Multi-state modeling of biomolecules

355:. It asks questions about whether a combinatorially complex model, once specified, is computationally tractable, given the large number of states and the even larger number of possible transitions between states, whether it can be stored electronically, and whether it can be evaluated in a reasonable amount of computing time. This problem is called the "computation problem". Among the approaches that have been proposed to tackle combinatorial complexity in multi-state modeling, some are mainly concerned with addressing the specification problem, some are focused on finding effective methods of computation. Some tools address both specification and computation. The sections below discuss rule-based approaches to the specification problem and particle-based approaches to solving the computation problem. A wide range of computational tools exist for multi-state modeling. 729:, and their progress is tracked through the course of the entire simulation. Because particle-based rule evaluation keeps track of individual particles rather than populations, it comes at a higher computational cost when modeling systems with a high total number of particles, but a small number of kinds (or pools) of particles. In cases of combinatorial complexity, however, the modeling of individual particles is an advantage because, at any given point in the simulation, only existing molecules, their states and the reactions they can undergo need to be considered. Particle-based rule evaluation does not require the generation of complete or partial reaction networks at the start of the simulation or at any other point in the simulation and is therefore called "network-free". 343: 334:. The reaction rate is the same for two molecules that differ only in features which do not affect this reaction. Thus, the number of parameters will be much smaller than the number of reactions. (In the coffee shop example, adding an extra shot of espresso will cost 40 cent, no matter what size the beverage is and whether or not it has milk in it). It is such "local rules" that are usually discovered in laboratory experiments. Thus, a multi-state model can be conceptualised in terms of combinations of modular features and local rules. This means that even a model that can account for a vast number of molecular species and reactions is not necessarily conceptually complex. 713: 330:) quickly leads to a large number of possible beverages (24 in this case; each additional binary choice will double that number). Although it is difficult for us to grasp the total numbers of possible combinations, it is usually not conceptually difficult to understand the (much smaller) set of features or modifications and the effect each of them has on the function of the biomolecule. The rate at which a molecule undergoes a particular reaction will usually depend mainly on a single feature or a small subset of features. It is the presence or absence of those features that dictates the 646:

simulation methods implementing numerical integration of ordinary and partial differential equations or the Gillespie stochastic algorithm, all possible pools of molecules and the reactions they undergo are defined at the start of the simulation, even if they are empty. Such "generate-first" methods scale poorly with increasing numbers of molecular states. For instance, it has recently been estimated that even for a simple model of CaMKII with just 6 states per subunits and 10 subunits, it would take 290 years to generate the entire reaction network on a 2.54 GHz Intel

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the user can specify a rule-based model, but can designate some species to be treated as populations (rather than particles) in the subsequent simulation. This method combines the computational advantages of particle-based modeling for multi-state systems with relatively low molecule numbers and of population-based modeling for systems with high molecule numbers and a small number of possible states. Specification of HPP models is supported by BioNetGen, and simulations can be performed with NFSim.

1105: 474:, where molecules are represented as nodes (or collections of nodes) and chemical bonds as edges. A reaction rule, then, corresponds to a graph rewriting rule. BNGL provides a syntax for specifying these graphs and the associated rules as structured strings. BioNetGen can then use these rules to generate ordinary differential equations (ODEs) to describe each biochemical reaction. Alternatively, it can generate a list of all possible species and reactions in 451:. This description of the system in terms of modular rules relies on the assumption that only a subset of features or attributes are relevant for a particular reaction rule. Where this assumption holds, a set of reactions can be coarse-grained into one reaction rule. This coarse-graining preserves the important properties of the underlying reactions. For instance, if the reactions are based on chemical kinetics, so are the rules derived from them. 766:(for instance, for holoenzymes consisting of several subunits), and reactions can be "neighbor-sensitive", i.e. the probability of a reaction for a given entity is affected by the value of a state flag on a neighboring entity. These properties make StochSim ideally suited to modeling multi-state molecules arranged in holoenzymes or complexes of specified size. Indeed, StochSim has been used to model clusters of 806: 455:

specified model into a dedicated simulation engine. However, many solutions to the specification problem also contain a method of interpreting the specified model. This is done by providing a method to simulate the model or a method to convert it into a form that can be used for simulations in other programs.

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One example of a particle-based simulator that allows for a representation of cellular compartments is SRSim. SRSim is integrated in the LAMMPS molecular dynamics simulator and allows the user to specify the model in BNGL. SRSim allows users to specify the geometry of the particles in the simulation,

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simulator used mainly to model chemical reactions and other molecular transitions. The algorithm used in StochSim is different from the more widely known Gillespie stochastic algorithm in that it operates on individual entities, not entity pools, making it particle-based rather than population-based.

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In population-based approaches, one can think of the system being modeled as being in a given state at any given time point, where a state is defined according to the nature and size of the populated pools of molecules. This means that the space of all possible states can become very large. With some

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Enumerating all possible states is a lengthy and potentially error-prone process. For macromolecular complexes that can adopt multiple states, enumerating each state quickly becomes tedious, if not impossible. Moreover, the addition of a single additional modification or feature to the model of the

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that can be used to encode macromolecules with internal states and binding sites and to specify rules by which they interact. The κ-calculus is merely concerned with providing a language to encode multi-state models, not with interpreting the models themselves. A simulator compatible with Kappa

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It is clear that an explicit description, which lists all possible molecular species (including all their possible states), all possible reactions or transitions these species can undergo, and all parameters governing these reactions, very quickly becomes unwieldy as the complexity of the biological

