Logical and Mathematical Models of Growth and Development of Microbial Population Processes

LOGICAL AND MATHEMATICAL MODELS OF GROWTH AND DEVELOPMENT OF MICROBIAL POPULATION PROCESSES

v.

G. Popov*, G. A. Ugodchikov* and I. N. Blokhina**

'Main Board for Microbiological Industry of the USSR Council of Minist ers, 18 L esteva St , Moscow M -162, USSR "Research Institut e of Epid emiology and Microbiology, 44 Gruzinskaya St ., Gorky , USSR

Abstract. The conception of the system approach conformably to setting and the solution of problems for studying grovnh and development microbial pop ulation processes, their mathematical descriptj_on has been developed. The Iaicrobial populetion was considered as a complex system. All r e sea.rch stages of growth and development r.1icrobial p8pulation processes have been systematized and schematized by means of the developed approach on cellular, populational and biocenotical levels. The logical model (it was constructed on the basis of researcher's hypothesis) the s;y s tem scheme of the process the mathematical model the experimental control of the model the control of adequo.cy model the use of the model for controllir~ process. It has been shovm that both original and available mathematical models of growth and development of microbial population processes on cellular, populational and biocenotical levels can be constructed on the basis of the system approach. Automatized algorithm of constructing ma thematical models for growth and development of microbial population processes has been formulated. Kea1'lords. Mathematical model, logical model, system, modelling, mo eIs, control, system scheme. INTRODUCTION

and control of the microbiological sJrnthesis processes from a single point of view is quite urgent.

At present there exists a great number of different approaches for the construction of rnathel.'latical models of growth and development of microbial population processes. The majority of the existing models consider the influence of one or as maximum two exogenous substrata but in reality the dynamics of microbiological synthesis processes can be influenced by quite a greater number of exogenous variable. Besides, the available structural models considering the dynamics of intracellular processes describe single parts of biological inertia TIhereas in reali ty there can be an arbitrary number of them.

The present paper includes: - the special theory of systems, worked out on the basis of the system approach; - the examples of its usage for the mathematical modelling of the process of grovnh and development of microbial populations on cellular,populational and biocenotical levels. By the system we mean: - on the cellular level - averaging according to ensemble microbial cell; with exogenous substrata input to the system and the purpose products of the microbiological synthesis taken out from the output of the system (Figo1a);

For the proper description of microbial population behaviour, for the better study of the dynamics of their growth and development,taking into consideration the great variety of the connections existing, the elaboration of the new scientific approach providing setting,analysing and solving the problems of cognition 185

V. G. I'opov , C. A. Ugodchikov and I. N. Blokhina

186

- on the )opulational level - the totality of microbial cells averaGed according to the homogenious parts of the population of miCYOorg2.Ilisms of a single species with the initial exogenous s~bstrata input, e~d the final products of the substance transformation taken out from the output. The researcher is interested in dynaInics of the accumulation of the products in question (Fie. 1b); - on biocenotical level - the tot a lity of illcrobial cells averaged acccrdinc to the populations of the microorgan.isr.1s of the different species with the exogenous substrata going into the inputs and pur pos e products taken from the output (Figo 1c).

~)

Cellular level

b) Populational level; heterogeneity within the range of one species of microorganisms

c) Biocenotical level; heterogeneity within the ranee of different species of microorganisms. Fig. 1. Levels of biological system constructions-viz. nricrobial pOlmlations: a) cellular, b) populational, c) biocenotical. 1he system in every specific case is supposed to be composed of the set of subsystems with two inputs and one output (Fig. 2a). The flow of substance of which the product is being synthesized in output of the given subsys tem goes to the structural input and the control signal from the subsystem regulating the synthesis of the product on the output of system in question goes to the control input. Only those subsystems are being analysed whose constants of time of products accwnulation in the outputs are of the same order with the constants of time of purpose product accumulation. These products accumulated in the output of these sybsystems determine such

property as biological inertia and they are called key products.

a)

