Systems and Control group, Wageningen University, Wageningen, P.O. Box 17, 6700 AA, The Netherlands

Laboratory of Systems and Synthetic Biology, Wageningen University, Wageningen, Dreijenplein 10, 6703 HB, The Netherlands

Abstract

Background

In this paper the dynamics of the transcription-translation system for XlnR regulon in

Results

Simulation and systems analysis showed significant influence of activating and repressing feedback on metabolite expression profiles. The dynamics of the D-xylose input function has an important effect on the profiles of the individual metabolite concentrations. Variation of the time delay in the feedback loop has no significant effect on the pattern of the response. The stability and existence of oscillatory behavior depends on which proteins are involved in the feedback loop.

Conclusions

The dynamics in the regulation properties of the network are dictated mainly by the transcription and translation degradation rate parameters, and by the D-xylose consumption profile. This holds true with and without feedback in the network. Feedback was found to significantly influence the expression dynamics of genes and proteins. Feedback increases the metabolite abundance, changes the steady state values, alters the time trajectories and affects the response oscillatory behavior and stability conditions. The modeling approach provides insight into network behavioral dynamics particularly for small-sized networks. The analysis of the network dynamics has provided useful information for experimental design for future

Background

The filamentous fungus

The XlnR regulon is activated by D-xylose in the culturing media

For the XlnR regulon, literature information on the network structure was used as a basis for the simulations. To our knowledge, currently very little has been done on modeling the dynamics of the XlnR regulon and also on time course profiling of the genes that constitute the XlnR regulon in

Generally, in the study of biological networks, positive feedback (PFB), negative feedback (NFB)

In computational systems biology, numerous studies have been done on genetic network reconstruction using time course data but little attention has been given to understanding the network dynamics. It is crucial to understand or at worst have a fuzzy idea of a biological network dynamics if one is to gain deeper insight into the biological network dynamics and functionality.

This paper concerns the analysis of the network dynamic behavior, the effect of feedback loops and the conditions under which oscillatory responses in metabolite expression may be exhibited are investigated. Modeling of the XlnR regulon is explored by using nonlinear differential equations and Hill functions for the transcription and linear reaction kinetics for the translation process. To ensure that detailed aspects of the transcription-translation model formalism are captured, some assumptions are incorporated in the modeling.

Applications of dynamical systems in modeling transcription regulatory networks can be found in

Methods

Regulation mechanism for the XlnR regulon

In the model organism

Gene regulation can take place at different stages of the central dogma of molecular biology (DNA→ RNA → Protein). These stages include among others transcription, translation and post-translational modifications (PTMs) of the associated protein. In Figure

The XlnR regulon scheme

**The XlnR regulon scheme** The XlnR regulon induced by D-xylose in the presence or absence of CreA. The representations P1 and TP are the proteins from the

Transcription model

Commonly, hyperbolic functions and the sigmoid class of functions are used to represent the kinetics of gene regulation

Let the vector **z** = [_{1},…,_{n}^{Τ}_{i}_{i}

where ^{–}(_{i}_{i}^{+}(_{i}_{i}_{i}_{i}_{i}

According to Hasper

where _{i}_{1} = 1/_{i}_{i}_{i}_{i}_{1} - effective affinity constant for gene 1 activating gene _{is}_{id}**x**_{0} - vector of initial mRNA concentration, _{i}**b** = [_{1},…,_{n}^{Τ}**u** = [_{1},…,_{n}^{Τ}

Translation model

A system of linear differential equations to model the protein abundance (translation process) is then considered. The linear model representations (3) are used to capture the dynamics of the translation process with both the production and degradation terms being linear.

where _{i}_{i}_{i}_{1} ≈ _{2} ≈ … ≈ _{n}_{i} = 0 for all

The steady state values in (4) are based on the assumption that, for a small time window the change in concentration of the input stimulus and metabolite concentrations remain nearly unchanged. The model specifications for the transcription and translation process describe the rates of change of concentration of the genes and proteins. Overall, the system of 2

System stability

The interesting case to analyze is the systems behavior in the absence of the inhibitor, CreA. Let us denote the equilibrium concentrations of mRNA and protein quantities by the vectors

Let ^{2n} → ℝ^{2n} be a set of smooth functions (with _{1},…,_{2n})) that capture the XlnR regulon system dynamics. In this case we have _{1} = _{1},…,_{n}_{n}_{n+1} = _{1},…,_{2n} = _{n}

This Jacobian matrix is used to assess the regulon stability and to identify which parameters dictate the transcript abundance. First, consider a case of three genes and three proteins,

where **x** = [_{1}, _{2}, _{3}]^{Τ}**z** = [_{1}, _{2}, _{3}]^{Τ}

where

for

In the case of this regulon, the derived characteristic polynomial turns out to be the same as the determinant, i.e.

The formulations of the Jacobian matrix and the eigenvalue spectra can be extended to an

for ^{+}, a positive integer. In the network without feedback, it turns out that the trace of the Jacobian matrix is equal to the sum of all the eigenvalues (i.e.

