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Dimensional analysis yields the general second-order differential equation underlying many natural phenomena: the mathematical properties of a phenomenon's data plot then specify a unique differential equation for it.

Kepner GR - Theor Biol Med Model (2014)

Bottom Line: This yields a differential equation that describes the relationship among the physical variables governing the phenomenon's behavior.Complex phenomena such as the Standard Normal Distribution, the Logistic Growth Function, and Hill Ligand binding, which are characterized by data plots of distinctly different sigmoidal character, are readily analyzed by this approach.It provides an alternative, simple, unifying basis for analyzing each of these varied phenomena from a common perspective that ties them together and offers new insights into the appropriate empirical constants for describing each phenomenon.

View Article: PubMed Central - HTML - PubMed

Affiliation: Membrane Studies Project, PO Box 14180, Minneapolis, MN 55414, USA. kepnermsp@yahoo.com.

ABSTRACT

Background: This study uses dimensional analysis to derive the general second-order differential equation that underlies numerous physical and natural phenomena described by common mathematical functions. It eschews assumptions about empirical constants and mechanisms. It relies only on the data plot's mathematical properties to provide the conditions and constraints needed to specify a second-order differential equation that is free of empirical constants for each phenomenon.

Results: A practical example of each function is analyzed using the general form of the underlying differential equation and the observable unique mathematical properties of each data plot, including boundary conditions. This yields a differential equation that describes the relationship among the physical variables governing the phenomenon's behavior. Complex phenomena such as the Standard Normal Distribution, the Logistic Growth Function, and Hill Ligand binding, which are characterized by data plots of distinctly different sigmoidal character, are readily analyzed by this approach.

Conclusions: It provides an alternative, simple, unifying basis for analyzing each of these varied phenomena from a common perspective that ties them together and offers new insights into the appropriate empirical constants for describing each phenomenon.

Show MeSH
Logistic growth. The red dashed line gives the coordinates slope (P / t). The black dashed line gives the slope at this point (dP / dt).
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Figure 8: Logistic growth. The red dashed line gives the coordinates slope (P / t). The black dashed line gives the slope at this point (dP / dt).

Mentions: Various equations have been developed to model general biological growth as well as population growth [9,10]. Typically, a first-order D.E. is postulated using the growth velocity, dP / dt. The approach developed here is the first to take up the idea presented by Ginzburg [11] that the second derivative of the population with respect to time (d2P / dt2), the growth acceleration, might be a useful variable for describing population growth. This common phenomenon (FigureĀ 8) exhibits sigmoidal behavior differing mathematically from the sigmoidal Hill equation in having a finite value at zero time, the initial population P0.


Dimensional analysis yields the general second-order differential equation underlying many natural phenomena: the mathematical properties of a phenomenon's data plot then specify a unique differential equation for it.

Kepner GR - Theor Biol Med Model (2014)

Logistic growth. The red dashed line gives the coordinates slope (P / t). The black dashed line gives the slope at this point (dP / dt).
© Copyright Policy - open-access
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4530561&req=5

Figure 8: Logistic growth. The red dashed line gives the coordinates slope (P / t). The black dashed line gives the slope at this point (dP / dt).
Mentions: Various equations have been developed to model general biological growth as well as population growth [9,10]. Typically, a first-order D.E. is postulated using the growth velocity, dP / dt. The approach developed here is the first to take up the idea presented by Ginzburg [11] that the second derivative of the population with respect to time (d2P / dt2), the growth acceleration, might be a useful variable for describing population growth. This common phenomenon (FigureĀ 8) exhibits sigmoidal behavior differing mathematically from the sigmoidal Hill equation in having a finite value at zero time, the initial population P0.

Bottom Line: This yields a differential equation that describes the relationship among the physical variables governing the phenomenon's behavior.Complex phenomena such as the Standard Normal Distribution, the Logistic Growth Function, and Hill Ligand binding, which are characterized by data plots of distinctly different sigmoidal character, are readily analyzed by this approach.It provides an alternative, simple, unifying basis for analyzing each of these varied phenomena from a common perspective that ties them together and offers new insights into the appropriate empirical constants for describing each phenomenon.

View Article: PubMed Central - HTML - PubMed

Affiliation: Membrane Studies Project, PO Box 14180, Minneapolis, MN 55414, USA. kepnermsp@yahoo.com.

ABSTRACT

Background: This study uses dimensional analysis to derive the general second-order differential equation that underlies numerous physical and natural phenomena described by common mathematical functions. It eschews assumptions about empirical constants and mechanisms. It relies only on the data plot's mathematical properties to provide the conditions and constraints needed to specify a second-order differential equation that is free of empirical constants for each phenomenon.

Results: A practical example of each function is analyzed using the general form of the underlying differential equation and the observable unique mathematical properties of each data plot, including boundary conditions. This yields a differential equation that describes the relationship among the physical variables governing the phenomenon's behavior. Complex phenomena such as the Standard Normal Distribution, the Logistic Growth Function, and Hill Ligand binding, which are characterized by data plots of distinctly different sigmoidal character, are readily analyzed by this approach.

Conclusions: It provides an alternative, simple, unifying basis for analyzing each of these varied phenomena from a common perspective that ties them together and offers new insights into the appropriate empirical constants for describing each phenomenon.

Show MeSH