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It is easy to imagine a biological system where some components are complex multi-state molecules, whereas others have few possible states (or even just one) and exist in large numbers. A hybrid approach has been proposed to model such systems: Within the Hybrid Particle/Population (HPP) framework,

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and parameters that apply to many types of molecular species into one reaction template. It might also add a set of conditions that govern reaction parameters, i.e. the likelihood or rate at which a reaction occurs, or whether it occurs at all. Only properties of the molecule or complex that matter

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The existence (or potential existence) of such large numbers of molecular species is a combinatorial phenomenon: It arises from a relatively small set of features or modifications (such as post-translational modification or complex formation) that combine to dictate the state of the entire molecule

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In addition, several types of modifications can co-exist, exerting a combined influence on a biological macromolecule at any given time. Thus, a biomolecule or complex of biomolecules can often adopt a very large number of functionally distinct states. The number of states scales exponentially with

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ML-Rules is similar to React(C), but provides the added possibility of nesting: A component species of the model, with all its attributes, can be part of a higher-order component species. This enables ML-Rules to capture multi-level models that can bridge the gap between, for instance, a series of

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Of course, not all of the possible states of a multi-state molecule or complex will necessarily be populated. Indeed, in systems where the number of possible states is far greater than that of molecules in the compartment (e.g. the cell), they cannot be. In some cases, empirical information can be

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Principles of particle-based modeling. In particle-based modeling, each particle is tracked individually through the simulation. At any point, a particle only "sees" the rules that apply to it. This figure follows two molecular particles (one of type A in red, one of type B in blue) through three

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The combinatorial complexity of signaling systems involving multi-state proteins poses two kinds of problems. The first problem is concerned with how such a system can be specified; i.e. how a modeler can specify all complexes, all changes those complexes undergo and all parameters and conditions

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of the model at the simulation stage, and thereby saves time and computational power. The simulation follows each particle, and at each simulation step, a particle only "sees" the reactions (or rules) that apply to it. This depends on the state of the particle and, in some implementation, on the

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from a protein might depend on the conformational state of the protein, but not on its subcellular localization. An implicit description would therefore list two dissociation processes (with different rates, depending on conformational state), but would ignore attributes referring to subcellular

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Some rule-based specification systems and their associated network generation and simulation tools have been designed to accommodate spatial heterogeneity, in order to allow for the realistic simulation of interactions within biological compartments. For instance, the Simmune project includes a

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Many rule-based specification methods exist. In general, the specification of a model is a separate task from the execution of the simulation. Therefore, among the existing rule-based model specification systems, some concentrate on model specification only, allowing the user to then export the

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such multi-state systems poses two problems: The problem of how to describe and specify a multi-state system (the "specification problem") and the problem of how to use a computer to simulate the progress of the system over time (the "computation problem"). To address the specification problem,

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Methods have been proposed to reduce the size of the state space. One is to consider only the states adjacent to the present state (i.e. the states that can be reached within the next iteration) at each time point. This eliminates the need for enumerating all possible states at the beginning.

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The Stochastic Simulator Compiler (SSC) allows for rule-based, modular specification of interacting biomolecules in regions of arbitrarily complex geometries. Again, the system is represented using graphs, with chemical interactions or diffusion events formalised as graph-rewriting rules. The

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Instead, reactions are generated "on-the-fly" at each iteration. These methods are available both for stochastic and deterministic algorithms. These methods still rely on the definition of an (albeit reduced) reaction network - in contrast to the "network-free" methods discussed below.

482:. One can also make use of BioNetGen's own ODE-based simulation software and its capability to generate reactions on-the-fly during a stochastic simulation. In addition, a model specified in BNGL can be read by other simulation software, such as DYNSTOC, RuleMonkey, and NFSim. 578:

Thus, by only considering states and features important for a particular reaction, rule-based model specification eliminates the need to explicitly enumerate every possible molecular state that can undergo a similar reaction, and thereby allows for efficient specification.

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spatial component: Users can specify their multi-state biomolecules and interactions within membranes or compartments of arbitrary shape. The reaction volume is then divided into interfacing voxels, and a separate reaction network generated for each of these subvolumes.

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Some of the best-known classes of simulation approaches in computational biology belong to the population-based family, including those based on the numerical integration of ordinary and partial differential equations and the Gillespie stochastic simulation algorithm.

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MCell allows individual molecules to be traced in arbitrarily complex geometric environments which are defined by the user. This allows for simulations of biomolecules in realistic reconstructions of living cells, including cells with complex geometries like those of

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within the spine. Although other types of calcium-regulated molecules were included in the simulations, only CaMKII molecules are visualized. They are shown in red when bound to calmodulin and in black when unbound. The simulation compartment is a reconstruction of a

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and movement of cells. Models specified in ML-Rules can be simulated using the James II simulation framework. A similar nested language to represent multi-level biological systems has been proposed by Oury and Plotkin. A specification formalism based on molecular

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Even with "on-the-fly" network generation, networks generated for population-based rule evaluation can become quite large, and thus difficult - if not impossible - to handle computationally. An alternative approach is provided by particle-based rule evaluation.

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and MCell. Modelers can thus choose from a variety of tools; the best choice depending on the particular problem. Development of faster and more powerful methods is ongoing, promising the ability to simulate ever more complex signaling processes in the future.

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Even if a large reaction system can be successfully generated, its simulation using population-based rule evaluation can run into computational limits. In a recent study, a powerful computer was shown to be unable to simulate a protein with more than 8

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of complexes, and hence, of multi-state molecules. This can in some cases be circumvented by adjusting the diffusion constants of ligands that interact with the complex, by using checkpointing functions or by combining simulations at different levels.

