Authors
Johannes W. Dietrich
Abstract
Introduction
Homeostasis and allostasis are preconditions of independent life. Therefore, the underlying motifs of feedback control systems are among the essential and most basic biological information processing structures. Traditionally, the relation between structure and behaviour of control loop systems used to be investigated through linear and time-invariant (LTI) models. However, LTI models do not match well the natural properties of living systems. It is, therefore, hardly surprising that this class of models doesn’t contribute too much to our understanding of biological phenomena or to medical decision making. However, the development of more realistic models is faced by obstacles, e.g. by complexities in the mathematical formulation of nonlinear systems and by the fact that elements and signals of biological control loop systems may be materialised in a very heterogeneous manner.
Methods
Proceeding from requirements of endocrine systems we developed an “evidence-based” modelling platform. The main goal was that it should be possible to match the elements of the corresponding models to empirically verified biochemical and physiological data in a bijective manner.
Basic elements of this approach include (1) analog signal memory with intrinsic adjustment (ASIA) elements, which translate production rates to concentrations by coupling secretion to distribution and elimination, (2) saturation kinetics based on the Michaelis-Menten (MiMe) formalism, which map enzymatic mechanisms and signal-transduction processes and (3) non-competitive divisive inhibition (NoCoDI) for the description of negative feedback effects and antagonistic structures.
ASIA elements are able to predict the equifinal concentration of a biological signalling substance as y(inf) = alpha * x(inf) / beta depending on an input signal x(inf), the reciprocal apparent volume of distribution (alpha = 1 / VD) and a clearance exponent beta = ln(2) / half-life. The transitional behaviour of ASIA elements can be described by means of exponential functions.
MiMe kinetics describe the nonlinear dependence of an output signal y(t) on an input signal x(t) with
y(t) = G * x(t) / (D + x(t)), where G represents the maximum response of the subsystem and D the input leading to a half-maximal response (e.g. a dissociation constant). Apart from enzymatic processes MiMe kinetics are also useful for modelling stimulatory receptor-mediated signal transduction processes.
The integration of stimulating (x1(t)) and inhibiting (x2(t)) input signals may be modelled by means of a NoCoDI element with y(t) = G * x1(t) / ((D + x1(t)) * (1 + x2(t) / KI)), where KI represents a second dissociation constant (of the inhibiting pathway).
In addition to mathematical formulations, this modelling platform was also implemented as a reusable class library (CyberUnits) for Object Pascal-based IDEs in order to facilitate the development of computer simulations.
Results
The elements described above may be coupled to a closed formalism (MiMe-NoCoDI model), which predicts the system’s equilibrium behaviour as a quadratic equation. If k parallel feedback control systems are to be modelled, the solution gets the form of an algebraic equation of grade k + 1. The CyberUnits class library facilitates rapid implementations of computer simulations based on the MiMe-NoCoDI platform.
This platform can be used to describe a large number of different feedback loops, as they occur e.g. in endocrine, biochemical and neurophysiological systems. It was successfully applied to thyroid homeostasis, the insulin-glucose system and the hypothalamus-pituitary-adrenal axis. Flexible implementations in the form of computer simulations are possible with the CyberUnits library. GUI-based didactical demo implementations have been written for different computing platforms including macOS, Linux and Windows.
Discussion
The MiMe-NoCoDI platform is able to integrate biochemical, pharmacological and cybernetic models in a uniform representation of closed feedback loops. Mapping the sub-models to biochemical and physiological processes allows for both easy parameterisation for a large number of biological processing structures and vertical translation between molecular processes and the level of the whole organism.
This platform provides a closed theory for a considerable class of biological motifs. Limitations of this methodology are in the number of possible parallel loops since the resulting equilibrium equations are no longer directly solvable if k > 4.
References
- Dietrich JW, Boehm BO. Equilibrium behaviour of feedback-coupled physiological saturation kinetics. In: Trappl R (Editor). Cybernetics and Systems 2006. Austrian Society for Cybernetic Studies. DOI 10.13140/2.1.2400.2568.
- Dietrich JW, Landgrafe-Mende G, Wiora E, Chatzitomaris A, Klein HH, Midgley JE, Hoermann R. Calculated Parameters of Thyroid Homeostasis: Emerging Tools for Differential Diagnosis and Clinical Research. Front Endocrinol (Lausanne). 2016;7:57. DOI 10.3389/fendo.2016.00057. PMID 27375554; PMCID PMC4899439.
- Berberich J, Dietrich JW, Hoermann R, Müller MA. Mathematical Modeling of the Pituitary-Thyroid Feedback Loop: Role of a TSH-T(3)-Shunt and Sensitivity Analysis. Front Endocrinol (Lausanne). 2018 Mar 21;9:91. DOI 10.3389/fendo.2018.00091. PMID 29619006; PMCID PMC5871688.
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