How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order $\Omega^{-3/2}$ for reaction systems which do not obey detailed balance and at least accurate to order $\Omega^{-2}$ for systems obeying detailed balance, where $\Omega$ is the characteristic size of the system. Hence the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order $\Omega^{-1/2}$ and variance estimates accurate to order $\Omega^{-3/2}$. This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.Comment: 39 pages, 3 figures, accepted for publication in J. Chem. Phy

A stochastic automaton shows how enzyme assemblies may contribute to metabolic efficiency

<p>Abstract</p> <p>Background</p> <p>The advantages of grouping enzymes into metabolons and into higher order structures have long been debated. To quantify these advantages, we have developed a stochastic automaton that allows experiments to be performed in a virtual bacterium with both a membrane and a cytoplasm. We have investigated the general case of transport and metabolism as inspired by the phosphoenolpyruvate:sugar phosphotransferase system (PTS) for glucose importation and by glycolysis.</p> <p>Results</p> <p>We show that PTS and glycolytic metabolons can increase production of pyruvate eightfold at low concentrations of phosphoenolpyruvate. A fourfold increase in the numbers of enzyme EI led to a 40% increase in pyruvate production, similar to that observed <it>in vivo </it>in the presence of glucose. Although little improvement resulted from the assembly of metabolons into a hyperstructure, such assembly can generate gradients of metabolites and signaling molecules.</p> <p>Conclusion</p> <p><it>in silico </it>experiments may be performed successfully using stochastic automata such as HSIM (Hyperstructure Simulator) to help answer fundamental questions in metabolism about the properties of molecular assemblies and to devise strategies to modify such assemblies for biotechnological ends.</p

Tailorability, Multifunctionality, and Scalability in Mechanical Metamaterials and Related Materials

This thesis provides, through a collection of publications, an analysis of existing materials and/or constructs in an endeavor to examine what anomalous mechanical properties can be brought out and studied from these existing materials and/or constructs, or evolutions of them. The known and developed structures were studied through a combination of mathematical models, molecular computational techniques, namely forcefield simulations and density functional theory (DFT) simulations, and finite elemental analysis simulations. The results achieved through these techniques allowed for in-depth analyses of the materials and constructs, and their mechanisms of operation were successfully identified and considered in terms of underlying actions. This approach allowed for a novel understanding of such anomalous mechanical properties, and how they may be tailored to requirements and therefore possibly applied. Poly(phenylacetylene) networks were reconsidered, and novel networks made from penta- and tetra- substituted poly(phenylacetylene) sheets were discovered. These networks were studied for mechanical properties, including anomalous properties, and were found to buckle when loaded in off-axis directions. This buckling property, which is normally seen as an undesirable property, was analysed, and considered for its advantages and engineerability within the field of auxetics. Furthermore, the pores these networks exhibit due to the nature of the substitutions were discussed in terms of possible nanodelivery and nanofiltration applications. Honeycombs, including re-entrant, standard, and hybrid configurations were studied to understand the effect of the component materials on the properties of the systems. It was found that through configuring the thermal expansion properties of the comprising ligaments, one could produce systems with positive, zero, and negative thermal expansion. This allows for greatly increased tailorability and possible applicability of such systems. The anomalous properties of boron arsenate (BAsO4) with 4̅ symmetry were considered, being a material known to exhibit a negative Poisson’s ratio and negative linear compressibility under certain conditions. Also, this crystal had not yet been studied in detail in order to thoroughly understand and quantify the underlying actions which present these properties. These underlying actions could allow for future designs and applications to better Formatted: HighlightFormatted: Highlight v apply the properties discovered in this remarkable crystal. Work was also carried out on an equivalent macroscale model which was studied through finite element analysis (FEA) and mathematical modelling in order to generalise the model into a structure which is not effected by the chemistry of the atoms within the crystal, but is instead made of materials at macroscale