853:. Each slot stands for a particular modification, and any number of slots can be assigned to a molecule. Each slot can be occupied by a particular state. The states are not necessarily binary. For instance, a slot describing binding of a particular 761:

representing a particular modification. Reactions can be made dependent on a set of state flags set to particular values. In addition, the outcome of a reaction can include a state flag being changed. Moreover, entities can be arranged in geometric

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is a software suite that provides both specification and simulation capacities. Rule-based models can be written down using a specified syntax, the BioNetGen language (BNGL). The underlying concept is to represent biochemical systems as

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through a series of time steps until a specified end time. One way to classify simulation algorithms is by looking at the level of analysis at which the rules are applied: they can be population-based, single-particle-based or hybrid.

489:(MWC) type regulation mechanism, whose interactions are governed by their internal state, as well as by external modifications. A very useful feature of ANC is that it automatically computes dependent parameters, thereby imposing 4147: 784:

Another particle-based stochastic simulator that can read BNGL input files is RuleMonkey. Its simulation algorithm differs from the algorithms underlying both StochSim and DYNSTOC in that the simulation time step is variable.

51:, the Allosteric Network Compiler and others. To tackle the computation problem, they have turned to particle-based methods that have in many cases proved more computationally efficient than population-based methods based on 4137: 788:

The Network-Free Stochastic Simulator (NFSim) differs from those described above by allowing for the definition of reaction rates as arbitrary mathematical or conditional expressions and thereby facilitates selective

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An extension to StochSim includes a particle-based simulator DYNSTOC, which uses a StochSim-like algorithm to simulate models specified in the BioNetGen language (BNGL), and improves the handling of molecules within

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processor. In addition, the model generation step in generate-first methods does not necessarily terminate, for instance when the model includes assembly of proteins into complexes of arbitrarily large size, such as

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steps in a hypothetical simulation following a simple set of rules (given on the right). At each step, the rules that potentially apply to the particle under consideration are highlighted in that particle's colour.

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describe changes in molecular concentrations over time in a deterministic manner. Simulations based on differential equations usually do not attempt to solve those equations analytically, but employ a suitable

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Danos V, Feret J, Fontana W, Harmer R, Krivine J (2007). Rule-Based Modelling of Cellular Signalling. Proceedings of the Eighteenth International Conference on Concurrency Theory, CONCUR 2007, Lisbon, Portugal

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Stiles JR, Bartol TM (2001). Computational Neuroscience: Realistic Modeling for Experimentalists. In: De Schutter, E (ed). Computational Neuroscience: Realistic Modeling for Experimentalists. CRC Press, Boca

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relationships between neighboring subunits or members of a macromolecular complex. One drawback of this method for representing protein complexes, compared to Meredys, is that MCell does not allow for the

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Hogg, J. S., Harris, L. A., Stover, L. J., Nair, N. S., & Faeder, J. R. (2013). Exact hybrid particle/population simulation of rule-based models of biochemical systems. arXiv preprint arXiv:1301.6854.

868:: A "state dimension" and one or more "spatial dimensions". The "state dimension" is used to describe the multiple possible states making up a multi-state protein, while the spatial dimension(s) describe 93:

that can act as complex computational devices. These networks rely on the ability of single proteins to exist in a variety of functionally different states achieved through multiple mechanisms, including

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Chylek LA, Stites EC, Posner RG, Hlavacek WS (2013) Innovations of the rule-based modeling approach. In Systems Biology: Integrative Biology and Simulation Tools, Volume 1 (Prokop A, Csukás B, Editors),

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Some particle-based simulation packages use an ad-hoc formalism for specification of reactants, parameters and rules. Others can read files in a recognised rule-based specification format such as BNGL.

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The slot-and-state syntax in MCell can also be used to model multimeric proteins or macromolecular complexes. When used in this way, a slot is a placeholder for a subunit or a molecular component of a

158:, and for several proteins and/or nucleic acid species to assemble into larger complexes. A molecular species with several of those features can therefore exist in a large number of possible states. 575:

Models involving multi-state and multi-component species can also be specified in Level 3 of the Systems Biology Markup Language (SBML) using the multi package. A draft specification is available.

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John, M., Lhoussaine, C., Niehren, J., & Versari, C. (2011). Biochemical reaction rules with constraints. In Programming Languages and Systems (pp. 338-357). Springer Berlin Heidelberg.

864:, and the state of the slot will indicate whether a specific protein component is absent or present in the complex. A way to think about this is that MCell macromolecules can have several 418:

complex under investigation will double the number of possible states (if the modification is binary), and it will more than double the number of transitions that need to be specified.

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used to rule out certain states if, for instance, some combinations of features are incompatible. In the absence of such information, however, all possible states need to be considered

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as well as interaction sites. It is therefore especially good at simulating the assembly and structure of complex biomolecular complexes, as evidenced by a recent model of the inner

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Tschernyschkow S, Herda S, Gruenert G, Döring V, Görlich D, Hofmeister A, et al. (September 2013). "Rule-based modeling and simulations of the inner kinetochore structure".

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show that it can express the stochastic π calculus. They also provide a stochastic simulation algorithm based on the Gillespie stochastic algorithm for models specified in

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localization, because they do not affect the rate of ligand dissociation, nor are they affected by it. This specification rule has been summarized as "Don't care, don't write".

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to a given reaction (either affecting the reaction or being affected by it) are explicitly mentioned, and all other properties are ignored in the specification of the reaction.

