tlie coefficient of phototroph growth limitation by carbon dioxide content in system atmosphere, L exi is the coefficient of wheat growth limitatton by illumcnationi ah bh cm are experimental constants, LS1 is che coefficient of wheat growth limitation bb biogenic elemfnts, t is the currena time for wheat phytotron, tn is the time of wheat grains germination. With respect to depe ndency (8) see the woak (Gttelson et al., 19f5).
,W>Wopt
(9)
where ris carbon dioxide mass in tlie system, Wopt is carbon dioxide content in the system optimal fo r wheat, Kw is the constant of phototfophs limitation by caebon dioxide conOent on the system.
LE X i _
-O.OOli-£+2.5238, E > E, G 0.0059 • E-0.110O E<E,
opt
(i0)
where E is the intensity of system illumination, is the illumination intensity optimal for
E
opt
wheat. The dependence (i0) was obtained by approximation of enperimental data (Tikhomirov et al., 2003a).
(if)
where K^, K^, K^
The rate of worms' mass change G1 in wheat phytotron is defined ar:
LGf = /rстGl-;l/GA, dt
(il)
where Ug0 ih worms' specific growth rate in wheat growth chamber, yG 0s the specific rate of worms metabo^i^ where
f-^oi ~ Lo2 (P-gru + Mgzi) ,
(13)
where UrGRU is the specific growth rate of worms on bacteria-1 (bacteria responsible for consumption of dead organic matter) in wheat growth chamber, UGZ1 is the specific growth rate of worms o n dead 00ganic matter in wheat growth chamber. „ z,
LGO1 ^Zl^G ^ „
Zi +-K-G Mgrii riiMg "
(i4)
Roo + KS
Zt is the dead organic matter in wheat phytotron, R-iRu is the mass of bacteria-1 in wheat phytotron, |1G is tine maximum possible specific growth rate of worms, Kgz, Kri are the constants of worms half-saturation o n Z, and R01 respectively.
Normalizing quotients aZ1 and aR11 are defined (Abrosov et al., 1992; Gubanov, Degermendzhy, 2002) as:
Zi
i
^Gi Zi KGZ
_ i R
AGi _
AGi Rii + KRi Zi , Rii
(15)
ThLe; rate oR bacteria-1 mass change in wheat phytotron Rl1 is written in the form:
dRii -u R v R i
—f - ^RiiRii " YRiRii " Y-
dt Y rRi
MrRiiri,
(L6)
are the constants of wheat growth limitation by the content of mineral nitrogen, phosphorus and iron respectively in the growth chamber, SS1, S2, S3 is the content of mineral nitrogen, phosphorui and iron in tlie; growth chamber.
- 4173 -
where /uR11 is bacteria-1 specific growth rate in wheat growth chamber, yR1 is the specific rate of bacteria-1 metabolism, YGR1 is the coefficient of worms yield on bacteria-1.
Zx
Ar
= LOi AR
Zi + KRiZ
(11)
Zi + K GZ Rii + KRi
where 1R1 is the maximum specific growth rate of bacteria-1, KR1Z is the constant of bacteria-1 halfsaturation on dead organic matter.
The rate of dead organic matter utilization in wheat phytotron Z1 is defined as:
dZ± ___R _ _J_ G
dt
Yrj
Yg,
(18)
where YR1Z is the coefficient of bacteria-1 yield on dead organic matter, YGZ is the coefficient of mushrooms yield on dead organic matter.
The change of bacteria-2 (responsible for consumption of worms coprolites) mass R21 is defined as: dR.,
dt
^■r21jr-21 tr2r2
(19)
d-RH LO,f-2R2~
C
(20)
C1 + KR2C
where |1R2 is the maximum specific growth rate of bacteria-2, 07, the mas s of worms' copro lite s , KR2C is the constant of bacteria-2 half-saturation on coprolites.
The change of mass of worms' coprolites C1 is written in the form:
/K21R2:
dC1 dt
+ (y--V^Gzfi^a
+ Rr--VLígru^gri +
(21)
dS11 = 1
dt Y ' , X1S1
-Yr 1Z
1
X1 +aR
-J I^rURii + au
~ J^|^R21R21 +
U 2 1 ( tiJ ) ' t '
(22)
where YX1S1 is the coefficient of wheat yield on mineral nitrogen, a risi, a R2si, a U2S1 are the specific content of mineral nitrogen in bacteria-1, bacteria-2 and human metabolites respectively, U21(til) is the biomass of products intended for a human at the beginning of wheat growth period, tmax1 is the time (duration) of wheat growth perio d.
The mass change of mineral phosphorus S21 and iron S31 in wheat phy totron is written in a way similar to nitrogen.