According to Aro

Feedback in the network

Numerous transcription systems are known to include genes that regulate their own expression values _{1}) has to be modified accordingly. The adapted equation is given by

where

is the repressor Hill function and _{A}_{A}**S**_{1} = {**S**_{2} = {**S**_{1} ⋃ **S**_{2} = {1,…,**S**_{1} and **S**_{2}, respectively. The effect of the D-xylose and the feedback loop is modeled as additive. Equation (11) also specifies the build up of proteins and repression or activation of the

xlnR gene promoter activity under feedback

Let us define the promoter activities by the expressions (13) and (14). The promoter activity corresponding to the case when an activating protein is involved in the feedback loop is represented by the term Γ_{A} and that for the case of a repressing feedback effect denoted by Γ_{R}.

The extracts from the denominator functions are given by (15) and (16), respectively.

These terms are used in the calculations for the activating and repressing promoter activity for the XlnR regulon. For the sake of illustrations, two target genes were considered (i.e. values of _{RL}_{AL}**S**_{1} and **S**_{2} of unit elements which index the proteins that are responsible for regulating the

Existence of oscillatory behavior

The eigenvalue spectra from the derived Jacobian matrix can be used for this analysis. The presence of at least a pair of eigenvalues with complex parts implies the existence of oscillatory behavior. The PTMs in the feedback loop may produce oscillatory behavior depending on the individual attributes of the target genes and the consequent proteins in the feedback loop.

We observed that in the absence of a feedback loop, the system dynamics is dictated by the degradation parameters. Active degradation of proteins or mRNA is a major part of many metabolic and stress response systems

Using the adapted model (11), the computed entry in the (1, _{1} with respect to the variable of interest within each of the sets **S**_{1} and **S**_{2}, we then have the more compact expression

This term corresponds to the repressing proteins, and the terms given by

for the activating proteins. The parameters _{AL}_{RL}_{1} ∈ ℝ\ {0}, a parameter that intrinsically represents the auto-regulation effect of the

where

are the conjugate roots. The eigenvalues _{5}(·) and _{6}(·) may take on values from the real space, ℝ or the complex space, ℂ. From (21) and (22) we observe that oscillation can only be obtained if the condition _{1} < –(_{1} – _{1d})^{2}/4_{1} for _{1} > 0 is satisfied. This condition on _{1} signifies a contribution from a NFB loop in the XlnR regulon network. Notice that the expression (_{1} – _{1d})^{2} > 0 for all values of _{1} and _{1d}. This finding adds to consolidate the findings by Tiana

The presence or absence of oscillatory behavior is insufficient for drawing conclusions about stability in system responses. Stability, using (21) and (22) exists if the conditions Re(_{5}(·)) < 0 and Re(_{6}(·)) < 0 are simultaneously fulfilled. Hence, the inequality

Solving the inequality (23) for _{1} leads to the condition _{1} <_{1}_{1d}/_{1}. A similar analysis for the existence of oscillatory behavior and stability dynamics can be done for the other proteins in the feedback loop, for example at position (1, 5) and (1, 6) or combinations in the matrix (17). However, although such analysis is conceptually simple, the analytic expressions are very complex to work with. Information about the stability and oscillatory behavior is obtained by numerical solutions. An example is considered to investigate the time evolution of gene activity and protein abundance in the XlnR regulon.

Bifurcation analysis

Bifurcation analysis relates to stability on the system parameters. Stability properties for the system without feedback are given by (10), where it was shown that the roots of the characteristic polynomial correspond to the degradation rate constants for the mRNA expression and protein abundance. As these constants are positive, the system is always stable. In the case that one of them equals to zero, then the system is critically stable.

For the network with feedback loop, consider the conjugate roots in (21) and (22) denoted by _{i}_{i}_{i}

or after working out becomes _{1} = _{1}_{1d}/_{1}. This example illustrates the case of a feedback in the cell at position(1, 4) of the matrix in (17). The analysis for the other entries of

Results

System specification

The analysis is illustrated by an example case. Consider a regulon network of three genes given a perturbation of D-xylose. The pulse perturbation takes place at time ^{Kt}_{1} = 1, _{1} = 2_{2} = 2.5_{3} = 1_{1d} = 0.5, _{2d} = 0.4, _{3d} = 0.3, _{2s} = 5, _{3s} = 6, _{21} = 0.1, _{31} = 0.1, _{1} = _{2} = _{3} = 0.5, _{1} = 1, _{2} = 1 and _{3} = 1.

Stability and response analysis - without feedback

The expression for the characteristic polynomial, ** P**(·) in (10) is independent of the translation rate parameters

In Figure _{R1} = 1/_{1d} ≈ 2 hours is noticed for the master regulator and for the target genes, _{R1} <_{R2}, _{R3}. The relaxation time is an approximation for the time required for the system to relax into steady state. This represents the time it takes a system to react to a persistent external input (D-xylose).

