Characteristics of transcriptional activity in nonlinear dynamics of genetic regulatory networks

Abstract

Microarray measurements of mRNA abundances is a standard tool for evaluation of transcriptional activity in functional genomics. The methodology underlying these measurements assumes existence of a direct link between transcription levels, that is, gene-specific mRNA copy numbers present in the cell, and transcription rates, that is, the numbers of gene-specific mRNA molecules synthesized per unit of time. In this paper, the question of whether or not such a tight interdependence may exist is examined in the context of nonlinear dynamics of genetic regulatory networks. Using the equations of chemical kinetics, a model has been constructed that is capable of explicitly taking into consideration nonlinear interactions between the genes through the teamwork of transcription factors. Jacobian analysis of stability has shown that steady state equilibrium is impossible in such systems. However, phase space compressibility is found to be negative, thus suggesting that asymptotic stability may exist and assume either the form of limit cycle or of a chaotic attractor. It is argued that in rapidly fluctuating or chaotic systems, direct evaluation of transcription rates through transcription levels is highly problematic. It is also noted that even if a hypothetical steady state did exist, the knowledge of transcription levels alone would not be sufficient for the evaluation of transcription rates; an additional set of parameters, namely the mRNA decay rates, would be required. An overall conclusion of the work is that the measurements of mRNA abundances are not truly representative of the functionality of genes and structural fidelity of the genetic codes.

Keywords: asymptotic stability; genetic regulatory network; jacobian eigenvalues; microarrays; nonlinear dynamics; transcription factors.

Figure 1.

Typical spectra of eigenvalues. Top row: kinetic rates are distributed uniformly (beta with shapes 1 and 1). Bottom row: distribution is unimodal (beta shapes 3 and 3. Left column: rates are constant and equal to the averages of those in right column.

Figure 2.

Spectra of eigenvalues for drastically different distributions of kinetic rates. Top row: kinetic rates are unimodal with very little variations (beta with shapes 10 and 10. Bottom row: distribution is bimodal with sharp peaks at zero and one (beta with shapes 0.3 and 0.3. Left column: rates are constant and equal to the averages of those in right column.

Figure 3.

Spectra of eigenvalues for drastically different distributions of kinetic rates. Top row: kinetic rates are unimodal with sharp peak at the right end of the interval (beta with shapes 3 and 0.3). Bottom row: kinetic rates are unimodal with sharp peak at the left end of interval (beta with shapes 0.3 and 3). Left column: rates are constant and equal to the averages of those in right column.

Figure 4.

Top row: kinetic rates are distributed uniformly (beta with shapes 1 and 1). Bottom row: distribution is unimodal (beta shapes 3 and 3. Left column: rates are constant and equal to the averages of those in right column. The difference with Figure 1 is that the link density here is 100.

Figure 5.

Top row: kinetic rates are distributed uniformly (beta with shapes 1 and 1). Bottom row: distribution is unimodal (beta shapes 3 and 3. Left column: rates are constant and equal to the averages of those in right column. The difference with Figure 1 is that the link density here is 1000.

Figure 6.

All four pictures: distribution is unimodal (beta shapes 3 and 3. Top row: link density is 1. Bottom row: link density is 1. Left column: rates are constant and equal to the averages of those in right column.

Figure 7.

All four pictures: distribution of kinetic rates is unimodal (beta shapes 3 and 3), and link density is 100. Top row: stoichiometric coefficients are distributed exponentially with scale = 0.1. Bottom row: stoichiometric coefficients are distributed as gamma with shape = 2 and rate = 20. Both distributions produce the same mean = 0.1. Left column: kinetic rates are constant and equal to the averages of those in right column.

Figure 8.

Histograms of the beta distributions for various combinations of shape parameters.