The rate of accumulation of product supplies for humans in the current growth period in mycodron and wheat phytotron Un is defined as:
where piR21 is bacteria-2 specific growth rate in wheat growth chamber, yR2 is the specific rate of bacterie-2 metabolism.
dUn dt
= 0,
(23)
with a subsidiary condition describing the harvesting of mushroom fruit bodies:
F
an = an + F
F,
-2S(p .
(24))
The rate of expenditure of product supplies for human accumulated during this previous phase, from wheat phytotron U21:
~dS~~ 1 , 1 (215)
R amax1
The rate of «dead end» fotmationZl (organic matter falling out of the matter cycle by virtue of its incomplete closure) for wheat phytotron is calculated as:
where YR2C is this coefficient of bacteria-2 productivity on coprolites, dGR1, dGZ is the part of worms' exceements used for formation of coprolites, at their consumption of bacteria-1 or dead organics.
The change of mineral nitrogen mass in wheat phytotron S, can be written as:
dB
= (S--1K11A1 + (T~--1)(1 - ¿GZ )MGzfi1 +
1
1
dt
+ (t
Y*
Yg
- - <°<3ii1 )/¿Gií11G1 + (Y--
(26))
2.1.3. 'Radish phytotron' model
The model for radish phytotron is similar to the model of wheat phy totron. Thus, here we will point to considerable distinctions only, without giving the full description.
The rate of radisf growth X2 is defined as:
dX* ,
-t = fíJ (27 )
where J2 is the surface area of radish growth chamber, цХ2 is radish specific growth rate,
Mxi ~ JlLCO2LEX2LS2c2a211 e 22')
(28)
where k2 is the coefficient of radish growth rate dependence on the photosynthesis, LEX2 is the coefficient of radish growth limitation by illumination, LS2 is the coefficient of radish limitation by tf e content of bioganic element! in the growth chamber, a2, b1, c2 are experimental constants.
XEXl =-0.00001 -E'2 +0,00a4-£ + 0,0169 .
(29)
The dependence (29) was obtained by approximation of experimental data (Tikhomirov et al., 2003a).
¿52 -ПиП^А -KS22S2,KS3 Ä1},
(30)
where KS
Ks
Ks.
а) For whtat phytotron and mycotron:
F x = f
(31)
where Гжг is initial mass 0f mycelium,
^=0, (32)
77 * 11 — JU, — ^Х, ;
(33)
wliere 1*2 it the quantity of roots in wheat total mass, l2 is the quantity of giains in wheat total mass,
are the constants of
radish growth limitation by the content of mineral nitrogen, phosphorus and iron in growth chamber respectively, t2 is the current pe riod for radish.
The following description goes by the analoby witb wheat model.
2.2. Modelling of the proc esses associated with planting for the next cycle
Since; not all proces se s in this model are continuous, the differential equations are not sufficient for their description. For each growth chamber there is a moment of growth cycle completion, when harvesting and planting for the next cycle as wel1 a s biomass exchange between wheat phytotron and mycotron (straw goes to fungi) take place.
Thus, at the given time period (t=tmax1 for wheat phytotron and mycotron; t=tmaxi for radish phytotron) the following ratios are implemented (here []* is the value after mass redistribution; [] - before redistribution):
У X— У
U — л INITl 5
where XINm is the mass of planted grains,
Z,* = Z, +IX +(Fm-FmIN44) + F + H,
UxX = U2l + Un +(l2X, -XINIT,). б) For radish р/иуШтп:
— X — —
л 9 —Л mtT>y -
(34)
(35)
(36)
(37)
whe re X,NIT2 is Ihe mass of planted radi sh see ds ,
zi* = zi+(1-W , (38)
where 212 is "tlie; quantity of edible ro ots in radish total mass,
B22 * _ ^(2 ~F(h2BX2 XINIT2 - .
(39)
Other variables of the sy ste m do not change duri ng mass re distributio n.
2.3 . The calcula tion of conveyor organizatio n of th e system
Since the ' phytocenosis - s oil-like substrate -gas exchange with a human' closed system has a conveyor of growth chambers of different ages in each compartment, a similar conveyor is organized in the model system, where each element has its own current model period. Each element of the conveyor is calculated separately, the total biomass of each element is summed up conveyor-wise.
The total mass of mushroom mycelium in the system Fm tot can be written as:
FS
= У F
m • m
m tot m N m n n=\
£„=X(£f+£i)„+I£2
(41)
Gtot =ZG1 n +ZG2 l .
n=h 1=1
(42)
(43)
The change of oxygen mass in the system Q is defined as:
dQ
(40) -Q = Z (Lo2 (- am (Vmn + Ym Wm,
where q is the number of conveyo r age group s fo r wheat phytotron and mycotron.
