50
A. F. BARTHOLOMAY
e. Time Scales t. In connection with the compositional, structural and
functional aspects of a biosystem, explicit attention has already been called
to the importance of the time factor in defining these components of the
system meaningfully. In all of these cases time is featured as the independent
variable of any mathematical representation, in the following sense: given
any measurable quantity connected with any component or activity of the
system, it amounts to some kind of a function of time. Thus, anticipating the
notion of mathematical model, if a deterministic approach is chosen, such a
quantity becomes an ordinary analytic function of time; in the stochastic approach, it is treated as a random function of time as the independent variable.
And in fact, in all models an ultimate mathematical aim is to deduce the
exact nature of the variation of the variable with time as independent variable.
The importance of time in an elaboration of the environmental component has been accentuated particularly by WADDINGTON [23] and taken up
again somewhat later by GOODWIN [14]. In fact, in their studies, time becomes the basis for classifying the different biological activities (v.i.) in such
a way as to delineate between a system and its environment. Thus, three
different magnitudes of time stretches or scales are differentiated with respect
to which different orders of biological activities or functions are defined.
The "shortest" phenomena are the biochemical and metabolic processes;
so-called "epigenetic" or developmental processes taking place over much
longer time stretches are next in the expanding order of classification; and
finally the "longest" phenomena are the genetic and evolutionary processes.
From this point of view then, given the appropriate time-scales over which
the various activities of the target system transpire, related activities
occurring over adjacent time-scales would lead inversely to the reconstruction of the collection of systems to be considered in the environment.
The involvement of systems in the environment chosen in this way in a
mathematical model representation of the biosystem could in many cases
be limited simply to an inclusion in the set P of parameters associated with
the system. In this sense, the environmental component E may be thought
of as including the parametric sets P.
The preceding formulation in biomathematical terms of the concept of
biosystem e; serves to call attention to specific aspects of the "domain"
over which a mathematical model will be defined. It is therefore an integral
part to the total conception of mathematical model, just as the precise
specification of a probability space must precede the construction of a
probability measure on a given set of random events. In fact this analogy is
worth re-emphasizing for, as the sequel shows, the construction of a mathematical model of a biosystem is similar, conceptually, to imposing a "measure" on a well-specified space or collection of sets. In this case, the underlying space refers to some or all of the biomathematically described components indicated.
A. F. BARTHOLOMAY
e. Time Scales t. In connection with the compositional, structural and
functional aspects of a biosystem, explicit attention has already been called
to the importance of the time factor in defining these components of the
system meaningfully. In all of these cases time is featured as the independent
variable of any mathematical representation, in the following sense: given
any measurable quantity connected with any component or activity of the
system, it amounts to some kind of a function of time. Thus, anticipating the
notion of mathematical model, if a deterministic approach is chosen, such a
quantity becomes an ordinary analytic function of time; in the stochastic approach, it is treated as a random function of time as the independent variable.
And in fact, in all models an ultimate mathematical aim is to deduce the
exact nature of the variation of the variable with time as independent variable.
The importance of time in an elaboration of the environmental component has been accentuated particularly by WADDINGTON [23] and taken up
again somewhat later by GOODWIN [14]. In fact, in their studies, time becomes the basis for classifying the different biological activities (v.i.) in such
a way as to delineate between a system and its environment. Thus, three
different magnitudes of time stretches or scales are differentiated with respect
to which different orders of biological activities or functions are defined.
The "shortest" phenomena are the biochemical and metabolic processes;
so-called "epigenetic" or developmental processes taking place over much
longer time stretches are next in the expanding order of classification; and
finally the "longest" phenomena are the genetic and evolutionary processes.
From this point of view then, given the appropriate time-scales over which
the various activities of the target system transpire, related activities
occurring over adjacent time-scales would lead inversely to the reconstruction of the collection of systems to be considered in the environment.
The involvement of systems in the environment chosen in this way in a
mathematical model representation of the biosystem could in many cases
be limited simply to an inclusion in the set P of parameters associated with
the system. In this sense, the environmental component E may be thought
of as including the parametric sets P.
The preceding formulation in biomathematical terms of the concept of
biosystem e; serves to call attention to specific aspects of the "domain"
over which a mathematical model will be defined. It is therefore an integral
part to the total conception of mathematical model, just as the precise
specification of a probability space must precede the construction of a
probability measure on a given set of random events. In fact this analogy is
worth re-emphasizing for, as the sequel shows, the construction of a mathematical model of a biosystem is similar, conceptually, to imposing a "measure" on a well-specified space or collection of sets. In this case, the underlying space refers to some or all of the biomathematically described components indicated.