~

b)

rl

>

I

>

I

I

x

t

~y

I

y

> t

1-1

Fig. 2 '£he system and subsyster.1s a) The sUbsystem: 1 - the structural input, 2 - the control input , 3 - the out put; b) the structure - functional scheme of the s yst em: - enzyme concentrations, key products of 2-nd and 3-d levels and substrate as well. 1he following levels of subsystems hierarchy is introduced: 1. the sybsystems in the oU-~Imt of which the enzymes 2.re synthesized; 2. the subsystems to the control input of which signals go from the outputs of the first level subsystems; 3. the subsystems to the control inputs of which sienals go from the outputs of the 2-nd or 3-d level sUbsystems.

The concentrations of the key products in the outputs of the a bove mentioned subsystems are denoted by 4 ,'7 , x respectively. The set of the exogenous substrata is denoted by ~ (Fig. 2b). The notion of the structurefunctional scheme of the system is introduced. This notion means the set of subsystems connected by inputs and outputs and forming a system in the whole. The process of choosing the structure-functional scheme of the system is fonnalized by means of construction of the logical model of the system. The logical model is constructed on the basis of the rerearche's hypothesis what substrata and intermediate combinations the purpose products

Logical and

~lathematical

,~, re synthesized from and also wrJ.l3.t phisiologicaly active molecules regulate the processes of transformation of above mentioned combinations into the purpose products.

~lod e ls

1&7

stance of '.'::lich the "roduct is ~rJ-ntllesized

'·,·,·:"tb. the l "c.-:;e of for-

!:iing c0r.trolled ty the ,si ver: enzv:;:e; W ' and VI i - cere the rate of fot}:ine controlled '~;)' the Liven enzy::w 2..>'ld t:1C i-th prod.uct on the outllUt of the sulJsyste::l, respectively"; r\. .~ :

is ~~~(:

of proportiohality. :1:ere ['.;.nd below for the function Fi (fly} W ) the follo'::ing Yi expression is adopted coeff~cient

FigoJ. 'I'he eXa.r:Lple of a relatively isolated system. The logical model of the system in every particular problem is constructed onlv after the vectors of variables ~n the input and on the output of the system are calculated; the lo£;ical model in ever
Fifl'{ ( .--Wy 1

where 11 y

.

1

=

1'\.4> 1

Fi (fly 1 -

INy 1. )

is the flow of the sub-

\

, f J\y,

lJ,

If

/> Wf,

nY, <; 'd

Yi

enz;)~e.

lilir~ilctun

pr:Lllciple

c) subsystems nre noninterc!tQ.11ce 2.1:;le - minimum prj,llcj.~le

Rule 2. Any substrate is spent both to synthesize products on the outDuts of the subsystem of the 2-nd and 3-d levels and to provide the activity of these combinations, which are formed of substrate in question.

v~ 1

1

Il'{ , \..j Y "

The size of the flow is directly proportional to the rate of S;)~­ thesis of the precursor w!.lcn the product in question is formed not of the exo£;enous substrate 1mt of the key .9roduct; othervlise, the size of the flow is eoual to the rate of ~oing t~e exogenous substrate from tLe culture li~uid to the scope of the given

Rule 1. If i-th key product on the output of the subsystem of the 2-nd or 3-d hierarchy level is generated in two or more ways, then in the case of their non-interchangeability, the rate of forming the key product is equal to the minimum of the rates of forming them in two (or more) ways; if these ways are interchangeable, then the rate of forming i-th key product will be eoual to the maximum of the rates of forming them in two (or more) ways (Fig. 4).

