Deterministic and Stochastic Models of Biological Systems
55
where k is a positive constant (fixed for a given set of environmental parameters, say), together with an initial condition x (0) = xo > 0, constitutes
a mathematical model of the "kinetics" of the process.
Note that in this procedure we have, however, represented a discrete
quantity, the number of molecules or the number of organisms, by a continuous mathematical quantity. In very large systems no error may result,
though in small systems a significant error could result. Such a simplification does result in making the process of the Calculus directly available
to the study of the kinetics of the system, so that we are in a position to
trace out implications of this representation of the underlying natural
process by using a well-known mathematical methodology. Thus, in this
case, the more direct relationship of x to t is obtained in a formal way by
solving or integrating the differential equation of that model to obtain:
x=xoe- kt
(6)
and, by applying the formal rules of algebra, this equation can be put in a
convenient "linear form":
log x ,= log Xo - kt,
(7)
showing log x as a linear function of t and k as the slope of the straight line.
And so we would expect that, if all we have assumed about the transformation
T is true, then a semilogarithmic plot of the data should approximate a
straight line.
As pointed out in the preceding, however, the representation given by
equations (6) or (7) should be enlarged to accommodate statistical or curvefitting procedures aimed e.g. at estimating from experimental data the socalled rate constant k, usually taken as the index of the strength of such a
transformation. In this case this amounts to replacing (6) e.g. by the more
realistic (stochastic) equation:
x = xoe- kt + e (t)
(8)
where e is in theory a random function of time such as a Gaussian random
process. In dealing with experimental fluctuations such an extension seems
to be quite generally and tacitly subsumed.
Speaking more generally, the utilization of such models thus involves
the compounding of the original model-construction procedure f-l with
some "statistical" procedure Ii:
(9)
which results in imbedding 9]( in a practical statistical framework. In this,
~m = 9J( x \13 refers to the probabilistically, or statistically, enriched form of the
original model ~JL Once this is done, then the methodology of statistics
55
where k is a positive constant (fixed for a given set of environmental parameters, say), together with an initial condition x (0) = xo > 0, constitutes
a mathematical model of the "kinetics" of the process.
Note that in this procedure we have, however, represented a discrete
quantity, the number of molecules or the number of organisms, by a continuous mathematical quantity. In very large systems no error may result,
though in small systems a significant error could result. Such a simplification does result in making the process of the Calculus directly available
to the study of the kinetics of the system, so that we are in a position to
trace out implications of this representation of the underlying natural
process by using a well-known mathematical methodology. Thus, in this
case, the more direct relationship of x to t is obtained in a formal way by
solving or integrating the differential equation of that model to obtain:
x=xoe- kt
(6)
and, by applying the formal rules of algebra, this equation can be put in a
convenient "linear form":
log x ,= log Xo - kt,
(7)
showing log x as a linear function of t and k as the slope of the straight line.
And so we would expect that, if all we have assumed about the transformation
T is true, then a semilogarithmic plot of the data should approximate a
straight line.
As pointed out in the preceding, however, the representation given by
equations (6) or (7) should be enlarged to accommodate statistical or curvefitting procedures aimed e.g. at estimating from experimental data the socalled rate constant k, usually taken as the index of the strength of such a
transformation. In this case this amounts to replacing (6) e.g. by the more
realistic (stochastic) equation:
x = xoe- kt + e (t)
(8)
where e is in theory a random function of time such as a Gaussian random
process. In dealing with experimental fluctuations such an extension seems
to be quite generally and tacitly subsumed.
Speaking more generally, the utilization of such models thus involves
the compounding of the original model-construction procedure f-l with
some "statistical" procedure Ii:
(9)
which results in imbedding 9]( in a practical statistical framework. In this,
~m = 9J( x \13 refers to the probabilistically, or statistically, enriched form of the
original model ~JL Once this is done, then the methodology of statistics