Ordinary Differential Equations

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Subsections

Ordinary Differential Equations

There is a long history of using systems of ordinary differential equations (ODEs) to model the reaction kinetics of regulatory systems. These approaches have several advantages. In principle, their more detailed representation of regulatory interactions provides a more accurate representation of the physical system under investigation. Additionally, there is a large body of dynamical systems theory that can be used to analyse such models. The primary disadvantage of ODE approaches is that they can be much more computationally intensive to analyse and solve than discrete models, especially for realistically sized systems.

Assumptions

Biological processes are almost inevitably highly complicated, and most mathematical models of gene regulation make two simplifying assumptions. The first of these is that the control of gene expression resides in the regulation of gene transcription. This assumption is known to be incorrect, as control may also be exercised at a number of other levels, including the post-transcriptional processing and translation of RNA and the control of RNA and protein degradation. While models have been developed that do investigate some of these processes, they are rarely integrated into a comprehensive framework. The second assumption is that genes are expressed and proteins produced at a continuous rate. Again, this assumption does not always hold. In some systems where the number of molecules involved is very small, the production and movement of individual molecules may be important, and there may be a degree of randomness. Stochastic approaches to modelling have been developed that reduce reliance upon this assumption, these are described in Section 6 below.

Model description

The basis for many ODE descriptions of regulatory systems is chemical rate equations, which describe the relationship between the rate of a reaction and the concentrations of the reactants. For example, consider a simple regulatory system in which a transcription factor X associates with an empty binding site Y₀ to give a bound site Y₁ at some rate k₁ and dissasociates at some rate k_-1. A bound site results in transcription and the production of a product P and an empty binding site (Y₀) at rate k₂. This system may be represented by the following rate equations:

$\begin{eqnarray} X + Y_0 & \rightleftharpoons^{k_1}_{k_{-1}} & Y_1 \\ Y_1 & \rightarrow^{k_2} & P + Y_0 \end{eqnarray}$

which can then be translated into the following set of differential equations:

$\begin{eqnarray} \frac{dx}{dt} & = & {-k_1}{x}{y_0} + {k_{-1}}{y_1} \\ \f... ... {k_{-1}}{y_1} - {k_2}{y_1} \\ \frac{dp}{dt} & = & {k_2}{y_1} \end{eqnarray}$

Introducing a number of assumptions: that the total number of bound and unbound sites is constant, y₀ + y₁ = b; and that the number of transcription factors is significantly higher than the number of binding sites, x $\gg$ b, such that all of the binding sites will generally be occupied, this set of equations can be simplified to:

$\begin{eqnarray} \frac{dx}{dt} & = & \frac{{-K_{max}}{x}}{k_n + x} \\ {K_{max}} & = & {k_2}{b} \\ k_n & = & \frac{{k_{-1}}+{k_2}}{k_1} \end{eqnarray}$

These equations correspond to the Michaelis-Menten kinetic scheme and describe a situation where the rate of expression increases with transcription factor availability up to some limiting value [38] (see Figure - graph).

Early work investigating the existence and properties of various steady, periodic and chaotic solutions to these sets of equations has been summarized in [132]. The equations above can be generalised to a set of reaction-rate equations in which the concentration of a gene product is described in terms of the concentrations of the other elements of the system:

$\begin{eqnarray} \frac{dx_i}{dt} & = & f_i(\mathbf{x}) \end{eqnarray}$

where x is the vector of gene product concentrations and f_i is an update function. This form of equation can be extended to include the influence of external input signals, product degradation and time delays:

$\begin{eqnarray} \frac{dx_i}{dt} & = & f_i(\mathbf{x}, \mathbf{u}) \\ \fr... ...)) \\ \frac{dx_i}{dt} & = & f_i(\mathbf{x}) - {\gamma_i}{x_i} \end{eqnarray}$

where u is a vector of input signals, $\tau$ is a time delay and $\gamma_{i}^{}$ is the degradation rate of product i. Another possible extension is to model transcription and translation as indpendent processes, in which the production of messenger RNA depends upon the concentrations of protein transcription factors and the production of proteins depends on the concentrations of messenger RNAs:

$\begin{eqnarray} \frac{dr_i}{dt} & = & f_i(\mathbf{p}) \\ \frac{dp_i}{dt} & = & g_i(\mathbf{r}) \end{eqnarray}$

where p and r are vectors of protein and mRNA concentrations respectively. In eukaryotic organisms, protein and mRNA are each produced in different cellular compartments and must be transported between them. An advantage of this approach is that it allows time delays due to mRNA and protein transport to be explicitly incorporated into a model [112].