Rule 3. The rate of synthesis of any enz~~e is controlled in accordance with the follovnng autoregulation: a) the synthesis of the enzyme is accelerated with the increase of the flow of substance of the 2-nd or 3-d level subsystem, with the rate controlled by given enzymes; b) synthesis of the enzyme is inhibited when the rate of forming the product increases (on the output of the 2-nd level subsystem) which is controlled by the given enzyme. Thus, the rate of synthesis of the i-th enzyme on the output of the 1-st level subsystem is:

)

d) subsystems are interchangeablemaximum principle Fig.4 Subsystem chains. a) unramified; b) divergent; c) [illd d) convergent. Rule 4.

ihe rute of synthesis

of CUe;)' ~; :rodLct en the outputs

of the subsystem of the 2-nd or the 3-d level is limited by the concentration of that key product \yhich controls tD.is rate, i.e. expressions for t he rate of synthesis of the i-th product on the output of the subsystem of 2-nd and 3-rd level are:

v. c. Popov, C. A. Ugodchikov and I. N. Blokhina

188

- ':ii th the control by the key product forr:'icd on the output of the subsysten of the 2-nd level; - \'/i th tr.e control by the key product formed on the output of the s ubsystem of the 3-d level, where A 4>, ' Ay" A x j - the values of the activity of the enz~neD of the key products of the 2-nd and 3-d levels, respectively, i and j - the indices of the numeration of the key products. Rule 5. The rate of the product decrease (due to its further i)articiuation in the transformation of substances in microbial cells) on two subsystem output of an arbitrary level is directly proportional to the rate of the synthesis of the product on the output of that Dubsystem on the construction of which the given product is spent. Rule 6. The values of activities A~i,AYi ,Axj and specific rates of the transport (the rates of transport per unit of population biomass) of exogenous substrata into the cells are the functions of substance concentration and also of exogenous substrate one which are fOrIi',ed on the output of the 1-st, 2-nd and 3-d level subsystemso Then VIe analyse the exruaples of the construction of mathematical nodels of growth and development processes of the microbial population on cellular, populational end biocenotic levels: 1. Cellular Level a) t:onosubstrate destructurized model. Suppose the system has one input (the growth on the limiting exogenous ,mbstrate the concentration of which is Si) and one output. The purpose product is the biomass of the microbial population,it is formed directly from the exogenous substrate; the synthesis of the purpose product is controlled by the enzyme, its content per emit of biomass is consj.dered to be constant. Biomass in the given case is the key product of the 2-nd level. The structure-functional scheme of the system is shovm in Fig.5ao The mathematical model is as follows cfY = Ay

I

( ~, ) Y Y

Y

-RfBlf.

FiC.5 ;:;;tructure - function a l scheme of systems. a) Accordine; to J.Lonod's model; b) according to model II; c) according to f:lodel I l l . Depending on the tyrye of nonlinearities of Ay(S) a different destructurized mOnosubstrute mathematical r.lOdels are obtained and with J\..:{,S,) = Ao s :1>\, the system of equations becon1es ~onod's model (I). b) Stl'ucturized model, takine intoconsideration tEe RNA influence on the dynamics of microbial cell biomass accUJilulation. ~uppose the system has one input and one output; the purpose product is the biomasB of the microbi~.l population; the synthesis of the purpose product is controlled by RNA, its accumulation dynamics is regulated by means of the enz;yme 4 t controlling the slowest stage of the ENA synthesis. In this case the biomass is the product on the output of the 3-d level subsystem and its concentration is Kt and RKA is the product on the output of the 2-nd level subsystem and its concentration is Y, The biomass and RNA are formed directly from the exogenous substrate the concentration of which is ~ I Structure-functional scheme of the system is shown in Fig. 5b. 'i 'he r.lathematical model is as follo'Ns: dX;_- A y (~)Y "", ! cH elY, cIt

d
dS' = _E.A (::,)Y · E.Y

dt

c)

4tt5P

dt

Jt {

a)

,

~~. ,,- t,A y (;o,)Y,-E. y A.p(S,)1>.

Logical and Mathematical Model s ~he pre~lent lIlathematical model perlllits to describe preeXIJOnential phases of the biomass and Rl'IA accun!Ulation dynamics (2). Presume that the content of the enzyme per unit of biomass is constant (i.e. ; ' = f\. ) then, the structure-function~l scheme will be as depicted in Fig. 5c and the mathematical model will be as follows:

dX'

eft

III

=

Ay (~,)Y,

dY , = Act> l ~,)X., n<

{ cH

d~..

