How molecular competition influences fluxes in gene expression networks

Abstract

Often, in living cells different molecular species compete for binding to the same molecular target. Typical examples are the competition of genes for the transcription machinery or the competition of mRNAs for the translation machinery. Here we show that such systems have specific regulatory features and how they can be analysed. We derive a theory for molecular competition in parallel reaction networks. Analytical expressions for the response of network fluxes to changes in the total competitor and common target pools indicate the precise conditions for ultrasensitivity and intuitive rules for competitor strength. The calculations are based on measurable concentrations of the competitor-target complexes. We show that kinetic parameters, which are usually tedious to determine, are not required in the calculations. Given their simplicity, the obtained equations are easily applied to networks of any dimension. The new theory is illustrated for competing sigma factors in bacterial transcription and for a genome-wide network of yeast mRNAs competing for ribosomes. We conclude that molecular competition can drastically influence the network fluxes and lead to negative response coefficients and ultrasensitivity. Competitors that bind a large fraction of the target, like bacterial σ(70), tend to influence competing pathways strongly. The less a competitor is saturated by the target, the more sensitive it is to changes in the concentration of the target, as well as to other competitors. As a consequence, most of the mRNAs in yeast turn out to respond ultrasensitively to changes in ribosome concentration. Finally, applying the theory to a genome-wide dataset we observe that high and low response mRNAs exhibit distinct Gene Ontology profiles.

Figure 1. Sigma cycle in E. Coli.

The sigma cycle in bacterial transcription refers to gene regulation by competitive association of promoter-specific transcription factors -sigma factors- with RNA polymerase (RNAP; –[9]). The sigma factors compete for binding to RNAP after each round of RNA synthesis, and determine the promoter-specificity of transcription initiation (see for instance for more mechanistic detail). In E. coli, seven different types of sigma factors exist, each directing transcription of a specific set of genes . Most of the growth-related and housekeeping genes expressed in the exponential phase of the growth of a cell population are transcribed by the RNAP-holoenzyme containing σ⁷⁰ (σ^D). σ⁵⁴ (σ^N) confers specificity to RNAP for transcribing genes regulated by the availability of nitrogen and some stress response genes , . σ²⁸ (σ^F) is needed for the expression of the flagellum and chemotaxis genes . σ³⁸ (σ^S) accumulates during the stationary phase and directs expression of genes related to stress management and maintenance . The other three sigma factors, σ³² (σ^H), σ²⁴ (σ^E) and σ¹⁹ (σ^FecI), act in heat shock response, extra- cytoplasmic stress and iron-transport respectively . Although other regulatory factors play a role (the alarmone ppGpp, anti-sigma factors, etc.; cf. [9]), the global pattern of gene transcription is believed to be determined largely through sigma factor competition .

Figure 2. Relation between a sigmoidal (Hill type) input-output relation and the corresponding response coefficient.

The reaction rate normalized to 1 () and the response coefficient towards a change in substrate concentration S () are plotted as a function of the substrate concentration (assuming excess substrate over enzyme).

Figure 3. Basic network diagram of two molecules competing for a common target molecule.

(A). The reaction scheme. In the reversible binding steps (with rates α₁ and α₂) the free competitors c₁ and c₂ form a complex with the target t (to produce complexes tc₁ and tc₂, respectively). In the irreversible production steps (with rates β₁ and β₂) the products (p₁ and p₂) are generated and the free competitor and the target are recycled. In the analogy to the bacterial transcription network the competitors are the sigma factors and the target is RNA polymerase. In the analogy to the translation network the competitors are the mRNAs and the target is the ribosome.(B). The reactions and rate equations for competitor i. The assumptions underlying these equations are discussed in the Introduction.

Figure 4. Flux responses in a network of three competitors.

