Technical aspects of modelling; networks and properties of individual reactions
Dynamic modelling of yeast biochemical networks employs discrete models such as Boolean networks and Petri nets, as well as continuous models, such as ordinary differential equation (ODE) systems. Simulation techniques for network dynamics can be deterministic or stochastic. ‘Deterministic’ means that if one state is known, then all following states are known in principle, while stochastic simulation calculates the probability of successive states. One may describe the dynamics of the same pathway with different approaches, which is illustrated in Figure 2 using the MAP kinase cascade as an example.
Figure 2. Mitogen-activated protein kinase (MAPK) cascades are ubiquitous in signalling cascades. (A) They consist of three or four different proteins that specifically catalyse the phosphorylation of the subsequent proteins. According to their roles, these kinases are called MAP kinase (MAPK), MAP kinase kinase (MAPKK) and so on, here denoted with MK1 to MK4, respectively. The dephosphorylation is ensured by phosphatases that are often less specific, but can also be very specific to certain targets. In some cases, the MAP kinases bind to a scaffold protein, forming a complex. The dynamics of such a pathway depend on the specific model and its parameters and the input function, i.e. the activation of upstream components (here MK4). (B) Boolean model. (C–E) Various ODE models. Parameters: k = 1, p = 1, Vmf = Vmb = 1, Kmf = Kmb = 0.1
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Boolean models in their basic form are deterministic models that assign discrete values to the state of a compound and discrete time points for updating this state. Update rules are based on a network of interactions between the nodes, i.e. the compounds. They are often used to describe gene expression events. A value of 0 denotes the off-state of a gene (no or low expression) and a value of 1 means the on-state (high expression). Boolean modelling, for example, has been applied to model yeast cell cycle progression4.
A Petri net is a mathematical representation of a (biochemical) network that consists of places (here indicating compounds), transitions (here indicating reactions), and directed arcs, which run between places and transitions. Places contain a number of tokens indicating the number of objects. A biochemical reaction is computed as the firing of a transition, i.e. the redistribution of tokens from one place to another via a transition according to specified rules.
ODE systems are the most frequently applied technique for modelling of dynamic processes56. They describe the temporal changes of compound concentrations that are due to production, degradation, modification or transport. The rate of change of the concentration ci of the ith compound reads:
where r is the number of reactions in the network, vj the rate of the jth reaction and nij the stoichiometric coefficient of ci in reaction j. For convenience, one may put all concentrations, rates and coefficients in vectors or matrices, such as:
The rates are functions of the concentrations c of compounds or modifiers, such as enzymes, and of parameters p (including kinetic constants and fixed concentrations of external compounds):
These rates can be described by different types of kinetic laws. Those rate laws are in turn the result of a modelling effort and are based on a set of assumptions, as indicated for the specific rate laws discussed below. Most of the rate laws are based on the mass action law35, stating that the rate of a reaction is proportional to the probability of collision of its reactants, which in turn is proportional to their concentrations ci to the power of molecularity mi:
where means the product overall for i = 1, .., n. Here, mi is an integer. It is zero if ci is not a substrate of vj, and 1, 2, … otherwise.
In experiments, rates often do not show the linear dependence of rate on concentrations, as assumed in mass action laws. This fact is acknowledged in power law kinetics used in the S-systems approach90, 91. Here, the rate reads:
where the concentrations and rates are normalized to some standard value denoted by superscript 0 and gi, j is a real number instead of an integer, as in equation (4). The normalized rates and concentrations are dimensionless quantities. The power law kinetics can be considered as a generalization of the mass action rate law. The exponent gi, j is equal to the concentration elasticities εj, i,42 i.e. the scaled derivatives of rates with respect to substrate concentrations:
In traditional enzyme kinetics, one considers separate steps of reactant binding to and release from the enzyme and even isomerization of enzyme–reactant complexes, which leads to the classical Michaelis–Menten expression3, 68. For one reactant and irreversible conversion, this reads:
where Vmax, j is the maximal rate of reaction j and Km, (i, j) is the Michaelis constant of reactant i in reaction j, i.e. the concentration of reactant i leading to half-maximal rate. A basic assumption is that the enzyme concentration Ej is much lower than the reactant concentrations, thereby allowing for a quasi-steady state of the enzyme-bound complexes. Here, the rate is proportional to the enzyme concentration, which is hidden in the maximal rate (Vmax, j = Ej·kcat, with kcat turnover number of enzyme j). This approach can be extended to cases with several reactants, reversible conversion of substrate into products, various sequences of reactant binding, or the effect of modifiers (see textbooks such as Cornish-Bowden14 and Klipp et al.55). Simplifications of complicated rate laws for multiple reactants with a high number of parameters lead to convenience kinetics67.
In metabolic modelling studies, approximative kinetic formats are used (for a recent review, see Heijnen38). One assumes that the rate is proportional to the enzyme concentration. The rates [equation (3)] are normalized with respect to a reference state, which is usually a steady state. This leads to the general expression:
where the superscript 0 denotes the values in the reference state and Ec is the matrix of concentration elasticities given in equation (6). One example is so-called ‘linear-logarithmic kinetics’:
where I is the r × r identity matrix. Another example is an approximation of the power law kinetics:
Sigmoid dependence of rates on reactant concentrations can sometimes be observed, e.g. if the enzyme has several subunits. They are described by Hill kinetics
(where n is called the Hill coefficient) or by more detailed kinetic laws, such as those suggested by Monod and colleagues69 and Koshland and colleagues60.
Most of these kinetic expressions have been developed for metabolic reactions catalysed by enzymes and they are subject of specific assumptions. Currently they are used to describe various types of reactions and protein–protein or protein–nucleic acid interactions. In some cases, their appropriateness remains to be proved. This can be done as soon as their respective kinetics can be assessed experimentally.
The stoichiometric matrix N introduced in equation (2) contains the information about the network structure in the sense of producing and degrading reactions. It does not describe activating or inhibiting interactions by compounds that are not reactants of the considered reaction. Such information is stored in the rate laws and hence in the elasticity coefficients (equation 6). Analysis of the matrix N provides two additional results. First, solving the steady state equation in terms of the rates vj is mathematically equivalent to solving the matrix equation N·K = 0. Every admissible set of fluxes, i.e. rates in steady state, is a linear combination of the columns vectors of K, which is a representation of the Kernel space of N, i.e. the space of solutions of the matrix equation. Solving the matrix equation G·N = 0 provides all conservation relations of the system at any state expressed as G·c = const.
ODE systems for biochemical networks can be simulated by using general purpose mathematical tools, but there exist also a number of specific tools that simplify this task (for an overview, see Klipp et al.56). Note that some of these tools allow switching from a quasi-continuous and deterministic simulation to a discrete and stochastic simulation of one and the same model, provided that all rates follow mass action kinetics.