Jt "

...,

- ~)(

A

'

y ~ ::',

)Y

<-

E.

1\ ~ '\ ~ '£"-.p ,,-Sdl'L
In the cane when Ay(S,) =A y _ %_'- , A 5, K:;, A et> ( "-) ,J," 'Po S., 1I.s< the matheeatical model III is one of the variants of the model D.Ramkt'islma (J). The results of the comparison of theoretical calculations and experimental date. of the model 11 are given in Fie. 6. 0

y

::,

QO

qo

800

10

10

400

~o

50

;0

'0

20

'0

::','

.....

200

i

2 ,

~ 5 ~

t,h

' 2 ! > " '" b

t,h

l ,

189

rease of the antioxidant activity of cells E.coli 111-17 and accumulation of lipid peroxy oxidation products in culture liquid.

11. Fopulational Level. I.ionosubstrate structurized model of the biomass accumulation dynamics conSidering the heterogeneity of bacteria according to the rate of their growth.Suppose the system has one input on which the limiting substrate goes,its concentration is ~. and two outputs; the purpose products are the biomass of two groups of cells homogeneous according to the rate of their growth. Presume that the biomass accumulation process of each group of cells is controlled by the concentration of ribosomes,synthesis kinetics of which is determined by the corresponding enzyme.Then the purpose products are the products of the 3-d level and their concentrations are x, and 'X1respectively. The ribosome concentrations of the 1-st and the 2-nd group of cells are Y, and Y, respectively; in analogy the enzyme concentration are 4, and 4, The structure-functional scheme of the system is depicted in Fig.7.

"'. 5 b t,h

Fig.6 Relative results of calculations by model 11 and gxperimental data (e.g • .l!.. coil M-I7). The a pplication of the special theory of systems for construction of the mathematical models of e;rowth of microe population on the cellular level permits to describe the following effects: I) passing from the consumption of the exogenous limiting substrate to the utilization of its intracellular pool under conditions of the cultive.tion in the glucose-salt medium '/iith the limitation of the growth of bacteria by phosphorus or macneSil.L'!1 (e.g. E.coil M-I7) (4). 2) the extreme character of behaviour of e.verage size bacterium cells in Glucose-salt medium Vii th the limi ttinc level for 8~-;1.monium nitrogen (5). 3) the relation between the dec-

Fig.7 Structure - functional scheme of system,implying heterogeneity of bacteria according to growth rate. The mathematical model enables to describe both the accumulation dynamics of bacterium biomasses under the batch cultivation and the dependance of the parameters of steady state conditions on specific rate of dissolution under continuous cultivation in chemostat conditions. The heterogeneity enables to explain the experimentally obtained fact that the bacterium growth under the continuous cultivation is 1,5 faster than it is under the batch cultivation. The use of the special theory of systems allo':ls to obtain struc-

v. c. Popov, C. A. Ugodchikov and I. N. Blokhina

190

turized variants of mathematical models, considering the following effects a) the dying off a~d lysis processes influence on the kinetics of the bacterium biomass accumulations; b) the influence of the peroxide oxidation of lipids on the bacterial population growth. I l l . 3iocenotical Level.

structurized mathematical model describing "the oredator - Drey" interaction of two snecies of microorganisms.suppose the system has one input on which the limiting substrate with the concentration S, 80es and one output from which dead" predator" with the concentration y~ are taken out. Assume also that alive "predator" Y, are tr8-Ylsformed into dead one at a rate deterrruned by the enzyme LP, and the accu.r:lUlation process of the alive" predator" biomass is controlled by enzyme
!:~onosubstrate

tee

The structure - functional scheme of the system is presented in Fig. 8a. Assume that 4>, 4>< 4>" c: y, = c" Y< = C:<-'Y~ = 3> where c " c< , c." are constants,

then the structure - functional scheme (l<'ig .8b) and the mathematical model are obtained. From the latter nith certain assumDtion for nonlinearity the models o~ V.Volterra (7), of A.~.Kolmogorov (8), and of other researchers are obtained. a')

h)