Comparison of the flux response coefficients calculated with expressions (2)-(4) (depicted by the big dots), to the values obtained with the basic model (cf. Figure 2) for three competitors, simulated at different target levels (10-1800 molecules/cell). The latter data points are represented by the lines. The two approaches give identical results. By analogy with the sigma factor example in the main text the total competitor levels were taken to be 700 molecules/cell for competitor 1 (σ⁷⁰), 370 molecules/cell for competitor 2 (σ⁵⁴) and 110 molecules/cell for competitor 3 (σ²⁸). Other parameters needed for simulation were the reaction rate constants: k_α1,f = 24, k_α2,f = 8, k_α3,f = 11 (molecules⁻¹.min⁻¹), k_α1,r = 3, k_α2,r = 6, k_α3,r = 30 (min⁻¹), and k_β1’ = k_β2’ = k_β3’ = 5 (min⁻¹). The values were selected to fit the sigma factor example at a total target concentration of 700 molecules/cell (cf. Table 1). (A). Flux responses of all individual competitors towards changes in target concentration. (B). Flux response of all individual competitors towards changes in competitor 1. (C). Competitor saturation (tc_i/C_i) as a function of the total target level. The saturation over all competitors is also indicated. (D) The concentration of free target and the response coefficient of the free target concentration to the total target concentration as a function of the total target concentration.

Figure 5. Flux responses in a network of two mRNAs competing for ribosomes.

(A). Comparison of the flux response coefficients calculated with expressions (8)-(10) (depicted by big dots), with the values obtained with a model of translation (based on [23]) of two mRNA competitors at different ribosome (target) levels (log-scaled: 0.1–220 µM). The latter data points are represented by the lines. The dashed-dotted line represents the flux response coefficient of the translation of mRNA 1 towards changes in mRNA 2. The dashed line represents the flux response coefficient of the translation of competitor 2 towards changes in competitor 2. The parameters used for the simulation were given the following values: length of ORF = 240 codons, ribosome width  =  12 codons, [competitor 1]  =  0.2 µM, [competitor 2]  =  2 µM. The initiation rate constants were 0.2 (µM.min)⁻¹ (competitor 1) and 5 (µM.min)⁻¹ (competitor 2), the elongation and termination rate constants were 50 min⁻¹ (competitor 1) and 20 min⁻¹ (competitor 2). (B). Comparison of the fluxes (in µM.min⁻¹) simulated in the same conditions. The dashed-dotted line represents the flux of the translation of mRNA 1, the dashed line represents the flux of the translation of competitor 2.

Figure 6. The whole-cell distribution of translation flux response coefficients towards changes in the total ribosome level.

The yeast genome-scale datasets published by Siwiak and Zielenkiewicz and by Brockmann et al. were used to produce a histogram of the distributions of values for two cases. The first distribution (blue bars) was produced by including genome-scale ribosome occupancy values in the calculations; the second (purple bars) was produced assuming ribosome occupancies equal to 1 (more details on the calculations in the Methods section). The vertical line was positioned at  = 1.

Figure 7. Gene Ontology mapping (in terms of biological process) of high and low response groups.

Histogram indicating the relative frequency (as %) of GO classes in high and low response gene sets. The genes corresponding to the mRNAs with the 5% highest (‘High, ’blue bars) or lowest (‘Low’, red bars) response coefficients were pooled and mapped with the GO Slim Mapper tool (http://www.yeastgenome.org/cgi-bin/GO/goSlimMapper.pl), based on the ‘Yeast GO-Slim Process’ GO set. This is a set of high level GO terms that best represent the major biological processes found in S. cerevisiae. These terms have been selected by S. cerevisiae Genome Database (SGD) curators based on annotation statistics and biological significance. The corresponding percentages for the whole yeast genome (as % of 6310 genes annotated at the moment of analysis, i.e. 4-3-2011, in the SGD) are represented by green bars (‘Genome’).

Figure 8. Gene Ontology mapping (in terms of molecular function) of high and low response groups.

Histogram indicating the relative frequency (as %) of GO classes in high and low response gene sets. The genes corresponding to the mRNAs with the 5% highest (‘High’, blue bars) or lowest (‘Low’, red bars) response coefficients were pooled and mapped with the GO Slim Mapper tool (http://www.yeastgenome.org/cgi-bin/GO/goSlimMapper.pl), based on the ‘Yeast GO-Slim Function’ GO set. This is a set of high level GO terms that best represent the major biological functions that are found in S. cerevisiae. These terms have been selected by S. cerevisiae Genome Database (SGD) curators based on annotation statistics and biological significance. The corresponding percentages for the whole yeast genome (as % of 6310 genes annotated at the moment of analysis, i.e. 4-3-2011, in the SGD) are represented by green bars (‘Genome’).