A number of different functions have been used for f_i, the updating function. A common feature is their sigmoidal shape, which experimental evidence has suggested is plausible. Possibile updating functions include the hill curve and the logistic function, respectively:

$\begin{eqnarray} f(x_j, \theta_{ij}, m) & = & \frac{x^{\phantom{j}m}_j}{{x^{... ...hantom{j}m}_{ij}}} \\ f(x_j, m) & = & \frac{1}{1 + e^{-mx_j}} \end{eqnarray}$

where m > 0 is a steepness parameter and $\theta_{{ij}}^{}$ > 0 is a threshold for the influence of x_j on x_i.

Due to nonlinearity of the updating functions, analytical solutions are not normally possible. In some cases, qualitative properties can be established, such as existance of steady styates, limit cycles and critical points [131]. The analysis of feedback dynamics carried out by Thomas [125] (described in Section 4.2) can be extended to continuous systems.

Another approach is to simplify the equations by replacing non-linear sigmoidal functions with step functions, or some other form of piecewise-linear function as described in Section 4.3 above.

Finally, it is sometimes possible to use numerical techniques to solve sets of equations. In numerical simulation, the exact solution of an equation is approximated by calculating values for each of the state variables at a series of discretised time steps. A number of systems have been characterised and solved in this manner, some of which are described below. A significant problem with the numerical approach is the lack of measurement of the various kinetic parameters in a system. The number of systems for which detailed parameter values are known is very small, and the size of most systems makes it unfeasible to obtain in vitro or in vivo measurements of many parameter values. Some researchers have dealt with this problem by searching the parameter space of a system for combinations that allow the qualitative behaviour to be reproduced [138]. Another possible solution is to use the rapidly increasing amounts of available gene expression data to estimate parameter values, as described in the section on `reverse engineering' below.

Applications

Model systems

There exist only a small number of systems for which sufficient experimental data has been obtained to enable accurate models to be built. One of the best characterised systems is phage- $\lambda$ [95]. This system has been the subject of a number of mathematical models [109,102], including a hybrid model [82] and a stochastic model [9].

Other areas of modelling include the circadian clock [76], and the cell cycle [130]. Further models are reviewed in [57].

Reverse engineering of network structure

Many of the approaches to modelling and simulation described above focused on either characterising a small, well-known regulatory system, or exploring the possible behaviour of a particular class of model networks. The relatively recent development of high-throughput experimental techniques in molecular biology has opened a new avenue of investigation. For the first time, there exists sufficient data to potentially enable network structure and dynamics to be inferred automatically with little or no a priori knowledge. DNA microrarrays can be used to generate thousands of measurements of gene expression levels during the course of a single experiment. The reverse engineering approach begins with the assumption that the interactions between these genes can be modelled as a network and aims to infer these interactions from the expression data.

Two of the main problems hampering reverse engineering efforts are the highly complex, combinatorial nature of the problem, and the relatively poor information content of the available data. Whereas microarrays are capable of collecting data on a large number of genes, the number of data points for each gene is typically very small. Furthermore the data is typically very noisy.

Both discrete and continuous modelling formalisms have been used for the task of network induction, as well as a number of different approaches to parameter learning. A recent overview of the different models and learning strategies used is provided by van Someren and colleagues [134].

Forward engineering of novel networks

As some of the basic control modules in regulatory networks become more well understood, the construction of synthetic networks in vitro has become possible [56]. These novel networks not only have many potential therapeutic uses, they also allow understanding of regulatory processes to be refined. Systems constructed so far include a toggle switch [45] and an oscillator [41]. In addition, suites of networks have been created by randomly combining low level modules, allowing the combinatorial possibilities of synthetic networks to be explored [52].

The role of modelling in this process is to enable the behaviour of complex networks to be predicted via simulation before the circuit is implemented in vitro. Modelling formalisms with a high level of biological fidelity are therefore preferred, and several quantitative and semiquantitative approaches incorporating both deterministic and stochastic dynamics [65].

Next: Neural network models Up: Continuous models Previous: Continuous models

Nic Geard 2004-05-06