Fig.8

~tructure sche ~ es of

f~Ylctio~al

the systec of "predator - ,') rey" interaction. a) Enz~n e content is chansable b) ~nzyme content is stable. Some structurized and destructurized mathel;]atice.l models considerine the below listed effects can be constructed by ceans of the special theory of systems:

1. the corQctition of two species for liniting substrate; inter"the predator <-. -Dre"n .; action; host" inter3. lithe parasite action. ')

neaIlS of special theor;'l of s"j's'(; cn al1 oricinal Y.loclel of the p;1a.:.;ely sin process, considering the "re::. influence on the d~"J.1a­ nics bior:lass accm:mlation of cells se:mtive &'1d resintant to pbac e, can be conntructed (9).

DJ-

COlTCLU:JIOL 'l'bus on t:lO oaSlS of tl1e above nentionea. facts the followinG conclusion can ·oe drown: The nynten approach conception on se -ctinc c.nd solvin:::; the problems of studyinG nather.latical description of L,"rov/th and developnent microbial population procesnes is elaborated. It is based on the presentation of the microbie.l population as a complex
1. "onod, J. (1949).Tll e Cro\-lth of bac t erial cultuTes. An..'1. ~ ev. of I.~icrobiol., 111, 371-374. 2. uGodc:5il:ov, G. A. (I977). Tbe use of systen approach for developing mathenatical nodels of CrO\'lth and developnent microbial population processes. In :Uynanics of Biolop;ical G;/stems,v.1, GOl,ki.pp.5G-9I. 3. ~;'ru:l1:rishna, 1)., .'l..C.:r?redricson, r:.Tsuchi;;ra. (1%7). ?~1e d~moL1ics of microvial Growtb.Distributed uns'crnctured r:loclel. J.FerL1.'i'ecllnol. ,-ii, 203-209.

4.

Zheleztnova, = .~.,

G.A.Ug odcbikov

(19 00). ?he mathematical

nodel developraent a.nd tbe s t ud~l of bacteria 'uiomass Gro\"lth d;;marnic s talcinc into accoUl1t the effect of me'i;aoolisD interchanc;eable lmi ts. J .i.:icrobiol, 1,64-60 .

Lo g i ca l and Nathematical Nodels

5.Ugodchikov, G.A., J . ~ .I:no. j urov (1977). The nathenatical model of n icrob e bioness [7owth dynrunics tal:in,3 into account cell division process. In Dynamics of !Holoraica1 ~:;;Y3tC. H 8. Gor1-:: i. pp.91-95. 6. Illol~hina~ LIT., G.A. U,jouch il:ov (19 8 0). 7~)e G.~rll0.D.ics o i' dcvelopin:::; 0. !:licrobial popul at ion . In ;::i.cl'obiolo[;i co.l 3:rnth csi8 , Bio te clmolo :;;/ ccn d 13; oe nr,ine e rin r; . hilio.. n. G. Vol terra, V. (1976 ). The jjo.thematical Theor' of Stru ' :'le 0C lenc e, ,1. 8 . Kolmogorov, A.H. (1972). The qualitative :Jtudy of no.thematical modelc of population dynamics. In Protlems Of Cbbernetics. v.25, pp.1011 U.

9. UGodchikov, G.A.(I976). The mathematico.l model of e rm'lt]} dynani cB of phQ :~ e bi omas s talcine i nto QCC 01,mt a second bacterium :::;ro·:lth . In ~(he L1athemo.t ical l 'l1e or;y o'T13iolop; ical Procenses. These 8 of the r epor t 8 of I -8 t Confere nce. l:o.11nin:::;1'o.u. pp.34 6- 352 .

DISCUS SI ON

Paper 4.1 M. Re u ss: The o perat ion for estima tin g th e sizes o f mi crobial agg l omera t es (Kolmogo roff - Sca le) sugges t an influ ence of vi scos ity accord ing t o wh i ch R_v 3/4 . Th i s wo uld res ult in a tr emendous increase in agg l ome rate s i ze d u rin g the process be cause of inc r ease of vi scos it y.

