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We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.

Cancer is a complex disease, triggered by multiple mutations in various genes and exacerbated by a number of different behavioural and environmental factors. Some risk factors associated with possible onset and development of cancer are preventable, such as inappropriate diet, physical inactivity, smoking, and drinking [

Recent advances in microarray and high-throughput sequencing technologies have provided pathways for measuring the expression of thousands of genes and mapping most crucial genes and groups of genes controlling different types of cancer. The networks of interactions between DNA, RNA, proteins, and molecules are defined as

GRNs are usually formalised as networks (undirected or directed) where the nodes represent individual genes, proteins, and so forth and the edges correspond to some form of regulation between the nodes. In order to make progress in understanding the onset and development of cancer, as well as to develop effective drug targets, it is essential to be able to reconstruct GRNs pertinent to particular types of cancer from available data. Yeh et al. [

In the next section we survey and compare different approaches to model the dynamics of GRNs, making an emphasis on particular biological features that can be best represented by each of the methods. Section

In the analysis of gene regulatory networks and their dynamics, the first step is the identification of key modules or components and possible relations between them, which is often done by interrogating available expression data. Once the topology of the GRN has been fixed, the next step in modelling the dynamics is making realistic assumptions about specific rules that govern the expression of particular genes. Depending on the level of understanding of underlying processes, the complexity of the GRN under investigation, and the specific questions to be addressed, there are several methodologically different approaches that can be employed. Endy and Brent [

Some of the first models developed for modelling GRNs were the so-called

Significant advantage of using Boolean networks to model GRNs lies in the fact that they allow one to consider networks with a very large number of nodes. At the same time, there are several deficiencies in this approach. The first one concerns the fact that since the gene states only admit the values of ON or OFF, this formalism does not take into account intermediate stages of gene expression [

Due to intrinsic imprecision and uncertainty associated with gene expression data, it may be appropriate to move away from precise rules of Boolean logic in favour of machine learning techniques based on fuzzy logic. The basic idea is that, rather than trying to reconstruct some assumed fixed gene network topology, one considers the whole family of possible networks with all possible distributions of links between nodes. The problem lies in using actual data to assign appropriate probabilities to each of these configurations, so that for a given input the fuzzy network would provide an output that most resembles actual data. A significant advantage of fuzzy logic for inferring the structure of GRNs lies in their ability to rely on already available knowledge of biological relations between different nodes in the network and, at the same time, being able to recover important previously unknown connections. On the other hand, fuzzy methods for GRN inference are characterised by a high level of computational complexity.

To give a few examples, fuzzy approach has been used to analyse microarray data from the yeast cell cycle and to recover a set of GRNs, with

A very powerful and mathematically insightful methodology for analysis of GRNs is based on nonlinear ordinary or delay differential equations (ODEs or DDEs). In this approach, a gene regulatory network is represented by concentrations of different mRNAs and proteins, and the dynamics can be written as a system of ODEs or DDEs using the law of mass action for individual reactions [

In order to more accurately represent a switch-like behaviour of the gene expression, several authors have developed models of GRNs using piecewise-linear differential equations, in which the continuous Hill function is replaced by a discontinuous step function [

In terms of applications to cancer, ODE models have explained aberrant dynamics of the NF-

Another aspect that has to be properly accounted for in dynamical models is the fact that transcription and translation during gene expression often take place over nonnegligible time periods. Monk [

Experimental evidence suggests that significant stochastic fluctuations are observed during gene expression and regulation; hence, in many cases it is paramount to use stochastic models for studying GRN dynamics [

Besides being an intrinsic feature of biological dynamics, stochasticity has proved to be important in the context of engineered genetic switches [

Another way to approach stochasticity in the analysis and reconstruction of GRNs is by using so-called

The rest of this paper is devoted to consideration of the effects of transcriptional and translational time delays on the dynamics of GRNs. In the next section we introduce the time-delayed model of a two-gene activation-inhibition network together with its quasi-steady state simplification and establish the well-posedness of both models. Section

To motivate the analysis of time-delayed effects on gene regulatory dynamics, following Polynikis et al. [

Due to the fact that the dynamics of mRNA is normally much faster than that of related proteins, one can use a quasi-steady state assumption to simplify CNM (

While the transcription and translation may be faster than characteristic times associated with significant changes in protein concentrations (of the order of 5 minutes for transcription + translation and 1 hour for a 50% change in the concentration of translated protein for

Before proceeding with the analysis, one has to augment models (

We now prove that solution

Steady states

In order to gain some first insights into the role of transcriptional and translational delays on the dynamics of GRN, we focus on the behaviour of the delayed simplified nonlinear model (DSNM) (

One should note that

The critical value of the time delay

If

Figure

Stability boundary of the steady state

In Figure

Stability boundary of the steady state

In a similar way, the effects of the transcription rates

Stability boundary of the steady state

Figure

Numerical solution of DSNM system (

Linearisation of the full nonlinear DCNM model (

The characteristic equation (

Let

From now on, we will assume that the condition in Lemma

Without loss of generality, suppose that (

Suppose the conditions of Lemma

The conclusion of Lemma

Figure

Stability boundary of the steady state

Numerical solution of DCNM system (

In this review we have discussed various mathematical models for the analysis of GRNs and focussed on the role played by the transcriptional and translational time delays in the dynamics of a two-gene activator-inhibitor GRN. By reducing the model to the one with a single time delay, we have considered possible behaviour in the quasi-steady state approximation of very fast mRNA dynamics, which has resulted in a lower-dimensional system of DDEs. Due to the presence of time delays, even this simplified model is able to exhibit loss of stability of the positive equilibrium through a Hopf bifurcation and a subsequent emergence of sustained periodic oscillations, which was not possible in the absence of the time delays, as discussed in Polynikis et al. [

We have also studied the stability of a positive steady state in the full system and showed that this steady state can also undergo a Hopf bifurcation depending on the time delay and system parameters. Our analysis extends an earlier result of Polynikis et al. [

The work presented in the paper can be extended in several interesting and important research directions. One possibility would be to account for the fact that in most experiments the transcriptional and translational time delay are not fixed but rather obey some form of a delay distribution. Recent work on the effects on delay distribution on system dynamics [

As it has already been mentioned, in some cases gene expression behaviour is characterised by a switch-like behaviour that can be better modelled using piecewise-linear rather than continuous transcription functions [

Besides providing insights into the dynamics of GRNs, there are several practical ways in which models similar to the one described in this review are helpful in monitoring and treatment of cancer. GRN models based on differential equations coupled with other techniques, such as machine learning and Bayesian networks, have proved effective in identifying specific oncogenes that can be used as biomarkers or drug targets [

The authors declare that there is no conflict of interests regarding the publication of this paper.