ABSTRACT

No abstract

Systems biology has experienced dramatic growth in the number, size, and complexity of computational models. To reproduce simulation results and reuse models, researchers must exchange unambiguous model descriptions. We review the latest edition of the Systems Biology Markup Language (SBML), a format designed for this purpose. A community of modelers and software authors developed SBML Level 3 over the past decade. Its modular form consists of a core suited to representing reaction‐based models and packages that extend the core with features suited to other model types including constraint‐based models, reaction‐diffusion models, logical network models, and rule‐based models. The format leverages two decades of SBML and a rich software ecosystem that transformed how systems biologists build and interact with models. More recently, the rise of multiscale models of whole cells and organs, and new data sources such as single‐cell measurements and live imaging, has precipitated new ways of integrating data with models. We provide our perspectives on the challenges presented by these developments and how SBML Level 3 provides the foundation needed to support this evolution.

To elucidate the dynamic features of a biologically relevant large-scale reaction network, we constructed a computational model of minimal protein synthesis consisting of 241 components and 968 reactions that synthesize the Met-Gly-Gly (MGG) peptide based on an Escherichia colibased reconstituted in vitro protein synthesis system. We performed a simulation using parameters collected primarily from the literature and found that the rate of MGG peptide synthesis becomes nearly constant in minutes, thus achieving a steady state similar to experimental observations. In addition, concentration changes to 70% of the components, including intermediates, reached a plateau in a few minutes. However, the concentration change of each component exhibits several temporal plateaus, or a quasi-stationary state (QSS), before reaching the final plateau. To understand these complex dynamics, we focused on whether the components reached a QSS, mapped the arrangement of components in a QSS in the entire reaction network structure, and investigated time-dependent changes. We found that components in a QSS form clusters that grow over time but not in a linear fashion, and that this process involves the collapse and regrowth of clusters before the formation of a final large single cluster. These observations might commonly occur in other largescale biological reaction networks. This developed analysis might be useful for understanding large-scale biological reactions by visualizing complex dynamics, thereby extracting the characteristics of the reaction network, including phase transitions.computational modeling | protein translation | cell-free protein synthesis | quasi-stationary state | network analysis B iological systems contain and are driven by multiple molecular components and reactions that form a complex reaction network. To elucidate the fundamental rules or features underlying complex reaction networks, various abstract computational models have been constructed (1-4). For example, cellular automaton (2) and NK models (1) have been applied to extract the fundamental features in the patterning and evolution of biological systems, respectively. A simplified cellular model was used to reveal why expressed gene copy number follows the power-law distribution (Zipf's law) (3). Although these models capture only the basic features of biological systems, they have helped elucidate the complex properties of life mechanisms. Additionally, although models that enumerate all detailed cellular processes have been constructed (5-7), attempts to extract the features or rules from such detailed models have been performed rarely.A single enzymatic reaction can often be described by MichaelisMenten kinetics, but once reactions are connected to one other, it becomes difficult to understand and capture a complete description of the reaction dynamics because of its high dimensionality. Despite such complexity, many high-dimensional dynamic data obtained from a variety of numerical models exhibit common behavior (reviewed in ref. 8). The models...

Among various approaches in proving gauge independence, models containing an explicit gauge dependence are convenient. The well-known example is the gauge parameter in the covariant gauge fixing which is of course most suitable for the perturbation theory but a negative metric prevents us from imaging a dynamical picture. Noncovariant gauge such as the Coulomb gauge is on the contrary used for many physical situations. Therefore it is desirable to include both cases. More than ten years ago, Steinmann introduced a function (distribution) which can play this role in his attempt on discussing quantum electrodynamics (QED) in terms of the gauge invariant fields solely. The method is, however, broken down in the covariant case: the invariant operators are ill-defined because of 1/p 2 singularity in the Minkowski space. In this paper, we apply his function to the path integral: utilizing the arbitrariness of the function we first restrict it to be able to have a well-defined operator, and then a Hamiltonian with which we can build up the (Euclidean) path integral formula. Although the formula is far from covariant, a full covariant expression is recovered by reviving the components which have been discarded under the construction of the Hamiltonian. There is no pathological defects contrary to the operator formalism. With the aid of the path integral formula, the gauge independence of the free energy as well as the S-matrix is proved. Moreover the reason is clarified why it is so simple and straightforward to argue gauge transformations in the path integral. Discussions on the quark confinement is also presented.