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Computer Modelling of Large Systems in Their Biological Context
COMPUTER MODELLING OF LARGE SYSTEMS IN THEIR BIOLOGICAL CONTEXT P. Jirku Center of Biomathematics, Czechoslovak Academy of Sdences, 142 20 Prague, Czechoslovakia
Abstract. Biological m~tivations for the study of discrete cellular systems are being described. Systems in cellular level are to be understood as consisting from individuals the behaviour of them is described by a certain class of automata. From knowledge of local behaviour of elements of system we try to infer on global behaviour of the whole system. The notion of cellular system with time-variant neighbourhood function (interconnection structure) is being defined and a connection between structural, developmental and behavioural similarity of such systems is being studied. In the last part of the present paper we mention some biological applications namely an application to computer modelling of myocardial damage which has been built in collaboration with physiologists. Keywords . Automata theory;computer applications;discrete time systems;modelling;physiological models.
INTRODUCTION The notion of a complex system consisting of a large number of active interconnected individuals has become in recent years one of the fundamental notions in both computer science and computer oriented biology. The main area of biological sciences where this notion is widely applied is a study of living systems on a cellular level. But from a computer science point of view the term cellular refers rather to the elements of a system and an analogy with living cells is not necessarily supposed. A study of such large systems is concerned with their dynamics, i. e. with relation of local information about behaviour (coded in elements of a system). and its global behaviour. Thus we can consider global behaviour as a propagation of information throughout cellular space. The aim of this contribution is to demonstrate how very complex systems can be constructed from relatively simple elements and how the behaviour of such systems can be checked and controlled. BIOLOGICAL MOTIVATIONS Biological systems are formed by liv307
ing cells into colonies or tissues according to the quality of functional and infomational connections inbetween individual cells. These systems may then produce new individual cells or systems of cells.Our aim is to describe somegeneral rules governing the dynamics of such discrete systems, i.e. to characterize namely such notions as growth, differentiation, division, and death of individual cells, development and reproduction of configurations etc. It is essential from the point of view of informational exchange inbetween cells that a atate of each cell is determined (but not necessarily fully) by states of cells occurring in the neighbourhood of the given cell. It is known at every timepoint with which cells the given cell communicates. But this environment can often change (e.g. migration of cells in embryonic organism). Differentiation is to be understood as specialization of cells for various functions. A system of differentiated cells can be adequately characterized by a description of the behaviour of differentiated cells in the system using various classes of automata. But the fact that a multicellular organism of differentiated cells results from a single (embryonic) cell can be - in accordance with
P. Jirku
308
the latest development of molecular biology - described in more complicated way: this single cell is, as an automaton, equipped with a programme (coded in DNK). At every timepoint only some of its parts may be activated (in the analogous way as operator controls activity of structural genes in operon). These are the basic ideas that will be summed up in the concept of cellular organism. CELLULAR SYSTEMS Let us give more details about cellular systems in the form of mathematical definitions. By a cellular system we understand an arbitrary set of interconnected objects (cells) that can appear in certain states and that can change these states and transfer information between them. Behaviour of individual cell is described by local transition function. Time is to be understood as a discrete variable changes that appear in the system are performed in steps. From the point of view of informational transitions inbetween individual cells there is essential the fact that a state of each cell is determined by states of cells which are actually presented in the neighbourhood of the cell in question. Following notions concerning dynamics of systems can be defined for cellular systems with fixed neighbourhoods. (Such systems can be represented as labeled mUltigraphs on the set of cells. By behavioural homomorphism of cellular systems we understand a partial mapping of configurations of one system into configurations of second one persistent with respect to global transitions. By a structural homomorphism of two cellular systems we understand a mapping of cells into cells preserving neighbourhoods (i.e. graph structure) and persistent with respect to global transitions. The following theorem associating interconnection structure of systems with their behaviour has been proved by Jump and Kirtane (1974) for systems with fixed interconnection structure. If two systems (with fixed interconnection structure) are structurally homomorphic, then they are also behaviourally homomorphic. For generalization of that result to time-variant cellular systems (i.e. with variable interconnection structure) we give, at first, some necessary formal definitions.
