- Models Of Cell Metabolism; Its Regulation And Control
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Kinetic theory and mathematical model of cell metabolism
Kinetische Theorie und mathematisches Modell des Zellstoffwechsels
Helgoländer wissenschaftliche Meeresuntersuchungen volume 14, pages 78–109 (1966)
Kurzfassung
Obwohl der Stoffwechsel eine physikalisch und chemische äußerst komplizierte Prozeßfolge ist, kann seine Kinetik durch ein hydrodynamisches Modell veranschaulicht werden, in welchem das Niveau der Flüssigkeit dem chemischen Potential entspricht. Die Funktionen sowohl des Intensitätsfaktors als auch des Drosselfaktors sind jedoch stark nicht-linear, so daß eine kontinuierliche Analyse, die auf Differentialgleichungen der Kinetik beruht, Schwierigkeiten bereiten. In die Differentialgeichungen der Kinetik, beispielsweise von mRNA oder Repressor, werden daher binäre Parameter eingeführt, da die entsprechenden molekularkinetischen Überlegungen sonst sehr komplizierte mathematische Modelle ergeben, die es erschweren, qualitative Angaben über das Regelungssystem zu machen; dafür gestattet aber die vorgeschlagene Methode nützliche Vereinfachungen. Zur Simulation wird ein System von Analogrechner und elektromechanischem Relaiskreislauf herangezogen. Liegt die Ausbeute der Rechenmaschine, etwa an Repressormenge, über oder unter einem bestimmten Schwellenwert, so wird ein Impuls durch den Schmitt-Kreislauf gegeben. Der Impuls wird dem Relaiskreislauf zugeführt, welcher als molekularer Automat angesehen werden kann und das kontinuierliche System wie eine Fabrik durch einen Computer reguliert. Durch entsprechende Vereinfachung des Kreislaufs und durch Vernachlässigung der Verzögerung, die durch Analogmaschine und Schmitt-Kreislauf entsteht, wird das schon früher mitgeteilte Modell erhalten.
Summary
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1.
A differential equation of the kinetics of metabolic systems is formulated
$$\frac{{dn_i }}{{dt}} = \mathop \sum \limits_j J_{ji} - \mathop \sum \limits_k J_{ik} $$wheren i is the quantity of a metabolite in a comparment i,J ji the influx from j to i andJ ik the outflux from i to k.
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2.
The fluxesJ ji orJ ik can be described as the product of throttling factor × intensive factor. The thermodynamical nature of the intensive factor is considered and the idea of quasi-equilibrium introduced.
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3.
The notion of parametric action is outlined. Rate processes concerned with the flexibility of the throttling factor are considered as transmission of information; this facilitates an investigation of the parametric interaction or informational correlation in chemical reaction systems.
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4.
The flexibility of the throttling factor reveals that the equation of kinetics is non-linear. Any feedback system has to be represented by such non-linear equations.
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5.
There are some cases in which non-linear behaviour may be represented approximately by a step-function; therefore, finite theory may be useful in such systems.
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6.
Since the limit of applicability of the continuous analysis based on differential equations of kinetics is well appreciated, binary functions or parameters are introduced into the equation of kinetics in order to unify the two types of analysis, finite and continuous. A hybrid computing system is used for this purpose.
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7.
Kinetics of mRNA and repressor formation is especially considered and an appropriate hybrid system suggested. In simplifying this system the switching circuit model proposed previously is obtained.
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8.
A mathematical model of cell differentiation is proposed. A flip-flop circuit is assumed, composed of reactions at the genetic and other levels under the influence of inducible enzymes. The continuous kinetics in this model cell may function only under the control of such a molecular automaton.
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9.
A pattern of active (1) or inactive (0) states of DNA (101001 ...) may be superimposed upon the information at the genetic level. Cell differentiation is the most striking and permanently lasting variation of this pattern. In addition, temporal variations (repression and induction; open or closed states) which may be of physiological significance, are considered.
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10.
Energy for parametric action must be distinguished from energy for the response. The relation between entropy of activation and negative entropy of information, e. g. of DNA, is discussed.
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11.
In the appendices (I to III) thermodynamics of an open system are discussed; a thermodynamical function, such as a chemical potential in a dynamical system, is logically defined. Finally, the maximum principle and the importance of cybernetics are discussed.
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Sugita, M. Kinetic theory and mathematical model of cell metabolism. Helgolander Wiss. Meeresunters 14, 78–109 (1966). https://doi.org/10.1007/BF01611615
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DOI: https://doi.org/10.1007/BF01611615