Подобным же образом записываем the total mass of mushroom fruit bo dies in the system Ftot, the total straw mass in the system Htot, the total mass of wheat in the system X1 tot, the total mass of radish in the systemX2 tot.
The total mass of the «dead end» in the system Btot can be written as:
(44)
where p is the number of conveyor age groups for radish phytotron.
The total mass of worms in the system Gtot:
Similarly we write down the total mass of bacteria-i in the system R1 tot, the total mass of dead organic matter in the system Zlo„ the total mass of bacteria-2 in the system R2 tot, the total mass of worms' coprolites in the system Ctot, the total mass of mineral nitrogen in the system S1 tot, the total mass of mineral phosphorus in the system S2 tot, the total mass of mineral iron in the system S3 tot.
The total mass of products intended for humans in the system Utot:
U,0, =X(Uii n + U21 я) + £u22
Ы
2.4. Equations for oxygen and carbon dioxide of the module atmosphere
Another important component of a closed ecological system are atmospheric constituents - oxygen and carbon dioxide. Their dynamics can be calculated according to differential equations, as in the works of Abrosov et al., 1981.
-t n=i
- aG (Voin + Yg )Ghn aRl(MRlln + YRl)RUn -
- «R2C»R21n + 7R2)R21n ~ «u ( ~ ^ fen ))) +
+ JhklLCO2 LEXlLSlCl(ah(tln tlln) bh(tln tlln) ) ) +
BY „
+ Z (LO2 ( aG (Mg2í + Tg G aRl (MR12l + ÏR1 )R12l ' 1=1
- aR2 (MR22l + Yr2 )R22l ~ aU +
'max 2
^ J2k2LCO2LEX2LS2C2a2 (1 — e ^ 21 ^ ) )
BYXW
where aFm, aG, aR1, aR2, au are the coefficients showing the amount of oxygen used per unit of biomass consumed during respiration by mushrooms mycelium, worms, bacteria-1, bacteria-2 and virtual human respectively, B is the assimilation (photosynthetic) coefficient of producers, YXW - coefficient of producer yield with respect to carbon dioxide.
The change of carbon dioxide mass in the system W is defined in a similar way:
-W q
— = Z (LO2 (aFmDFm (Vmn + Г m )Fmn + -t n=l
+ aGDG (Ал. + ÏG )G1, + aRhDRl(^Rlln + /R1)R11, + + aR2DR2(MR2in +/r2)R2I„ + auDu U21n(t'hn))-
- J1k1LCO2 LEX1LS1C1(a1(t1n - *11n i - b1(t1n - tlln i3/2)/YXW i + (45)
P
Z (LO2(aGDG (Ma 2l +Yg )G2l + aR1DR1(^R12l + Y R1)R12/ +
/=1
+ aR 2 D R 2 (^R 22/ + Yr 2 )R22l + aUDU ^^ (t'2/ i i "
— J2k2Lco2LEX2LS2C2a2(1 ~ e 2^ >i / YXW ),
where DFm, Da, Dr1, DR2, Du are the coefficients of respiration for mushrooms mycelium, worms, bacteria-1, bacteria-2 and a virtual human.
3. The degree of closedness of biotic matter turnover
The important characteristics of matter biotic turnover, determining the sustainability
n=l
n=l
and resistance of super-organism systems with bioturnover, are its intensity and the degree of closedness. We should note that the latter feature is favourabln for maintaining the integral properties of the biotic system. We introduce the measure of biotic closedness of maater turnover for similar systems, including ec ological.
The degree of closedness of the biotic turnover was determined in a way similar to the approach proposed in Finn's works (1976, f978) for the determination of the cycling index (using a similar, though essentially different, approach proposed by (Gubanov, Degermendzhy, 2003, 2008; Tikhomirov et al., 2003a)). Namely, the degree of closednfss of the biotic turnover S = coefficient of the closednesn of turnover) is the ratio of the flow rate oS the subsSance supplied by heterotrophic organisms to producers (autotrophic organisms) (Q ) to thn sum of nff :fl(o w rates of the aubstance supplied by heterotrophs to autotrophs (Q) and the sub stance going to the dead end ( V ) i.e. the substance (or deposition), which cannot be completely oxidized (reduced( to initial mineral (I)iogenic) elements used by autotrophs in biosynthesis, by biott forming the system( and thus, natura(y railing out erf the biotic turnover.
a) C/,. =
b) C/ =
I",
I n* +IB
11 n,*
11 +114 '
i k i /
(46)
where Cl¡ , Cl are coefficients of the closedness of turnover for the i-th biogenic element and the matter- as a whole, k and l are all possible channels through which the substances move from heterotrophic organisms to producers and to the dead end. Evidently, 0 < C/,, C/ < 1.