Time-variant cellular systems. Let M be a family of sets called set of cells, i.e. M = {M t }tET. Let neighbourhood function y: M X T - - P fin (M)
be a par-
tial function such that for every tET the function Yt:Mt~ Pfin(M t ) is a total funtion on Mt. This function will be sometimes called developmental stage of the system. To each cell mEM we assigne an automaton A = (S, so' c), where S is a finite set of states, soES is so called quiescent state, 15: s* - - S is a local transition function such that for every n there ex-
ists
15
n moreover
Sand
Sn _
15
c(so' •.. 'so)
bourhoodstate function defined as follows
=
U 0
sO.
n
A
(in time
and neight)
}mEyt(m)
is
(1)
under the condition that for every mE MtnM t + l (2 )
Here st+l means the state of the cell m in time t+l. A mapping Ct:Mt~ S (such that the set
{m; mEM
and t we will call
(m) ~ So } Is finite) t configuration of the system in question at a given developmental stage. Let Conf be a set of all possible conS'Y t figurations at a given developmental stage t and let Conf = U confs tET ,Y t So then there exists one global transition relation 6 S Conf X Conf such that for every mEM and cEConf
c
6(C) (m) = o(h(m))
(3 )
at every time t (under the condition that the left side of the equation is defined) . The relation 6 can be understood as a set of functions 6. and therefore the ~
system as nondeterministic automaton. For a given configuration coEconf the relation 6 then determines a set of sequences (derivations) d = co,c l ' ... such that 6 (c ) n
(4)
These derivations form a tree with the roote cO. By behavioural homomorphism of two
Computer modelling of large systems cellular systems (with time-variant neighbourhoods) we uderstand a partial mapping 1jJ:Conf - - conf such that 2 l for every configuration cEConf holds: l If cEdom(1jJ) than 6 (c)Edom(1jJ) (if it l exists) and 6
2
(1jJ)
(c) = 1jJ(6 (c». l
( 5)
Two cellular systems we will call developmentally homomorphic if there exists a partial mapping ':M l ~- - M2 preserving neighbourhoods, i.e. for every mEM holds: '(yl(m» = Y2('(m» l and, moreover, there exists a mapping 4>: SI S2 such that
= 0 (4) (s 1) ... 4> (sk) ) .
(6)
proposition. Two cellular systems are deve1opmenta11y homomorphic i f they are structurally homomorphic at every developmental stage. Theorem. Two deve10pmenta11y homomorphic cellular systems are also behaviourally homomorphic. We give only a sketch of the proof because there is no essential difference from that in Jump and Kirtane (1974): It is sufficient to restrict ourselves to those configurations which are compatible with' and 4>, i.e. to configurations such that for every x,yEM the following condition l holds: If '(x) = '(y) then 4> (c (x» = = 4>(c(y». Now, starting from' ?,nd 4> we want to define a mapping 1jJ from conf into conf . Let cEconf • Then l 2 l 1jJ(c): M2 S2 we define as follows: For every mEM Let now mlEM 6
2
(1jJ(c» (m
2
l
1jJ
l
= 4> (c (m»
.
and m ='(m ). Then 2 l = 6
)
(c) (' (m»
2
(1jJ(c» ('(m
l
»·
Applying definitions of 6 2 , " 4>, 6 , 1jJ respectively, we obtain 1 6 (1jJ(c» ('(m » = 1jJ(t. (c»'{m ) l 2 l l 1jJ (6
1
(c) ) (m
2
) •
APPLICATIONS The theory of cellular systems has been applied in various areas by many authors, and many biological models hayebeen built and studied. Time-variant cellular systems as they have been defined here can be recognized as systems which develop and compute simultaneously. Many authors raised a great effort in investigation of special classes of
309
cellular systems in such a way that both main attributes (developing and computation) have been studied separately. A large number of works deals with fixed interconnection structure. As an example we mention here only investigation of self-reproduction and a well-known game Life. On the other hand developmental systems can serve as examples of systems with time-variant interconnection structure. In those systems nontraditional devices from automata theory and formal languages are used. Time variant systems are often understood as rewiting systems with sequential or parallel rewritings. System with simple interconnection structure (parallel rewritings over strings) were originally introduced by Lindenmayer (1968) to provide mathematical and computer models for the development of filamentous organisms. Later these systems were generalized to more complex structures, especially to labelled multigraphs, as e.g. in Ehrig and Kreowski (1976), Nagl (1976) and many other authors. A physiological model The present author, in collaboration with other mathematicians and physiologists, has applied the theory of cellular systems on a study of myocardium damage. A mathematical and computer model of prenecrotic processes in myocardium during experimental cardiomyopathies has been established. A program named Myoc.ard in whi tdl apart of myocardial tissue is represented by network of some thousand cells was written (in FORTRAN IV language). Each of the cells is represented by an automaton, transition function of this automaton is represented by a subroutine of the program which can be easily modified. The program from an initial configuration (artifitial damage of some cells) computes and prints derivate configurations. This computation simulates the origin and growth of a focus of myocardium damage . Every member of such a generative sequence of configurations can be tested individually and results can be compared with experimental findings. Fore more details see Jirku and others (1977). CONCLUSION We have defined the concept of time-variant cellular system and then we have discussed such derived notions as structural and developmental similarity of systems in the context of their behaviour. These notions have been found to be good foundations for mathematical and computer modelling techiques capable of simulating some biological processes on the cellular
P. Jirku
310
level as follows from short list of applications and also from a physiological model performed in our institute.