D-xylose consumption, gene expression, protein abundance and phase plots: without feedback

**D-xylose consumption, gene expression, protein abundance and phase plots: without feedback** (A): The simulated trajectory for D-xylose consumption. (B): Gene expressions profiles. (C): Proteins abundance plots. (D): Phase plot for gene expression showing variation of mRNA concentrations of the _{i}

Feedback in the network

Since the presence of CreA is a strong repressor that inhibits the

A comparison of the metabolite expression dynamics for the network with and without feedback loops in the absence of CreA is shown in Figure **System specification** were used for the simulation with the extra parameters from (11) being _{RL}_{AL}_{ls}

D-xylose consumption, gene expression, protein abundance and phase plots: with feedback

**D-xylose consumption, gene expression, protein abundance and phase plots: with feedback** (A): The simulated trajectory for D-xylose. (B): Gene expression profiles with solid lines (–) showing the expression profiles for the genes in the absence of CreA. The corresponding dotted lines (⋯) show the simulated effect of competitive feedback (with

Activating and repressing feedback

The expressions (18) and (19) have the potential to yield oscillatory behavior in the metabolite response profiles. The oscillatory behavior (if and when it exists) is purely governed by the values of the system mechanistic parameters. Such oscillatory behavioral patterns of gene expression may vary from organism to organism, and can be detected from time series data if enough samples are taken.

To assess the effect of time delays in the transcription and translation processes, some cases were simulated. The results of the expression time-dynamics for both the genes and proteins are shown in Figure

Effect of time delay on expression

**Effect of time delay on expression** (A)-(B): Plots showing the effect of variation in time delay in the feedback loops corresponding to the transcription and translation processes, respectively. The observed effect on the responses is small except for the slight deviation at the peak of the expression profiles.

xlnR gene promoter site activity

The competitive effect of the activators and repressors for the promoter binding sites was also simulated. The effect of which transcription factor (TF) (either an activator or a repressor) wins occupancy of a promoter binding site depends partly on the strength of the synthesis parameter _{ls}

xlnR promoter region activity

**xlnR promoter region activity** Plot of the _{A}, Γ_{R} and Γ_{A}Γ_{R} depending on the regulator. The term Γ_{A}Γ_{R} - is the combined affect of competitive binding to promoter region by activators and repressors. Plots (A) and (B) show the influence of weak (_{ls}_{ls}

The promoter is most active (activity around 50 – 80%) when the regulon is fully active. This corresponds with the time window at which the network is fully responsive to the external perturbation. We observe that the

Bifurcation and oscillatory behavior analysis

A range of values of activating and repressing parameters _{1}, _{2}, _{3}, respectively on entries (1, 4), (1, 5) and (1, 6) in expression (17) was considered for analyzing the stability behavior of the network. It was observed that NFB on _{1} gives a stable system and values of _{2} and _{3} below –977 results in an unstable systems, Figures _{1} > 1. This can be seen from Figure

Stability index curves

**Stability index curves** (A)-(C): Plots of the stability indices for various values of _{j}_{j}_{j}**System specification** and the pseudo steady state expression values at which the Jacobian is estimated.

An analysis of how the various feedback parameters affect the oscillatory behavior of the gene and protein expression was also considered. The results show that there exist threshold values (or a range of parameter values) for the feedback parameters _{j}_{j}_{j}_{1}) is a near reflection of the corresponding resultant oscillation curve (Figure

Oscillation index curves

**Oscillation index curves** (A)-(C): The plots indicate possible oscillatory behavior for various values of _{j}_{j}

Discussion

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

Feedback significantly affects the response of the output profiles for the metabolites and changes the final steady state values (Figure

According to Bliss

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

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

The effect of time delay on stability can be analyzed from a transfer function of the model in the ”

The adaptive filtering approach developed in

According to Balsa-Canto

Conclusions

The investigations in this paper considers the XlnR regulon as a dynamic system instead of a static system. Our study provides insight into the dynamic properties of the XlnR regulon. By studying this system, it has become more clear that the transcription and translation degradation rate parameters and the D-xylose consumption profile dictate most of the dynamics in the regulation properties of the network. The existence of oscillatory behavior depends on the conditions of the mechanistic parameters in the feedback loop - conditions that cannot always be generalized analytically and therefore, must be treated by numerical analysis. The role played by feedback in the network dynamics was found to be significant on the expression dynamics of genes and proteins. This means that the effect of the feedback should be considered in the model if there is sufficient supportive biological need or evidence from data. Just like for most biological systems, this is no doubt an important piece of information for the accurate modeling of biological network.

The analysis of the network dynamics has provided useful information for future

Authors’ contributions

LHdG provided the biological knowledge that was used for the model formulations. JO performed the modeling, data analysis and wrote the manuscript. GvS and AJBvB also contributed in calculations and critical review of the methods used in the analysis. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

This work is supported by the graduate school VLAG and the IPOP program of Wageningen University.

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