Complex biological system, mathematical

The power-law formalism is a mathematical language or representation with a structure consisting of ordinary nonlinear differential equations whose elements are products of power-law functions. The power-law formalism meets two of the most important criteria for judging the appropriateness of a kinetic representation for complex biological systems the degree to which the formalism is systematically structured, which is related to the issue of mathematical tractability, and the degree to which actual systems in nature conform to the formalism, which is related to the issue of accuracy. 

Some of the necessary mathematical concepts and tools can be adopted from other fields and applied to biological systems. Others must be fashioned specifically to deal with novel aspects of biological complexity. The development of a general formalism for the characterization and analysis of organizationally complex biological systems must begin with an appropriate mathematical description for their component parts and associative processes. We shall return to these issues below. 

Engineers are well suited to combine theoretical prediction with experimental measurement. Previous investigators (2, 3, 4, 5, 6, 7) have had considerable success by first mathematically simulating complex biological systems and then conducting well-planned, experimental programs to combine theory with experimental results. 

The behavior of complex biological systems Is determined by the nature of the underlying mechanisms and their abundant Interrelationships, as expressed mathematically In Eqn. (1). 

Our Interest In the Integrated behavior of complex biological systems has been pursued on two levels. On the first, we have concentrated on the development of mathematical methods specifically appropriate for complex biological systems. On the second, we have applied these methods to a number of cases representing several classes of systems. 

Spatio and Spatio-Temporal Patterns. An exotic form of diffusional encounter should be mentioned, which arises from sets of reaction-diffusions equations (48). In 1952, the mathematician Alan Turing postulated the existence of two-dimensional and three-dimensional spatio and spatio-temporal patterns for certain classes of reactive systems (49). The physical realization of these mathematical solutions has been observed in a variety of systems (50). It suffices to say that since these patterns have been observed in both simple chemical systems and complex biological systems, their possibility in homogeneous catalysis should certainly not be ruled out. In this regard, static spectroscopic cells may be particularly prone to such spatial variation because of the lack of mixing. 

One of the major uses of molecular simulation is to provide useful theoretical interpretation of experimental data. Before the advent of simulation this had to be done by directly comparing experiment with analytical (mathematical) models. The analytical approach has the advantage of simplicity, in that the models are derived from first principles with only a few, if any, adjustable parameters. However, the chemical complexity of biological systems often precludes the direct application of meaningful analytical models or leads to the situation where more than one model can be invoked to explain the same experimental data. 

The structure and mathematical expressions used in PBPK models significantly simplify the true complexities of biological systems. If the uptake and disposition of the chemical substance(s) is adequately described, however, this simplification is desirable because data are often unavailable for many biological processes. A simplified scheme reduces the magnitude of cumulative uncertainty. The adequacy of the model is, therefore, of great importance, and model validation is essential to the use of PBPK models in risk assessment. 

The structure and mathematical expressions used in PBPK models significantly simplify the true complexities of biological systems. If the uptake and disposition of the chemical substance(s) is... 

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