REFERENCES Arbib, M.A. (1969) . Theories of abstract automata. Prentice-Hall, London. 412 pp. Codd, E.F. (1968). Cellular automata. Academic Press, New York. 112 pp. Ehrig, H. and H.-J. Kreowski (1976). Parallel graph grammars. Automata, languages, development. NorhtHolland. Amsterdam. pp. 425-442. Herman, G.T. and G. Rozenberg (1975). Developmental systems and languages. North-Holland/American Elsevier. Amsterdam. 363 pp. Jump, J.R. and J.S. Kirtane (1974). On the interconnection structure of cellular networks, Inf. & Control 24, 74-91.
Jirku, P., D. Pokorny, J. Pilny and Z. Deyl (1977). Modelling of prenecrotic processes in myocardium during experimental cardiomyo pathies. Proc. of the Int. Symp. Simulation 77. Acta Press. Zurich. pp. 479-484. Nagl, M. (1976). Graph rewriting systems and their application in biology. Lecture notes in biomathematics 11, 135-156. Lindenmayer, A. (1968). Mathematical models for cellular interactions in development. Journal of theoretical biology 18, 280-299. Lindenmayer, A. and G. Rozenberg (Eds.), (1976). Automata, languages, development. North-Holland. Amsterdam. 529 pp.
Abstract. Biological m~tivations for the study of discrete cellular systems are being described. Systems in cellular level are to be understood as consisting from individuals the behaviour of them is described by a certain class of automata. From knowledge of local behaviour of elements of system we try to infer on global behaviour of the whole system. The notion of cellular system with time-variant neighbourhood function (interconnection structure) is being defined and a connection between structural, developmental and behavioural similarity of such systems is being studied. In the last part of the present paper we mention some biological applications namely an application to computer modelling of myocardial damage which has been built in collaboration with physiologists. Keywords . Automata theory;computer applications;discrete time systems;modelling;physiological models.
INTRODUCTION The notion of a complex system consisting of a large number of active interconnected individuals has become in recent years one of the fundamental notions in both computer science and computer oriented biology. The main area of biological sciences where this notion is widely applied is a study of living systems on a cellular level. But from a computer science point of view the term cellular refers rather to the elements of a system and an analogy with living cells is not necessarily supposed. A study of such large systems is concerned with their dynamics, i. e. with relation of local information about behaviour (coded in elements of a system). and its global behaviour. Thus we can consider global behaviour as a propagation of information throughout cellular space. The aim of this contribution is to demonstrate how very complex systems can be constructed from relatively simple elements and how the behaviour of such systems can be checked and controlled. BIOLOGICAL MOTIVATIONS Biological systems are formed by liv307
ing cells into colonies or tissues according to the quality of functional and infomational connections inbetween individual cells. These systems may then produce new individual cells or systems of cells.Our aim is to describe somegeneral rules governing the dynamics of such discrete systems, i.e. to characterize namely such notions as growth, differentiation, division, and death of individual cells, development and reproduction of configurations etc. It is essential from the point of view of informational exchange inbetween cells that a atate of each cell is determined (but not necessarily fully) by states of cells occurring in the neighbourhood of the given cell. It is known at every timepoint with which cells the given cell communicates. But this environment can often change (e.g. migration of cells in embryonic organism). Differentiation is to be understood as specialization of cells for various functions. A system of differentiated cells can be adequately characterized by a description of the behaviour of differentiated cells in the system using various classes of automata. But the fact that a multicellular organism of differentiated cells results from a single (embryonic) cell can be - in accordance with
P. Jirku
308