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Another tool that generates full reaction networks from a set of rules is the Allosteric Network Compiler (ANC). Conceptually, ANC sees molecules as allosteric devices with a

2132:"Miniature endplate current rise times less than 100 microseconds from improved dual recordings can be modeled with passive acetylcholine diffusion from a synaptic vesicle" 886:

A (by no means exhaustive) selection of models of biological systems involving multi-state molecules and using some of the tools discussed here is give in the table below.

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that can undergo several functionally significant modifications that are mutually compatible. Thus, they can exist in a very large number of functionally different states.

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Since it is not written in terms of reactions, but in terms of more general "reaction rules" encompassing sets of reactions, this kind of specification is often called

287: 3433:"SBML Level 3 Package Specification: Multistate, Multicomponent and Multicompartment Species Package for SBML Level 3 (Multi). Version 1, Release 01 (Draft, Rev 369)" 368:

The most naïve way of specifying, e.g., a protein in a biological model is to specify each of its states explicitly and use each of them as a molecular species in a

63:. Given current computing technology, particle-based methods are sometimes the only possible option. Particle-based simulators further fall into two categories: Non- 793:

of models. RuleMonkey and NFsim implement distinct but related simulation algorithms. A detailed review and comparison of both tools is given by Yang and Hlavacek.

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An overview of tools discussed that are used for the rule-based specification and particle-based evaluation (spatial or non-spatial) of multi-state biomolecules.

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receptors of four different kinds interact in groups of three, and each individual receptor can exist in at least two possible conformations and has up to eight

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in the same state are pooled together. Application of a specific rule reduces or increases the size of one of the pools, possibly at the expense of another.

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states of its neighbours in a holoenzyme or complex. As the simulation proceeds, the states of particles are updated according to the rules that are fired.

4142: 564:. A model (or part of a model) is represented as a Python programme. This allows users to store higher-order biochemical processes such as catalysis or 822:

is shown in red, the spine head and neck in gray, and the parent dendrite in yellow. The figure was generated by visualizing the simulation results in

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Grunert G, Dittrich P (2011). "Using the SRSim Software for Spatial and Rule-Based Modeling of Combinatorially Complex Biochemical Reaction Systems".

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J. Himmelspach and A. M. Uhrmacher, "Plug'n simulate," Proceedings of the 40th Annual Simulation Symposium. IEEE Computer Society, 2007, pp. 137-143.

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biochemical processes and the macroscopic behaviour of a whole cell or group of cells. For instance, a proof-of-concept model of cell division in

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who model or simulate such biomolecules, because it raises questions about how such large numbers of states can be represented and simulated.

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Sneddon MW, Faeder JR, Emonet T (February 2011). "Efficient modeling, simulation and coarse-graining of biological complexity with NFsim".

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on a biological model, any simulation software evaluates a set of rules, starting from a specified set of initial conditions, and usually

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governing those changes in a robust and efficient way. This problem is called the "specification problem". The second problem concerns

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MCell uses an ad-hoc formalism within MCell itself to specify a multi-state model: In MCell, it is possible to assign "slots" to any

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distinct molecular species, yet the authors note several points at which the model could be further extended. A more recent model of

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Shimizu TS, Aksenov SV, Bray D (May 2003). "A spatially extended stochastic model of the bacterial chemotaxis signalling pathway".

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Hlavacek WS, Faeder JR, Blinov ML, Posner RG, Hucka M, Fontana W (July 2006). "Rules for modeling signal-transduction systems".

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can undergo a variety of transformations, including protein binding, binding of other nucleic acids, conformational change and

95: 4177: 52: 712: 3483:"CaMKII activation and dynamics are independent of the holoenzyme structure: an infinite subunit holoenzyme approximation" 561: 56: 471: 297: 721:

In particle-based (sometimes called "agent-based") simulations, proteins, nucleic acids, macromolecular complexes or

431:, rather than explicit, ways of specifying a biological signaling system. An implicit description is one that groups 2830:"The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models" 1243:"BioNetGen: software for rule-based modeling of signal transduction based on the interactions of molecular domains" 1135: 3654:"The efficiency of reactant site sampling in network-free simulation of rule-based models for biochemical systems" 830:

Spatial particle-based methods differ from the methods described above by their explicit representation of space.

314:. In such cases, computational modeling can be used to uncover to what extent the different states are populated. 4162: 2647:"Specification, annotation, visualization and simulation of a large rule-based model for ERBB receptor signaling" 2476:"Purification and characterization of a calmodulin-dependent protein kinase that is highly concentrated in brain" 726: 553:

compiler then generates the entire reaction network before launching a stochastic reaction-diffusion algorithm.

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of which is computed from reaction rates and from the numbers of molecules, in accordance with the stochastic

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Angermann BR, Klauschen F, Garcia AD, Prustel T, Zhang F, Germain RN, Meier-Schellersheim M (January 2012).

3224:"Key role of local regulation in chemosensing revealed by a new molecular interaction-based modeling method" 123: 64: 2207:"Fast Monte Carlo Simulation Methods for Biological Reaction-Diffusion Systems in Solution and on Surfaces" 1120: 3065: 2878: 1988:"Meredys, a multi-compartment reaction-diffusion simulator using multistate realistic molecular complexes" 1534:

Lok L, Brent R (January 2005). "Automatic generation of cellular reaction networks with Moleculizer 1.0".

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The stochastic Gillespie algorithm changes the composition of pools of molecules through a progression of

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as macros and re-use them as needed. The models can be simulated and analysed using Python libraries, but

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rings. Each subunit can exist in at least two distinct conformations, and each subunit features various

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modelers have in recent years moved away from explicit specification of all possible states, and towards

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Monod J, Wyman J, Changeux JP (May 1965). "On the nature of allosteric transitions: A plausible model".