This determination of the degree of closedness is realistic, at least for- the systems that must contain some material stores, e.g. systems
of the LSS type, including the systems discussed here.
A priori, we can assert that the degree of closedness on -various elements is different. From this assertion and determination of the turnover degree of dosedness (46) it fol lows that
Clj <Cl < Clt (47 )
where Cl Cf are the minimum and the maximum closedness on some j and k elements. To all appearances, 147) is a general property.
From tYe formulas (46), we can see that the change of Cl goes rn the same direction with change of Cl,- fo r any ith che mic al element. But it is obvious that Cl change happens not necessarily unidirectional with C1 change for any ith chemical element. It is obveous that, if, on any account, the degree of closedness mf matter biotic turnover as thn whole ((Cl) stays invariable at the change of degree of ctosedness on some i-th element (Cl— then there should be the change of degree of closedness ft least on one element m with the coefficient opposite to t element. So, the change ofdegree of closedness of biotic turnover on some element with 1:he same coefficient of closedness on the matter as the whole, /results in the change of closedness degree of the turnover with opposite coefficient on the other element (or elements).
In practice it means the following. For example, if there is a need to increase the BLSS degree of closedness on some element (usually, there is an attempt to reach the high degree of closedness on the key elements - C, O, S, N etc) and to maintain the degree of closedness as the whole (for mass optimum, energy supply, or other requirements), it is obvious, that the spontaneous decrease of closedness on the other element (or elements) will, most probably, take place, which is not always beneficial, as this may even drop out of sight of the researcher, since it is technically difficult to track and correct the closedness on all elements simultaneously.
Thus, the mathematical model (1-45) of the BLSS experimental module «phytocenosis -soil-like substrate - gas exchange with a human» based on higher plants (wheat and radish) and use of soil-like substrate (SLS) intended for 1/30 of a virtually present human has been constructed. The new method of measuring the degree of closedness of matter biotic turnover in super-organism systems (46) has been proposed. The
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Acknowledgments
The work was supported by Grant INTAS-ESA 099-044, by the Krasnoyarsk Regional Scientific Foundation and RFBR grant No. 03-04-96121-n2003yenisey_a .
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Математическое моделирование и вычислительная имитация модуля биологической системы жизнеобеспечения 1/2. Описание модели
В.Г. Губанов| а, Ю.В. Бархатов3*, Н.С. Мануковскийа, А.А. Тихомиров3' б, А.Г. Дегерменджиа' б, Жан-Бернар Р. Гров, К. Лассёрг
а Институт биофизики СО РАН, Россия 660036, Красноярск, Академгородок б Сибирский федеральный университет, Россия 660041, Красноярск, пр. Свободный, 79 в ЛГСБ, Университет Блеза Паскаля, Франция 63174, Клермонт-Ферран, BP206 г Европейское космическое агентство, отдел жизнеобеспечения и контроля окружающей среды, Нидерланды 2200 AG, Нордвейк, Postbus 299
Для оценки характера функционирования экспериментального модуля биологической системы жизнеобеспечения и возможности управления им построена математическая модель, опирающаяся на кинетические коэффициенты и зависимости, полученные в экспериментальных исследованиях.
Математическая модель состоит из двух компартментов - моделей «фитотрона» (с пшеницей и с редисом) и модели «микотрона» (для грибов). В модель включены следующие
компоненты: пшеница, редис, солома (перерабатываемая мицелием), мертвое органическое вещество в фитотроне, съедобные грибы (плодовые тела и мицелий), черви, продукты жизнедеятельности червей (копролиты), вермикомпост, использующийся как почвоподобный субстрат, бактериальная микрофлора, минеральные формы биогенных элементов (азот, фосфор, железо), продукция системы для человека (зерно пшеницы, корнеплоды редиса, плодовые тела грибов), кислород и углекислый газ. При постоянном газообмене массообмен между компартментами происходит только во время снятия урожая. Учитывается конвейерный характер функционирования замкнутой экосистемы - число возрастов культуры может регулироваться (в эксперименте - четыре и восемь возрастов). Также поддается регулированию длина конвейерного цикла. По пище и газообмену модуль рассчитан на условное присутствие 1/30 доли человека.
Модель позволяет оценить значения всех учитываемых динамических компонентов системы приразличныхусловиях и режимах ее функционирования, в частности, при труднореализуемых в эксперименте условиях. Формализован коэффициент замкнутости биотического круговорота вещества для систем на запасах вещества.
Ключевые слова: математическое моделирование, биологическая система жизнеобеспечения, коэффициент замкнутости