the latest development of molecular biology - described in more complicated way: this single cell is, as an automaton, equipped with a programme (coded in DNK). At every timepoint only some of its parts may be activated (in the analogous way as operator controls activity of structural genes in operon). These are the basic ideas that will be summed up in the concept of cellular organism. CELLULAR SYSTEMS Let us give more details about cellular systems in the form of mathematical definitions. By a cellular system we understand an arbitrary set of interconnected objects (cells) that can appear in certain states and that can change these states and transfer information between them. Behaviour of individual cell is described by local transition function. Time is to be understood as a discrete variable changes that appear in the system are performed in steps. From the point of view of informational transitions inbetween individual cells there is essential the fact that a state of each cell is determined by states of cells which are actually presented in the neighbourhood of the cell in question. Following notions concerning dynamics of systems can be defined for cellular systems with fixed neighbourhoods. (Such systems can be represented as labeled mUltigraphs on the set of cells. By behavioural homomorphism of cellular systems we understand a partial mapping of configurations of one system into configurations of second one persistent with respect to global transitions. By a structural homomorphism of two cellular systems we understand a mapping of cells into cells preserving neighbourhoods (i.e. graph structure) and persistent with respect to global transitions. The following theorem associating interconnection structure of systems with their behaviour has been proved by Jump and Kirtane (1974) for systems with fixed interconnection structure. If two systems (with fixed interconnection structure) are structurally homomorphic, then they are also behaviourally homomorphic. For generalization of that result to time-variant cellular systems (i.e. with variable interconnection structure) we give, at first, some necessary formal definitions.
Time-variant cellular systems. Let M be a family of sets called set of cells, i.e. M = {M t }tET. Let neighbourhood function y: M X T - - P fin (M)
be a par-
tial function such that for every tET the function Yt:Mt~ Pfin(M t ) is a total funtion on Mt. This function will be sometimes called developmental stage of the system. To each cell mEM we assigne an automaton A = (S, so' c), where S is a finite set of states, soES is so called quiescent state, 15: s* - - S is a local transition function such that for every n there ex-
ists
15
n moreover
Sand
Sn _
15
c(so' •.. 'so)
bourhoodstate function defined as follows
=
U 0
sO.
n
A
(in time
and neight)
}mEyt(m)
is
(1)
under the condition that for every mE MtnM t + l (2 )
Here st+l means the state of the cell m in time t+l. A mapping Ct:Mt~ S (such that the set
{m; mEM
and t we will call
(m) ~ So } Is finite) t configuration of the system in question at a given developmental stage. Let Conf be a set of all possible conS'Y t figurations at a given developmental stage t and let Conf = U confs tET ,Y t So then there exists one global transition relation 6 S Conf X Conf such that for every mEM and cEConf
c
6(C) (m) = o(h(m))
(3 )
at every time t (under the condition that the left side of the equation is defined) . The relation 6 can be understood as a set of functions 6. and therefore the ~
system as nondeterministic automaton. For a given configuration coEconf the relation 6 then determines a set of sequences (derivations) d = co,c l ' ... such that 6 (c ) n
(4)
These derivations form a tree with the roote cO. By behavioural homomorphism of two
Computer modelling of large systems cellular systems (with time-variant neighbourhoods) we uderstand a partial mapping 1jJ:Conf - - conf such that 2 l for every configuration cEConf holds: l If cEdom(1jJ) than 6 (c)Edom(1jJ) (if it l exists) and 6
2
(1jJ)
(c) = 1jJ(6 (c». l
( 5)
Two cellular systems we will call developmentally homomorphic if there exists a partial mapping ':M l ~- - M2 preserving neighbourhoods, i.e. for every mEM holds: '(yl(m» = Y2('(m» l and, moreover, there exists a mapping 4>: SI S2 such that
= 0 (4) (s 1) ... 4> (sk) ) .
