Multi-state modeling of biomolecules refers to a series of techniques used to represent and compute the behaviour of biological molecules or complexes that can adopt a large number of possible functional states.
Biological signaling systems often rely on complexes of biological macromolecules that can undergo several functionally significant modifications that are mutually compatible. Thus, they can exist in a very large number of functionally different states. Modeling 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, modelers have in recent years moved away from explicit specification of all possible states, and towards rule-based modeling that allow for implicit model specification, including the κ-calculus, BioNetGen, 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 ordinary differential equations, partial differential equations, or the Gillespie stochastic simulation algorithm. Given current computing technology, particle-based methods are sometimes the only possible option. Particle-based simulators further fall into two categories: Non-spatial simulators such as StochSim, DYNSTOC, RuleMonkey, and NFSim and spatial simulators, including Meredys, SRSim and MCell.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 Raton. 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.
Introduction
Multi-state biomolecules in signal transduction
In living cells, signals are processed by networks of proteins 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 post-translational modifications, ligand binding, conformational change, or formation of new complexes. Similarly, nucleic acids can undergo a variety of transformations, including protein binding, binding of other nucleic acids, conformational change and DNA methylation.
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 the number of possible modifications, a phenomenon known as "combinatorial explosion". This is of concern for computational biologists who model or simulate such biomolecules, because it raises questions about how such large numbers of states can be represented and simulated.
Examples of combinatorial explosion
Biological signaling networks incorporate a wide array of reversible interactions, post-translational modifications and conformational changes. Furthermore, it is common for a protein to be composed of several - identical or nonidentical - subunits, 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.
For instance, it has been estimated that the yeast scaffold protein Ste5 can be a part of 25666 unique protein complexes. In E. coli, chemotaxis 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 methylation sites, resulting in billions of potential states. The protein kinase CaMKII is a dodecamer of twelve catalytic subunits, arranged in two hexameric rings. Each subunit can exist in at least two distinct conformations, and each subunit features various phosphorylation and ligand binding sites. A recent model incorporated conformational states, two phosphorylation sites and two modes of binding calcium/calmodulin, for a total of around one billion possible states per hexameric ring. A model of coupling of the EGF receptor to a MAP kinase cascade presented by Danos and colleaguesDanos 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 accounts for distinct molecular species, yet the authors note several points at which the model could be further extended. A more recent model of ErbB receptor signalling even accounts for more than one googol () distinct molecular species. The problem of combinatorial explosion is also relevant to synthetic biology, with a recent model of a relatively simple synthetic eukaryotic gene circuit featuring 187 species and 1165 reactions.
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 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 a priori. In such cases, computational modeling can be used to uncover to what extent the different states are populated.
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 or complex, in the same way that the existence of just a few choices in a coffee shop (small, medium or large, with or without milk, decaf or not, extra shot of espresso) 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 reaction rate. 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.
Specification vs computation
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 governing those changes in a robust and efficient way. This problem is called the "specification problem". The second problem concerns computation. 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.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), Springer.
The specification problem
Explicit specification
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 simulation framework that allows transitions from state to state. For instance, if a protein can be ligand-bound or not, exist in two conformational states (e.g. open or closed) and be located in two possible subcellular areas (e.g. cytosolic or membrane-bound), then the eight possible resulting states can be explicitly enumerated as:
bound, open, cytosol bound, open, membrane bound, closed, cytosol bound, closed, membrane unbound, open, cytosol unbound, open, membrane unbound, closed, cytosol unbound, closed, membrane
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 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.
Rule-based model specification
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 system increases. Modelers have therefore looked for implicit, rather than explicit, ways of specifying a biological signaling system. An implicit description is one that groups reactions 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 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.
For instance, the rate of ligand dissociation 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 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".