The model gives a better understanding of the rate limiting steps in the process of activating the XlnR regulon and therefore, helps to define the biological control points. Similarly this knowledge can be used to obtain strains that have enhanced xylanolytic enzyme production. These enzymes are industrially of importance as food and feed additives, but are also part of a system that is used to bleach paper pulp. Given that the transcription rate and degradation rates have been shown to be the key parameters that dictate the systems dynamics for the XlnR regulon; this information is important for designing and sampling of time course experiments. Once the transcripts are unstable, the proteins get quickly degraded; otherwise they remain stable. This observation is linked to the D-xylose uptake in fermentation experiments. The consumption of D-xylose also indirectly controls the regulation of the target genes and therewith the breakdown of sugars.

Simulations showed that the dynamics of the D-xylose input function considered in the examples has an important effect on the profiles of the individual metabolite concentrations. This is particularly dictated by the value of the parameters in the external input function *u*(*t*). The larger the value of the *K*, the faster the consumption of D-xylose. This depends on the chemical reactions taking place in any given cell, or the saturation levels of the individual compounds in a cell.

Feedback significantly affects the response of the output profiles for the metabolites and changes the final steady state values (Figure 3). Further simulations showed that variations of the time delay in the feedback loop (*τ* = 1, 2,…, 5 hours) have a small effect on the pattern of the response (Figure 4). The stability analysis subsection shows that the metabolite response dynamics exhibits no oscillatory behavior for a network without feedback loop. For a network with feedback loops, the results from numerical analysis showed that feedback conditions for which the system is stable or for which the system exhibits oscillatory behavior can be obtained (Figure 7 (A)-(C)). In modeling the feedback loop, time delay was accounted for and included in the model.

According to Bliss *et al*. [16] and Thomas and d’Ari. [35], including time delay in modeling biological networks is considered important because many biological systems exhibit some delays in their feedback loops. However, according to our finding (Figure 4), incorporating the time delay had no strong effect on the overall dynamics of the metabolite expression profiles.

The analysis shows that the existence or absence of oscillatory behavior is dictated by the numerical values of the individual mechanistic parameters. The conditions for oscillatory behavior follow from the eigenvalue spectra. The eigenvalue spectra analysis like that in (20) and the corresponding conditions for which all the eigenvalues are less than zero, gives also indication for the stability properties for the XlnR network with feedback loop. If all the eigenvalues satisfy the condition
for all entries, then the system is stable, otherwise it is unstable.

Two scenarios can be considered: one in which the proteins involved in the feedback loop are activating and the other in which the proteins are repressing. The details of the expected behavioral dynamics from such a system requires a case by case analysis (like in Figure 6 (A)-(C)) of the effects of the proteins in the feedback loop. A similar analysis can be extended to study the stability in case of a combined effect of any two or more proteins of interest. When the number of network components become large, obtaining explicit analytic solutions and expressions for the eigenvalues and other quantities of interest increasingly become complex - in which case the alternative of numerical methods can be used (see Figure 6 (A)-(C) ). Thomas and d’Adri [36], and Thomas *et al*. [35] investigated the properties of mathematical Boolean net Modeling Genetic Networks works - investigations that provided significant insight into genetic network dynamics. In their work they showed the importance of NFB loops for maintaining homeostasis in levels of gene products. Our analysis leads to the observation that having a NFB loop stabilizes the response of the metabolite expressions (Figure 6 (A)-(C)). However, there exists certain ranges of values of the strength of feedback effects that make the system unstable. This sets constraints to the feasible parameter for the system if instability is not observed. In some cases having a PFB loop yields a stable network response, Figures 6 (B) and (C). This result is in agreement with that found in a study by Maithreye *et al*. [34]. In their investigations they found that NFB loops provide stability and withstand considerable variations and random perturbations of biochemical parameters.

The effect of time delay on stability can be analyzed from a transfer function of the model in the ”*s*” domain, or by a transformation to the ”*z*” domain. In these cases the delay time is considered as a finite dimensional system. Stability analysis can be done by searching for stability properties in the ”*s*” domain or ”*z*” domain. Examples of other methods that deal with the delay times are given for state estimation in the work by Liu *et al*. [37] and Yu *et al*. [38].

The adaptive filtering approach developed in [38] is based on the adaptive synchronization setting, for estimating unknown delayed genetic regulatory networks with noise disturbance. Using this approach, no exclusive knowledge of system parameters is required, e.g. those lacking in the XlnR regulon and many other biological networks. Liu *et al*. [37] proposed an adaptive feedback control approach for simultaneously identifying unknown (or uncertain) network topological structure, unknown parameters of uncertain general complex networks with time delay from available mRNA data and estimation of protein concentration. The effectiveness and applicability of their approach was shown using *in silico* numerical simulations. In contrast to [37] and [38], we study the XlnR regulon dynamics and do not focus on the system structure identification and parameter estimation.

According to Balsa-Canto *et al*. [39], powerful mathematical analytic tools highlight the value for successful study of many biological systems. However, such success can mainly be attributed to the unrelenting endeavors for an in-depth understanding of both computational methods and the biological problems of interest. For the case of the XlnR regulon, our analysis provides a basis for understanding the behavioral dynamics of genes and proteins after network perturbation. This will form a basis for future wet-lab experiments, particularly with the genes from the XlnR regulon. Given that the metabolite expression dynamics are known, this study provides a basis for strategic thinking in line with experimental design. The modeling approach used in this paper provides good information for understanding network behavioral dynamics particularly for small-sized networks. This is illustrated by the XlnR regulon in which even the simplest of structures can yield interestingly complex dynamics. Therefore, a reasons for having limited our focus to the regulon dynamics. Having detailed information regarding the basal parameters and the other mechanistic parameters might further improve the analysis and investigations into the network dynamics. Nevertheless, with informed parameter guesses, simulation studies provide good information into the systems behavior.