(6)
proposition. Two cellular systems are deve1opmenta11y homomorphic i f they are structurally homomorphic at every developmental stage. Theorem. Two deve10pmenta11y homomorphic cellular systems are also behaviourally homomorphic. We give only a sketch of the proof because there is no essential difference from that in Jump and Kirtane (1974): It is sufficient to restrict ourselves to those configurations which are compatible with' and 4>, i.e. to configurations such that for every x,yEM the following condition l holds: If '(x) = '(y) then 4> (c (x» = = 4>(c(y». Now, starting from' ?,nd 4> we want to define a mapping 1jJ from conf into conf . Let cEconf • Then l 2 l 1jJ(c): M2 S2 we define as follows: For every mEM Let now mlEM 6
2
(1jJ(c» (m
2
l
1jJ
l
= 4> (c (m»
.
and m ='(m ). Then 2 l = 6
)
(c) (' (m»
2
(1jJ(c» ('(m
l
»·
Applying definitions of 6 2 , " 4>, 6 , 1jJ respectively, we obtain 1 6 (1jJ(c» ('(m » = 1jJ(t. (c»'{m ) l 2 l l 1jJ (6
1
(c) ) (m
2
) •
APPLICATIONS The theory of cellular systems has been applied in various areas by many authors, and many biological models hayebeen built and studied. Time-variant cellular systems as they have been defined here can be recognized as systems which develop and compute simultaneously. Many authors raised a great effort in investigation of special classes of
309
cellular systems in such a way that both main attributes (developing and computation) have been studied separately. A large number of works deals with fixed interconnection structure. As an example we mention here only investigation of self-reproduction and a well-known game Life. On the other hand developmental systems can serve as examples of systems with time-variant interconnection structure. In those systems nontraditional devices from automata theory and formal languages are used. Time variant systems are often understood as rewiting systems with sequential or parallel rewritings. System with simple interconnection structure (parallel rewritings over strings) were originally introduced by Lindenmayer (1968) to provide mathematical and computer models for the development of filamentous organisms. Later these systems were generalized to more complex structures, especially to labelled multigraphs, as e.g. in Ehrig and Kreowski (1976), Nagl (1976) and many other authors. A physiological model The present author, in collaboration with other mathematicians and physiologists, has applied the theory of cellular systems on a study of myocardium damage. A mathematical and computer model of prenecrotic processes in myocardium during experimental cardiomyopathies has been established. A program named Myoc.ard in whi tdl apart of myocardial tissue is represented by network of some thousand cells was written (in FORTRAN IV language). Each of the cells is represented by an automaton, transition function of this automaton is represented by a subroutine of the program which can be easily modified. The program from an initial configuration (artifitial damage of some cells) computes and prints derivate configurations. This computation simulates the origin and growth of a focus of myocardium damage . Every member of such a generative sequence of configurations can be tested individually and results can be compared with experimental findings. Fore more details see Jirku and others (1977). CONCLUSION We have defined the concept of time-variant cellular system and then we have discussed such derived notions as structural and developmental similarity of systems in the context of their behaviour. These notions have been found to be good foundations for mathematical and computer modelling techiques capable of simulating some biological processes on the cellular
P. Jirku
310
level as follows from short list of applications and also from a physiological model performed in our institute.
REFERENCES Arbib, M.A. (1969) . Theories of abstract automata. Prentice-Hall, London. 412 pp. Codd, E.F. (1968). Cellular automata. Academic Press, New York. 112 pp. Ehrig, H. and H.-J. Kreowski (1976). Parallel graph grammars. Automata, languages, development. NorhtHolland. Amsterdam. pp. 425-442. Herman, G.T. and G. Rozenberg (1975). Developmental systems and languages. North-Holland/American Elsevier. Amsterdam. 363 pp. Jump, J.R. and J.S. Kirtane (1974). On the interconnection structure of cellular networks, Inf. & Control 24, 74-91.
Jirku, P., D. Pokorny, J. Pilny and Z. Deyl (1977). Modelling of prenecrotic processes in myocardium during experimental cardiomyo pathies. Proc. of the Int. Symp. Simulation 77. Acta Press. Zurich. pp. 479-484. Nagl, M. (1976). Graph rewriting systems and their application in biology. Lecture notes in biomathematics 11, 135-156. Lindenmayer, A. (1968). Mathematical models for cellular interactions in development. Journal of theoretical biology 18, 280-299. Lindenmayer, A. and G. Rozenberg (Eds.), (1976). Automata, languages, development. North-Holland. Amsterdam. 529 pp.