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In vivo reaction space is constrained by complex structures which are made of entwined cytoskeletons and organelles; this create the difference between in vivo and in vitro in respect of molecular mobility, and it may affect reaction processes. Our motivation is to reveal the background mechanisms of the properties of molecular behaviors in vivo by numerical approach. For this object, we reassembled a pseudo-intracellular environment in 3D lattice space, and executed Monte Carlo simulation. By changing the relative amount of non-reactive obstacles in the simulation space, we tested the effect of the level of crowdedness to the molecular mobility and reaction processes. Our results showed that molecules demonstrated anomalous diffusion correlating to the restriction level of the reaction space. Reaction processes also showed distinct characteristics, that is increase of reaction rate at the beginning of reactions, with the decrease of the reaction rate at later time frame of reactions. Our results suggested that the anomalous behaviors at singe molecule level in vivo could bring an essential difference to the reaction processes and the results.

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BACKGROUND: With the increasing availability of high dimensional time course data for metabolites, genes, and fluxes, the mathematical description of dynamical systems has become an essential aspect of research in systems biology. Models are often encoded in formats such as SBML, whose structure is very complex and difficult to evaluate due to many special cases. RESULTS: This article describes an efficient algorithm to solve SBML models that are interpreted in terms of ordinary differential equations. We begin our consideration with a formal representation of the mathematical form of the models and explain all parts of the algorithm in detail, including several preprocessing steps. We provide a flexible reference implementation as part of the Systems Biology Simulation Core Library, a community-driven project providing a large collection of numerical solvers and a sophisticated interface hierarchy for the definition of custom differential equation systems. To demonstrate the capabilities of the new algorithm, it has been tested with the entire SBML Test Suite and all models of BioModels Database. CONCLUSIONS: The formal description of the mathematics behind the SBML format facilitates the implementation of the algorithm within specifically tailored programs. The reference implementation can be used as a simulation backend for Java™-based programs. Source code, binaries, and documentation can be freely obtained under the terms of the LGPL version 3 from http://simulation-core.sourceforge.net. Feature requests, bug reports, contributions, or any further discussion can be directed to the mailing list simulation-core-development@lists.sourceforge.net.

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MOTIVATION: The Systems Biology Markup Language (SBML) is currently supported by >230 software tools, among which 160 support numerical integration of ordinary differential equation (ODE) models. Although SBML is a widely accepted standard within this field, there is still no language-neutral library that supports all features of SBML for simulating ODE models. Therefore, a demand exists for a simple portable implementation of a numerical integrator that supports SBML to enhance the development of a computational platform for systems biology. RESULTS: We implemented a library called libSBMLSim, which supports all the features of SBML and confirmed that the library passes all tests in the SBML test suite including those for SBML Events, AlgebraicRules, 'fast' attribute on Reactions and Delay. LibSBMLSim is implemented in the C programming language and does not depend on any third-party library except libSBML, which is a library to handle SBML documents. For the numerical integrator, both explicit and implicit methods are written from scratch to support all the functionality of SBML features in a straightforward implementation. We succeeded in implementing libSBMLSim as a platform-independent library that can run on most common operating systems (Windows, MacOSX and Linux) and also provides several language bindings (Java, C#, Python and Ruby). AVAILABILITY: The source code of libSBMLSim is available from http://fun.bio.keio.ac.jp/software/libsbmlsim/. LibSBMLSim is distributed under the terms of LGPL. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

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The intracellular environment is known to be a crowded and inhomogeneous space. Such an in vivo environment differs from a well-diluted, homogeneous environment for biochemical reactions. However, the effects of both crowdedness and the inhomogeneity of environment on the behavior of a mobile particle have not yet been investigated sufficiently. As described in this paper, we constructed artificial reaction spaces with fractal models, which are assumed to be non-reactive solid obstacles in a reaction space with crevices that function as operating ranges for mobile particles threading the space. Because of the homogeneity of the structures of artificial reaction spaces, the models succeeded in reproducing the physiological fractal dimension of solid structures with a smaller number of non-reactive obstacles than in the physiological condition. This incomplete compatibility was mitigated when we chose a suitable condition of a perimeter-to-area ratio of the operating range to our model. Our results also show that a simulation space is partitioned into convenient reaction compartments as an in vivo environment with the exact amount of solid structures estimated from TEM images. The characteristics of these compartments engender larger mean square displacement of a mobile particle than that of particles in smaller compartments. Subsequently, the particles start to show confined particle-like behavior. These results are compatible with our previously presented results, which predicted that a physiological environment would produce quick response and slow exhaustion reactions.

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ABSTRACT: : In our previous study, we introduced a combination methodology of Fluorescence Correlation Spectroscopy (FCS) and Transmission Electron Microscopy (TEM), which is powerful to investigate the effect of intracellular environment to biochemical reaction processes. Now, we developed a reconstruction method of realistic simulation spaces based on our TEM images. Interactive raytracing visualization of this space allows the perception of the overall 3D structure, which is not directly accessible from 2D TEM images. Simulation results show that the diffusion in such generated structures strongly depends on image post-processing. Frayed structures corresponding to noisy images hinder the diffusion much stronger than smooth surfaces from denoised images. This means that the correct identification of noise or structure is significant to reconstruct appropriate reaction environment in silico in order to estimate realistic behaviors of reactants in vivo. Static structures lead to anomalous diffusion due to the partial confinement. In contrast, mobile crowding agents do not lead to anomalous diffusion at moderate crowding levels. By varying the mobility of these non-reactive obstacles (NRO), we estimated the relationship between NRO diffusion coefficient (Dnro) and the anomaly in the tracer diffusion (α). For Dnro=21.96 to 44.49 µm2/s, the simulation results match the anomaly obtained from FCS measurements. This range of the diffusion coefficient from simulations is compatible with the range of the diffusion coefficient of structural proteins in the cytoplasm. In addition, we investigated the relationship between the radius of NRO and anomalous diffusion coefficient of tracers by the comparison between different simulations. The radius of NRO has to be 58 nm when the polymer moves with the same diffusion speed as a reactant, which is close to the radius of functional protein complexes in a cell.

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In vivo environments are highly crowded and inhomogeneous, which may affect reaction processes in cells. In this study we examined the effects of intracellular crowding and an inhomogeneity on the behavior of in vivo reactions by calculating the spectral dimension (d(s)), which can be translated into the reaction rate function. We compared estimates of anomaly parameters obtained from fluorescence correlation spectroscopy (FCS) data with fractal dimensions derived from transmission electron microscopy (TEM) image analysis. FCS analysis indicated that the anomalous property was linked to physiological structure. Subsequent TEM analysis provided an in vivo illustration; soluble molecules likely percolate between intracellular clusters, which are constructed in a self-organizing manner. We estimated a cytoplasmic spectral dimension d(s) to be 1.39 ± 0.084. This result suggests that in vivo reactions initially run faster than the same reactions in a homogeneous space; this conclusion is consistent with the anomalous character indicated by FCS analysis. We further showed that these results were compatible with our Monte-Carlo simulation in which the anomalous behavior of mobile molecules correlates with the intracellular environment, leading to description as a percolation cluster, as demonstrated using TEM analysis. We confirmed by the simulation that the above-mentioned in vivo like properties are different from those of homogeneously concentrated environments. Additionally, simulation results indicated that crowding level of an environment might affect diffusion rate of reactant. Such knowledge of the spatial information enables us to construct realistic models for in vivo diffusion and reaction systems.