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simulators such as StochSim, DYNSTOC, RuleMonkey, and NFSim and spatial simulators, including Meredys,

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Faeder JR, Blinov ML, Hlavacek WS (2009). "Rule-Based Modeling of Biochemical Systems with BioNetGen".

3544:, Allard J, Wollman R (April 2012). "Cell polarity: quantitative modeling as a tool in cell biology". 1291:

Faeder JR, Blinov ML, Goldstein B, Hlavacek WS (2005). "Rule-Based Modeling of Biochemical Networks".

1104: 4030: 3774: 3665: 3608: 3553: 3494: 3235: 2962: 2773: 2582: 2332: 2274: 2218: 2143: 1669: 1488: 1300: 861: 819: 778: 486: 143: 107: 24: 3070: 2883: 2571:"Structural analysis and stochastic modelling suggest a mechanism for calmodulin trapping by CaMKII" 1429: 216:, for a total of around one billion possible states per hexameric ring. A model of coupling of the 1373: 1092: 1087: 993: 588: 448: 139: 60: 44: 39: 31: 227: 154:. Furthermore, it is common for a protein to be composed of several - identical or nonidentical - 3790: 3577: 3463: 3201: 3083: 2546: 2456: 2404: 2358: 2298: 1917: 1659: 1593: 1559: 1394: 1316: 857:

to a protein of interest could take the states "unbound", "partially bound", and "fully bound".

823: 624: 542: 3283:"Computational modeling of cellular signaling processes embedded into dynamic spatial contexts" 666: 466: 48: 4167: 4109: 4058: 3999: 3950: 3909: 3860: 3825: 3743: 3691: 3634: 3569: 3520: 3413: 3364: 3312: 3263: 3139: 2988: 2931: 2896: 2851: 2801: 2732: 2678: 2610: 2538: 2497: 2448: 2396: 2350: 2290: 2244: 2171: 2107: 2073: 2019: 1968: 1909: 1865: 1809: 1755: 1695: 1621: 1551: 1516: 1452: 1442: 1386: 1264: 1172: 1154: 810: 763: 432: 428: 377: 301: 290: 265: 4099: 4089: 4078:"Spatial rule-based modeling: a method and its application to the human mitotic kinetochore" 4048: 4038: 3989: 3981: 3940: 3899: 3891: 3852: 3817: 3782: 3762: 3733: 3725: 3681: 3673: 3624: 3616: 3561: 3510: 3502: 3455: 3403: 3395: 3354: 3346: 3302: 3294: 3253: 3243: 3193: 3185: 3129: 3119: 3075: 3031: 2978: 2970: 2923: 2888: 2841: 2791: 2781: 2722: 2712: 2668: 2658: 2600: 2590: 2528: 2517:"Structure of the autoinhibited kinase domain of CaMKII and SAXS analysis of the holoenzyme" 2487: 2438: 2388: 2340: 2282: 2234: 2226: 2205:

Kerr RA, Bartol TM, Kaminsky B, Dittrich M, Chang JC, Baden SB, et al. (October 2008).

2161: 2151: 2099: 2063: 2053: 2009: 1999: 1958: 1948: 1901: 1855: 1845: 1799: 1791: 1745: 1685: 1677: 1611: 1603: 1543: 1506: 1496: 1434: 1378: 1308: 1254: 1218: 1180: 1162: 1144: 850: 758: 609: 534: 173: 165: 1582:"Rule-based modelling and simulation of biochemical systems with molecular finite automata" 68: 2869:

Finney A, Hucka M (December 2003). "Systems biology markup language: Level 2 and beyond".

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Tolle DP, Le Novère N (2006). "Particle-Based Stochastic Simulation in Systems Biology".

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framework that allows transitions from state to state. For instance, if a protein can be

289:) distinct molecular species. The problem of combinatorial explosion is also relevant to 4034: 3778: 3669: 3612: 3557: 3498: 3335:"Efficient stochastic simulation of reaction-diffusion processes via direct compilation" 3239: 2966: 2777: 2645:

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2586: 2336: 2278: 2222: 2147: 1673: 1492: 1304: 4104: 4077: 4053: 4018: 3994: 3969: 3904: 3879: 3738: 3714:"Extracellular sheets and tunnels modulate glutamate diffusion in hippocampal neuropil" 3713: 3686: 3677: 3653: 3629: 3596: 3515: 3506: 3482: 3408: 3383: 3359: 3334: 3307: 3282: 3258: 3223: 3134: 3107: 2983: 2950: 2825: 2796: 2761: 2727: 2700: 2673: 2646: 2605: 2570: 2239: 2206: 2068: 2041: 2014: 1987: 1963: 1936: 1860: 1833: 1804: 1779: 1690: 1647: 1616: 1581: 1511: 1476: 1167: 1130: 854: 722: 565: 490: 323: 99: 86: 3856: 3620: 2927: 2492: 2475: 1750: 1733: 655:

filaments. In these cases, a termination condition needs to be specified by the user.

4126: 3821: 3541: 2166: 2131: 1937:"Simulation tools for particle-based reaction-diffusion dynamics in continuous space" 1832:

Colvin J, Monine MI, Gutenkunst RN, Hlavacek WS, Von Hoff DD, Posner RG (July 2010).

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Meier-Schellersheim M, Xu X, Angermann B, Kunkel EJ, Jin T, Germain RN (July 2006).

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states (e.g. open or closed) and be located in two possible subcellular areas (e.g.

3970:"A rule-based kinetic model of RNA polymerase II C-terminal domain phosphorylation" 3712:

Kinney JP, Spacek J, Bartol TM, Bajaj CL, Harris KM, Sejnowski TJ (February 2013).