V.v. Biryukov : You are r i gh t . But viscosity is not a con tr o ll ed vari ab l e . I n o ur alg o rithm we have hidden thi s parame t er in th e coeff i c i en t B (equation 27). This coe f ficient is t o be adapted during t he f e rm enta tion using experimental data. We didn 't intere s tin g in th e ex act v a lue of the s i ze of microbia l agglomerates. C. Cooney : !,Tould you connnent on the prob lem in u s ing respirati on rat e for con tr ol of l arge ferment o r s?

V.V. Biryukov: The r es p iration rate i s not directly con tr olled . It is used i n different co ntr o l algorithms as a parame ter , which g ives primary inf ormation fo r ca lcul ations .

ItCBP --<;*

19 1

Paper 4. 2 C. Coone'l : h'hat is lhe rC'laliunsh i p bet,,''''l'n the pe ri oJ for adapt i ve cc)ntt'l,l :lIld th e cell gro h'th rate? R. Lut t man: The l e ngtil of t h" Cldapt i c)n i ntl'rvals shou l d b" chl.ll)sl'n ClCl.'l.)Hi iIlg to the ce l l gro h'th r a l l' . ,\n a na l yt i c relati on , hOh'ever , i s not ye t kIlo"n . A. Halme : In the cost fun c ti on yo u included a term h'hi c h J esc ribes th e pt'otein o r RSA cOIl t l'nt . ThClt "as done in a con tinu ous way . l n p ract i cCl I pro duct i on , I th i nk , you cCln 'l do that i n such a ",ay , but prL'f erClb l 'l put some mini mum or mClx i mum li mi ts. Can yo u comment" R. Luttman: In th e c ost fU Il-,t i on l)nly the int e rmedi a t e pr OdUl'l , t he system vari ah l e hi omass takes pla ce . The des ir ed prod uct i s th e mass of pr o t e in. The pr o tein co nt ent of p r o duced bacte ri a is strnngly grO\, th rate d e p en d en t. So we tClk e int o acco un t a "qu ality" de p enden t hi o mass pri ce , that means th pr o fit of produ ced pr ote i n . On th e o th er hilnd i n an industri a l vie\" se ll i ng SCP f or human f ood , th e restr i c ti on of ce r ta in toxi c compoun ds , e . g . RNA , i s a probl em of r ecove r y . M. lnstalle : a) HO\, did yo u c huose th e 3 hours int e r va l of i dentification proce dur e? b) Is it not ne cessil ry to add noi se t o th e sys t em in o rd e r t o lmp rove it s i dentifiability? R. Luttman : Th e t hree hour s interva l was c hoosen a s it t u rned out t o be suff i c i en t for th e a ct u al pro cess . It co uld be shortened dO\,n t o a pprox. one ho ur, s in ce th at i s th e act u a l ti me needed for computati on of a com pl e t e OLFO-cyc l e . But, o f course, fa s ter l oo ps must be ru n se parate l y ,,, i th f aste r cyc l e ti mes . b ) Add iti on of noise was not necessa ry in th e pr ese nt case . Thi s i s du e t o t he fact that we don ' t have a cont inu o \l s fermentation but a batc h pr ocess . Therefo r e th e dynam i cs of th e pr ocess i s fully con tain ed i n t he meas ur ements . P . Hagand er : Di d yo u have any p roblems with initi a l conditio ns of the PDE when do ing the on - line es ti mat i on? R. Luttman: In th e pres en t case we d idn ' t obse rv e any pr oblems wit h initial cond iti ons . The initial cond ition s are tak en as saturated for t=O and as the id en tifi ed s tate s of th e pr oceed i ng i nt e rv a l s i n the ada ption int e rv als with t >O . However , if problems sho ul d occ ur, t here a r e tw o poss i bil it ies to ove r come