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Online ISSN 1751-8857 Print ISSN 1751-8849

IET Systems Biology

Volume 6, Issue 4, August 2012

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Volume 6, Issue 4

August 2012

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  • Modelling noise in biochemical reaction networks

    • Editorial: Modelling noise in biochemical reaction networks
      Linear noise approximation is valid over limited times for any chemical system that is sufficiently large
      • Author(s): E.W.J. Wallace ; D.T. Gillespie ; K.R. Sanft ; L.R. Petzold
      • Source: IET Systems Biology, Volume 6, Issue 4, p. 102 –115
      • DOI:  10.1049/iet-syb.2011.0038
      • Type: Article
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      • p. 102 –115 (14)
        The linear noise approximation (LNA) is a way of approximating the stochastic time evolution of a well-stirred chemically reacting system. It can be obtained either as the lowest order correction to the deterministic chemical reaction rate equation (RRE) in van Kampen's system-size expansion of the chemical master equation (CME), or by linearising the two-term-truncated chemical Kramers-Moyal equation. However, neither of those derivations sheds much light on the validity of the LNA. The problematic character of the system-size expansion of the CME for some chemical systems, the arbitrariness of truncating the chemical Kramers-Moyal equation at two terms, and the sometimes poor agreement of the LNA with the solution of the CME, have all raised concerns about the validity and usefulness of the LNA. Here, the authors argue that these concerns can be resolved by viewing the LNA as an approximation of the chemical Langevin equation (CLE). This view is already implicit in Gardiner's derivation of the LNA from the truncated Kramers-Moyal equation, as that equation is mathematically equivalent to the CLE. However, the CLE can be more convincingly derived in a way that does not involve either the truncated Kramers-Moyal equation or the system-size expansion. This derivation shows that the CLE will be valid, at least for a limited span of time, for any system that is sufficiently close to the thermodynamic (large-system) limit. The relatively easy derivation of the LNA from the CLE shows that the LNA shares the CLE's conditions of validity, and it also suggests that what the LNA really gives us is a description of the initial departure of the CLE from the RRE as we back away from the thermodynamic limit to a large but finite system. The authors show that this approach to the LNA simplifies its derivation, clarifies its limitations, and affords an easier path to its solution.
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      Non-linear corrections to the time-covariance function derived from a multi-state chemical master equation
      • Author(s): M. Scott
      • Source: IET Systems Biology, Volume 6, Issue 4, p. 116 –124
      • DOI:  10.1049/iet-syb.2011.0031
      • Type: Article
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      • p. 116 –124 (9)
        The time-covariance function captures the dynamics of biochemical fluctuations and contains important information about the underlying kinetic rate parameters. Intrinsic fluctuations in biochemical reaction networks are typically modelled using a master equation formalism. In general, the equation cannot be solved exactly and approximation methods are required. For small fluctuations close to equilibrium, a linearisation of the dynamics provides a very good description of the relaxation of the time-covariance function. As the number of molecules in the system decrease, deviations from the linear theory appear. Carrying out a systematic perturbation expansion of the master equation to capture these effects results in formidable algebra; however, symbolic mathematics packages considerably expedite the computation. The authors demonstrate that non-linear effects can reveal features of the underlying dynamics, such as reaction stoichiometry, not available in linearised theory. Furthermore, in models that exhibit noise-induced oscillations, non-linear corrections result in a shift in the base frequency along with the appearance of a secondary harmonic.
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      Simulation of mRNA diffusion in the nuclear environment
      • Author(s): M.R. Roussel  and  T. Tang
      • Source: IET Systems Biology, Volume 6, Issue 4, p. 125 –133
      • DOI:  10.1049/iet-syb.2011.0032
      • Type: Article
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      • p. 125 –133 (9)
        A mathematical model is devised to study the diffusion of mRNA in the nucleus from the site of synthesis to a nuclear pore where it is exported to the cytoplasm. This study examines the role that nuclear structure can play in determining the kinetics of export by considering models in which elements of the nuclear skeleton and confinement by chromatin direct the mRNA movement. As a rule, a dense chromatin layer favours rapid export by reducing the effective volume for diffusion. However, it may also result in a heavy tail in the export time distribution because of the low mobility of molecules that accidentally find their way deep into the dense layer. An anisotropic solid-state transport system can also assist export. There exist both an optimal ratio of the anisotropy and an optimal depth of the solid-state transport layer that favour rapid export. [Includes supplementary material]
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      Anomalous diffusion and multifractional Brownian motion: simulating molecular crowding and physical obstacles in systems biology
      • Author(s): T.T. Marquez-Lago ; A. Leier ; K. Burrage
      • Source: IET Systems Biology, Volume 6, Issue 4, p. 134 –142
      • DOI:  10.1049/iet-syb.2011.0049
      • Type: Article
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      • p. 134 –142 (9)
        There have been many recent studies from both experimental and simulation perspectives in order to understand the effects of spatial crowding in molecular biology. These effects manifest themselves in protein organisation on the plasma membrane, on chemical signalling within the cell and in gene regulation. Simulations are usually done with lattice- or meshless-based random walks but insights can also be gained through the computation of the underlying probability density functions of these stochastic processes. Until recently much of the focus had been on continuous time random walks, but some very recent work has suggested that fractional Brownian motion may be a good descriptor of spatial crowding effects in some cases. The study compares both fractional Brownian motion and continuous time random walks and highlights how well they can represent different types of spatial crowding and physical obstacles. Simulated spatial data, mimicking experimental data, was first generated by using the package Smoldyn. We then attempted to characterise this data through continuous time anomalously diffusing random walks and multifractional Brownian motion (MFBM) by obtaining MFBM paths that match the statistical properties of our sample data. Although diffusion around immovable obstacles can be reasonably characterised by a single Hurst exponent, we find that diffusion in a crowded environment seems to exhibit multifractional properties in the form of a different short- and long-time behaviour.
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      Influence of cell-to-cell variability on spatial pattern formation
      • Author(s): B. Greese ; K. Wester ; R. Bensch ; O. Ronneberger ; J. Timmer ; M. Hülskamp ; C. Fleck
      • Source: IET Systems Biology, Volume 6, Issue 4, p. 143 –153
      • DOI:  10.1049/iet-syb.2011.0050
      • Type: Article
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      • p. 143 –153 (11)
        Many spatial patterns in biology arise through differentiation of selected cells within a tissue, which is regulated by a genetic network. This is specified by its structure, parameterisation and the noise on its components and reactions. The latter, in particular, is not well examined because it is rather difficult to trace. The authors use suitable local mathematical measures based on the Voronoi diagram of experimentally determined positions of epidermal plant hairs (trichomes) to examine the variability or noise in pattern formation. Although trichome initiation is a highly regulated process, the authors show that the experimentally observed trichome pattern is substantially disturbed by cell-to-cell variations. Using computer simulations, they find that the rates concerning the availability of the protein complex that triggers trichome formation plays a significant role in noise-induced variations of the pattern. The focus on the effects of cell noise yields further insights into pattern formation of trichomes. The authors expect that similar strategies can contribute to the understanding of other differentiation processes by elucidating the role of naturally occurring fluctuations in the concentration of cellular components or their properties.
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