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Bray D (July 1995). "Protein molecules as computational elements in living cells".

1921: 1398: 1320: 388:-bound), then the eight possible resulting states can be explicitly enumerated as: 217: 111: 3927:

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In StochSim, each molecular species can be equipped with a number of binary state

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and ligand binding sites. A recent model incorporated conformational states, two

3597:"Binding and diffusion of CheR molecules within a cluster of membrane receptors" 2974: 1438: 835: 635: 352: 319: 182: 20: 19:

refers to a series of techniques used to represent and compute the behaviour of

4023:

Proceedings of the National Academy of Sciences of the United States of America

3459: 2766:

Proceedings of the National Academy of Sciences of the United States of America

2533: 2516: 2136:

Proceedings of the National Academy of Sciences of the United States of America

1681: 1477:"Scalable rule-based modelling of allosteric proteins and biochemical networks" 1111: 3189: 3036: 3019: 1953: 1607: 1223: 1206: 770: 750: 733: 369: 221: 213: 178: 4019:"CD4 and CD8 binding to MHC molecules primarily acts to enhance Lck delivery" 2425:

Hlavacek WS, Faeder JR, Blinov ML, Perelson AS, Goldstein B (December 2003).

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Ibrahim B, Henze R, Gruenert G, Egbert M, Huwald J, Dittrich P (July 2013).

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that allow for implicit model specification, including the κ-calculus,

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In Population-based rule evaluation, rules are applied to populations. All

478:, which can then be exported to simulation software packages that can read 4017:

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Stiles JR, Van Helden D, Bartol TM, Salpeter EE, Salpeter MM (June 1996).

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or complex, in the same way that the existence of just a few choices in a

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Knowledge (XXG) articles published in peer-reviewed literature (J2W)

1547: 560:, where model specification is embedded in the programming language 458:

An early rule-based specification method is the κ-calculus, a

1664: 1598: 804: 711: 652: 341: 162: 4138:

Knowledge (XXG) articles published in PLOS Computational Biology

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phosphorylation states) using ordinary differential equations.

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Knowledge (XXG) articles published in peer-reviewed literature

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sites, resulting in billions of potential states. The protein

3763:"Fast parallel algorithms for short-range molecular dynamics" 122:

the number of possible modifications, a phenomenon known as "

27:

that can adopt a large number of possible functional states.

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This article was adapted from the following source under a

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models can also be exported into BNGL, kappa, and SBML.

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cascade presented by Danos and colleagues accounts for

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Examples of multi-state models of biological systems

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system increases. Modelers have therefore looked for

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receptor signalling even accounts for more than one

171:

can be a part of 25666 unique protein complexes. In

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This is of concern for 3761:Plimpton S (March 1995). 3388:Molecular Systems Biology 3190:10.1017/s0960129512000199 1954:10.1186/s13628-014-0011-5 1608:10.1049/iet-syb.2010.0015 1224:10.1016/j.tcs.2004.03.065 1052:Human mitotic kinetochore 689:{\displaystyle 2^{8}=256} 528:secretion and diffusion, 413:unbound, closed, membrane 359:The specification problem 2059:10.1186/1471-2105-11-307 1851:10.1186/1471-2105-11-404 779:macromolecular complexes 732:This method reduces the 524:binding and activation, 410:unbound, closed, cytosol 282:{\displaystyle 10^{100}} 128:computational biologists 4044:10.1073/pnas.1010568107 3566:10.1126/science.1216380 3125:10.1186/1752-0509-5-166 3037:10.1093/comjnl/38.7.578 3020:"Stochastic π-calculus" 2787:10.1073/pnas.0809908106 2664:10.1186/1752-0509-6-107 2393:10.1126/science.1080010 2157:10.1073/pnas.93.12.5747 1383:10.1126/stke.3442006re6 583:The computation problem 407:unbound, open, membrane 401:bound, closed, membrane 124:combinatorial explosion 3986:10.1098/rsif.2013.0438 3787:10.1006/jcph.1995.1039 2718:10.1186/1752-0509-7-42 2005:10.1186/1752-0509-4-24 996:of allosteric proteins 827: 718: 690: 620:Differential equations 404:unbound, open, cytosol 398:bound, closed, cytosol 364:Explicit specification 347: 326:or not, extra shot of 283: 248: 152:conformational changes 106:, or formation of new 4178:Stochastic simulation 808: 715: 691: 634:reaction events, the 539:stochastic simulation 395:bound, open, membrane 345: 284: 249: 104:conformational change 4095:10.3390/cells2030506 3024:The Computer Journal 1536:Nature Biotechnology 1127:Terrence J Sejnowski 820:postsynaptic density 667: 487:Monod-Wyman-Changeux 392:bound, open, cytosol 266: 228: 21:biological molecules 4035:2010PNAS..10716916A 3779:1995JCoPh.117....1P 3670:2011PhBio...8e5009Y 3613:2002BpJ....82.1809L 3601:Biophysical Journal 3558:2012Sci...336..175M 3499:2012PhBio...9c6010M 3240:2006PLSCB...2...82M 3112:BMC Systems Biology 3080:10.1021/j100540a008 2967:2004BpJ....86.3510C 2955:Biophysical Journal 2778:2009PNAS..106.6453F 2705:BMC Systems Biology 2651:BMC Systems Biology 2587:2012PLoSO...729406S 2337:2001Natur.409..391E 2279:1995Natur.376..307B 2223:2008SJSC...30.3126K 2148:1996PNAS...93.5747S 1992:BMC Systems Biology 1674:2008PhRvE..78c1910Y 1586:IET Systems Biology 1493:2010PLSCB...6E0975O 1305:2005Cmplx..10d..22F 1093:Rule-based modeling 1088:Multiscale modeling 946:receptor signalling 891: 45:rule-based modeling 3400:10.1038/msb.2013.1 3299:10.1038/nmeth.1861 2893:10.1042/bst0311472 2096:Membrane Computing 2046:BMC Bioinformatics 1906:10.1038/nmeth.1546 1838:BMC Bioinformatics 1658:(3 Pt 1): 031910. 1313:10.1002/cplx.20074 889: 828: 818:. The area of the 719: 686: 348: 279: 244: 214:calcium/calmodulin 140:signaling networks 3730:10.1002/cne.23181 3064:(25): 2340–2361. 3018:Priami C (1995). 2444:10.1002/bit.10842 2387:(5610): 1189–90. 2231:10.1137/070692017 2113:978-3-642-18122-1 1652:Physical Review E 1448:978-1-934115-64-0 1079: 1078: 964:BioNetGen, PROMOT 928:CaMKII regulation 897:Biological system 851:molecular species 811:calcium signaling 500:. The authors of 291:synthetic biology 32:signaling systems 4185: 4163:Chemical bonding 4118: 4117: 4107: 4097: 4073: 4067: 4066: 4056: 4046: 4029:(39): 16916–21. 4014: 4008: 4007: 3997: 3980:(86): 20130438. 3965: 3959: 3958: 3948: 3924: 3918: 3917: 3907: 3875: 3869: 3868: 3840: 3834: 3833: 3805: 3799: 3798: 3758: 3752: 3751: 3741: 3709: 3700: 3699: 3689: 3658:Physical Biology 3649: 3643: 3642: 3632: 3592: 3586: 3585: 3538: 3529: 3528: 3518: 3487:Physical Biology 3478: 3472: 3471: 3443: 3437: 3436: 3428: 3422: 3421: 3411: 3379: 3373: 3372: 3362: 3330: 3321: 3320: 3310: 3278: 3272: 3271: 3261: 3251: 3219: 3210: 3209: 3175: 3166: 3160: 3157: 3148: 3147: 3137: 3127: 3103: 3092: 3091: 3073: 3053: 3042: 3041: 3039: 3015: 3009: 3006: 2997: 2996: 2986: 2946: 2940: 2939: 2911: 2905: 2904: 2886: 2877:(Pt 6): 1472–3. 2866: 2860: 2859: 2849: 2821: 2810: 2809: 2799: 2789: 2757: 2751: 2747: 2741: 2740: 2730: 2720: 2696: 2687: 2686: 2676: 2666: 2642: 2633: 2630: 2619: 2618: 2608: 2598: 2566: 2555: 2554: 2536: 2512: 2506: 2505: 2495: 2486:(20): 12735–44. 2471: 2465: 2464: 2446: 2422: 2413: 2412: 2376: 2367: 2366: 2348: 2346:10.1038/35053181 2316: 2307: 2306: 2287:10.1038/376307a0 2273:(6538): 307–12. 2262: 2253: 2252: 2242: 2217:(6): 3126–3149. 2202: 2193: 2189: 2180: 2179: 2169: 2159: 2127: 2118: 2117: 2091: 2082: 2081: 2071: 2061: 2037: 2028: 2027: 2017: 2007: 1983: 1977: 1976: 1966: 1956: 1932: 1926: 1925: 1889: 1874: 1873: 1863: 1853: 1829: 1818: 1817: 1807: 1775: 1764: 1763: 1753: 1729: 1720: 1717: 1704: 1703: 1693: 1667: 1643: 1630: 1629: 1619: 1601: 1577: 1568: 1567: 1531: 1525: 1524: 1514: 1504: 1487:(11): e1000975. 