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Quantitative and logic modelling of molecular and gene networks

Key Points

  • Computational models of molecular and gene networks are now commonplace. They are becoming larger and more complex, and are based on various approaches. Standard formats permit the sharing and reuse of models for different purposes.

  • Different types of representations of biological processes provide different levels of insight. The choice of representation affects the modelling and simulation methods, as well as the processing of data for model building and validation.

  • A model can be based on prior information gathered from the literature or pathway databases. Alternatively, models can be based on empirical data and the regulatory networks inferred from measurements.

  • Quantitative models can be developed at different levels of granularity, and such simulations provide quantitative temporal predictions.

  • Logic models are increasingly being used in cases in which a lack of quantitative information prevents the use of chemical kinetics approaches.

  • Modelling of entire cells requires the use of modular models based on different approaches and simulation procedures.

Abstract

Behaviours of complex biomolecular systems are often irreducible to the elementary properties of their individual components. Explanatory and predictive mathematical models are therefore useful for fully understanding and precisely engineering cellular functions. The development and analyses of these models require their adaptation to the problems that need to be solved and the type and amount of available genetic or molecular data. Quantitative and logic modelling are among the main methods currently used to model molecular and gene networks. Each approach comes with inherent advantages and weaknesses. Recent developments show that hybrid approaches will become essential for further progress in synthetic biology and in the development of virtual organisms.

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Figure 1: Granularity of time representation and variable values for various modelling approaches.
Figure 2: The four views of systems biology.
Figure 3: Network inference methods.

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Acknowledgements

The author is grateful to L. Calzone and M. Froehlich for extensive reading and correction of the manuscript, to L. Stephens and S. Edelstein for their corrections and advice, and to C. Chaouiya, P. Mendes, J. Saez-Rodriguez and D. Thieffry for help with the bibliography.

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Mathematical models

Descriptions of a system using mathematical concepts and language. Models are composed of a set of variables and a set of equations that establish relationships between the variables.

Numerical simulations

Reproductions of the behaviour of a system, obtained by iteratively computing the values of variables in a mathematical model over a certain number of time steps.

Parameter values

The temporal evolution of model variables (for example, protein concentrations) is affected by the values of other variables and by parameters such as dissociation constants, kinetic rate constants and reaction orders. Parameter values affect the dynamic behaviour of model variables.

Initial conditions

Values for the model variables at the start of numerical simulations. These initial conditions might affect the simulation results — for instance, in the case of systems with several stable states that can be reached from different trajectories.

Quantitative models

Mathematical models in which the values of the variables are determined by numerical analysis of the variables and parameters in the system.

Chemical kinetics

The study of rates of chemical processes and how they affect the evolution of chemical compounds in a system.

Open standards

Standards that are publicly available and that can be implemented without restriction by licensing terms. In computational biology, open standards are additionally developed by the community, and implementations are not subjected to fees.

Systems Biology Graphical Notation

(SBGN). A set of standardized symbols to represent the entities included in a biochemical network and their relationships. The notation is formed of three complementary languages to represent activity flows, entity relationships and process descriptions.

Bipartite graphs

Graphs that contain two types of nodes, in which nodes of one type are only connected to nodes of the other type. For example, in a metabolic network, nodes representing biochemical species connect to nodes representing reactions.

Systems Biology Markup Language

(SBML). A format to encode mathematical models that is used in systems biology. Although initially focused on non-spatial, reaction-based biochemical models, the language now features packages covering different modelling approaches. SBML is supported by software libraries in different programming languages and can be imported or exported by hundreds of modelling and simulation tools.

Biological network interference

A procedure whereby an unknown set of biological interactions and processes is deduced from the molecular phenotypes it produces: for instance, a list of gene expression, of molecular concentrations or of phenotypes on perturbation.

Information theoretic methods

Inference methods based on the information theory. Variables (nodes) are linked in a network if information about one variable (for instance, the distribution of its values) is affected by the knowledge of the values of the other.

Bayesian inference

A method of inference using Bayes' theorem to evaluate the probability of a network given a data set, as a function of the probability that this network produces the data set, the chance probability of this network and the chance probability of the data set.

Logic models

Mathematical models in which the discrete values of variables are determined by logical combinations of the values of other variables.

Ordinary differential equations

(ODEs). Equations describing the change of a variable in a system over time as a function of the values of other variables and parameters in the system. In a model of a biochemical systems, the ODEs are derived from the combination of the different processes in which the entity represented by the variable is involved.

Stochastic simulation

Simulation of a model in which each process has a certain probability to occur. Examples of stochastic simulations are solutions of stochastic differential equations in which noise factors are added to otherwise deterministic ordinary differential equations, and dynamic Monte Carlo simulations in which reaction rates are sampled from distributions.

Reaction order

The order of a reaction for a given reactant is defined as the exponent to which its concentration is raised in the rate law that characterizes the reaction. In the case of reactions taking place in a well-stirred, diluted medium, the reaction order of a molecular species is equal to its stoichiometry for this reaction.

Mass action law

A law stating that the velocity of a reaction is proportional to the concentration of the reactants it consumes raised to the power of their stoichiometry. For instance, the rate of a reaction consuming two molecules of A and one molecule of B will be proportional to [A]2 × [B].

Henry–Michaelis–Menten kinetics

A kinetic scheme used in enzymatic reactions. If the formation of an enzyme–substrate complex is faster than the formation of the enzymatic product or if the concentration of enzyme–substrate complex is constant, one can explicitly avoid representing the enzyme–substrate complex. The rate of formation of the enzymatic product is then proportional to the fraction of enzyme bound to the substrate: that is, [E] × [S] / (Km + [S]), where Km is the concentration of substrate necessary to achieve half the maximal reaction velocity.

S-systems

Modelling approaches for biochemical systems in which the creation and destruction of molecular species are expressed as products of the concentration of all of the molecular species in the systems raised to a phenomenological order (obtained by fitting the model to experimental data).

Global optimization

A branch of numerical analysis that deals with the global optimization of a function or a set of functions according to some criteria. Examples of global optimization problems in biological network modelling are parameter estimation and flux balance analysis.

Identifiable model

A model in which the values of its parameters can be unambiguously determined by the data sets available. A model is non-identifiable if alternative sets of parameter values can fit the data sets.

Attractors

Stable behaviour of a system, as reflected by a fixed trajectory in the space of all possible states of the system. Examples of attractors are periodic behaviours (for example, oscillations) and steady states.

Fuzzy logic

Approximate logic computation in which the variables can have partial truth values ranging from 0 (false) to 1 (true).

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Le Novère, N. Quantitative and logic modelling of molecular and gene networks. Nat Rev Genet 16, 146–158 (2015). https://doi.org/10.1038/nrg3885

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