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The Book in Retrospect and Prospect

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Book cover Stochastic Chemical Kinetics

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

The chapter summarizes the main concepts of the book related also to further possible developments.

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Notes

  1. 1.

    Its significance now noticed [11], as we cite it almost verbatim. As they basically write, Arányi and Tóth were the first to systematically study the master equation for enzyme kinetics. They considered the special case in which there is only one enzyme molecule with several substrate molecules in a closed compartment and showed that the master equation can then be solved exactly. The exact solution consists of the probability distribution of the state of the system at any time point. This is remarkable when one considers that it is impossible to solve the CCD model without imposing restrictions on the reaction conditions such as pseudo-first-order kinetics, or applying an approximation. From the exact solution of the probability distribution, Arányi and Tóth derived exact expressions for the time course of the mean substrate and enzyme concentrations and compared them with those obtained by numerical integration of the CCD model. Interestingly, they found differences of 20–30 % between the average substrate concentrations calculated using the CCD and CDS models for the same set of rate constants and for the case of one enzyme reacting with one substrate molecule. If the initial number of substrate molecules is increased to five, whilst keeping the same rate constants, then one notices that the difference between the CCD and CDS results becomes negligibly small. In general, it can be shown that the discrepancy between the two approaches stems from the fact that the mean concentrations, in chemical systems involving second-order reactions, are dependent on the size of the fluctuations in a CDS description and independent in the CCD description. The discrepancies become smaller for larger numbers of substrate molecules because fluctuations roughly scale as the inverse square of the molecule numbers. This important contribution by Arányi and Tóth went largely unnoticed at the time, because experimental approaches did not have the resolution for measuring single-enzyme-catalysed experiments to test the theoretical results.

  2. 2.

    It was interesting too see, how the relationship between bistability (i.e. three-stationarity) and bimodality was commented by [21]:“…It is often assumed that bistability of deterministic mass action kinetics is associated with bimodality in the steady-state solution to the master equation. However, this is often not the case-we can have bistability without obvious bimodality and bimodality without bistability. In fact, steady-state solutions to either the mass action kinetics or the master equation can be very misleading-we cannot ignore the dynamics.…” Cobb illustrated in 1978 [6] with the aid of a non-kinetic model that there is no one-to-one correspondence between the location of the equilibrium points and of the extrema of stationary distributions. Somewhat artificial kinetic examples were constructed [8] to support the view that all the four possible cases among uni- and multistationarity and uni- and bimodality may occur.

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Authors and Affiliations

  1. Center for Complex Systems Studies, Kalamazoo College, Kalamazoo, MI, USA

    Péter Érdi

  2. Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary

    Péter Érdi

  3. Department of Inorganic and Analytical Chemistry, University of Debrecen, Debrecen, Hungary

    Gábor Lente

Authors
  1. Péter Érdi
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  2. Gábor Lente
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Érdi, P., Lente, G. (2014). The Book in Retrospect and Prospect. In: Stochastic Chemical Kinetics. Springer Series in Synergetics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0387-0_4

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