1472: 1461: 1460: 1432: 1416: 1403: 1402: 1376: 1356: 1325: 1324: 1288: 1273: 1272: 1262: 1238: 1229: 1228: 1226: 1202: 1189: 1188: 1170: 1152: 1121:reviewer reports 1114: 1107: 892: 695: 693: 692: 687: 679: 678: 625:numerical solver 541:using a kinetic 288: 286: 285: 280: 278: 277: 253: 251: 250: 245: 243: 242: 166:scaffold protein 4193: 4192: 4188: 4187: 4186: 4184: 4183: 4182: 4173:Enzyme kinetics 4123: 4122: 4121: 4075: 4074: 4070: 4016: 4015: 4011: 3967: 3966: 3962: 3939:(24): 3067–74. 3926: 3925: 3921: 3877: 3876: 3872: 3842: 3841: 3837: 3807: 3806: 3802: 3760: 3759: 3755: 3711: 3710: 3703: 3651: 3650: 3646: 3594: 3593: 3589: 3552:(6078): 175–9. 3540: 3539: 3532: 3480: 3479: 3475: 3448:Curr. Bioinform 3445: 3444: 3440: 3430: 3429: 3425: 3381: 3380: 3376: 3345:(17): 2289–91. 3332: 3331: 3324: 3280: 3279: 3275: 3221: 3220: 3213: 3173: 3168: 3167: 3163: 3158: 3151: 3105: 3104: 3095: 3071:10.1.1.704.7634 3055: 3054: 3045: 3017: 3016: 3012: 3007: 3000: 2948: 2947: 2943: 2913: 2912: 2908: 2884:10.1.1.466.8001 2868: 2867: 2863: 2823: 2822: 2813: 2759: 2758: 2754: 2748: 2744: 2698: 2697: 2690: 2644: 2643: 2636: 2631: 2622: 2568: 2567: 2558: 2514: 2513: 2509: 2473: 2472: 2468: 2424: 2423: 2416: 2378: 2377: 2370: 2331:(6818): 391–5. 2318: 2317: 2310: 2264: 2263: 2256: 2204: 2203: 2196: 2190: 2183: 2142:(12): 5747–52. 2129: 2128: 2121: 2114: 2093: 2092: 2085: 2039: 2038: 2031: 1985: 1984: 1980: 1934: 1933: 1929: 1891: 1890: 1877: 1831: 1830: 1821: 1777: 1776: 1767: 1731: 1730: 1723: 1718: 1707: 1645: 1644: 1633: 1579: 1578: 1571: 1548:10.1038/nbt1054 1533: 1532: 1528: 1474: 1473: 1464: 1449: 1430:10.1.1.323.9577 1421:Systems Biology 1418: 1417: 1406: 1358: 1357: 1328: 1290: 1289: 1276: 1253:(17): 3289–91. 1240: 1239: 1232: 1204: 1203: 1194: 1143:(9): e1003844. 1124: 1110: 1108: 1101: 1084: 1035:T-cell receptor 884: 816:dendritic spine 803: 791:coarse-graining 747: 723:small molecules 710: 670: 665: 664: 661:phosphorylation 640:master equation 602: 585: 535:finite automata 460:process algebra 424: 366: 361: 340: 269: 264: 263: 234: 226: 225: 210:phosphorylation 206:phosphorylation 136: 116:DNA methylation 83: 78: 12: 11: 5: 4191: 4189: 4181: 4180: 4175: 4170: 4165: 4160: 4158:Cell signaling 4155: 4150: 4145: 4140: 4135: 4125: 4124: 4120: 4119: 4068: 4009: 3960: 3933:Bioinformatics 3919: 3884:Bioinformatics 3870: 3851:(2): 291–309. 3835: 3800: 3753: 3701: 3644: 3607:(4): 1809–17. 3587: 3530: 3473: 3454:(3): 315–320. 3438: 3423: 3374: 3339:Bioinformatics 3322: 3287:Nature Methods 3273: 3211: 3184:(2): 471–503. 3161: 3149: 3093: 3043: 3030:(7): 578–589. 3010: 2998: 2941: 2906: 2861: 2834:Bioinformatics 2811: 2772:(16): 6453–8. 2752: 2742: 2688: 2634: 2620: 2556: 2507: 2466: 2414: 2368: 2308: 2254: 2194: 2181: 2119: 2112: 2083: 2029: 1978: 1941:BMC Biophysics 1927: 1894:Nature Methods 1875: 1819: 1784:Bioinformatics 1765: 1738:Bioinformatics 1721: 1705: 1631: 1569: 1526: 1462: 1447: 1404: 1374:10.1.1.83.1561 1361:Science's STKE 1326: 1274: 1247:Bioinformatics 1230: 1191: 1100: 1097: 1096: 1095: 1090: 1083: 1080: 1077: 1076: 1074: 1071: 1068: 1065: 1062: 1061: 1059: 1056: 1053: 1050: 1047: 1046: 1044: 1041: 1038: 1032: 1029: 1028: 1026: 1023: 1020: 1011: 1008: 1007: 1005: 1000: 997: 991: 988: 987: 985: 982: 979: 976: 973: 972: 970: 965: 962: 959: 956: 955: 953: 950: 947: 941: 938: 937: 935: 932: 929: 926: 923: 922: 920: 917: 914: 911: 908: 907: 904: 901: 898: 895: 883: 880: 802: 799: 746: 743: 709: 706: 685: 682: 677: 673: 601: 598: 584: 581: 566:polymerisation 423: 420: 415: 414: 411: 408: 405: 402: 399: 396: 393: 378:conformational 365: 362: 360: 357: 339: 336: 276: 272: 241: 237: 233: 135: 132: 100:ligand binding 82: 79: 77: 74: 36:macromolecules 13: 10: 9: 6: 4: 3: 2: 4190: 4179: 4176: 4174: 4171: 4169: 4166: 4164: 4161: 4159: 4156: 4154: 4151: 4149: 4146: 4144: 4141: 4139: 4136: 4134: 4131: 4130: 4128: 4115: 4111: 4106: 4101: 4096: 4091: 4088:(3): 506–44. 4087: 4083: 4079: 4072: 4069: 4064: 4060: 4055: 4050: 4045: 4040: 4036: 4032: 4028: 4024: 4020: 4013: 4010: 4005: 4001: 3996: 3991: 3987: 3983: 3979: 3975: 3971: 3964: 3961: 3956: 3952: 3947: 3942: 3938: 3934: 3930: 3923: 3920: 3915: 3911: 3906: 3901: 3897: 3893: 3889: 3885: 3881: 3874: 3871: 3866: 3862: 3858: 3854: 3850: 3846: 3839: 3836: 3831: 3827: 3823: 3819: 3815: 3811: 3804: 3801: 3796: 3792: 3788: 3784: 3780: 3776: 3772: 3768: 3764: 3757: 3754: 3749: 3745: 3740: 3735: 3731: 3727: 3724:(2): 448–64. 3723: 3719: 3715: 3708: 3706: 3702: 3697: 3693: 3688: 3683: 3679: 3675: 3671: 3667: 3664:(5): 055009. 3663: 3659: 3655: 3648: 3645: 3640: 3636: 3631: 3626: 3622: 3618: 3614: 3610: 3606: 3602: 3598: 3591: 3588: 3583: 3579: 3575: 3571: 3567: 3563: 3559: 3555: 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