Gene network requirements for regulation of metabolic gene expression to a desired state

Abstract

Gene circuits that control metabolism should restore metabolic functions upon environmental changes. Whether gene networks are capable of steering metabolism to optimal states is an open question. Here we present a method to identify such optimal gene networks. We show that metabolic network optimisation over a range of environments results in an input-output relationship for the gene network that guarantees optimal metabolic states. Optimal control is possible if the gene network can achieve this input-output relationship. We illustrate our approach with the best-studied regulatory network in yeast, the galactose network. We find that over the entire range of external galactose concentrations, the regulatory network is able to optimally steer galactose metabolism. Only a few gene network parameters affect this optimal regulation. The other parameters can be tuned independently for optimisation of other functions, such as fast and low-noise gene expression. This study highlights gene network plasticity, evolvability, and modular functionality.

Figure 1. Identification procedure for a regulatory gene network capable to regulate a desired state of metabolic gene expression.

(A) Schematic overview of the metabolic and regulatory gene network and their inputs and outputs. Dynamics in the environment, in this example changes in substrate level s, lead to altered enzyme expression levels (as indicated by ) to restore fitness in the perturbed condition. These altered enzyme expression levels are achieved by the regulatory gene network that uses signalling metabolites as input. Note that these signalling metabolites are a function of the environmental change. (B) Optimal steering of a metabolic network by a regulatory gene network involves four steps: (1) Optimisation of metabolic performance. The metabolic network is optimised for an objective function under constraints. In this example, optimising the metabolic enzyme levels that lead to the highest steady state flux J under the constraint of a limited amount of resource, R. (2) The optimisation is performed for different environmental conditions (in this example different nutrient concentrations), yielding the relationship between the external substrate s and the optimal metabolite and enzyme concentrations. (3) From the metabolites signalling to the gene network are selected, to form –together with (s)– the optimal input-output relationship for the gene network. (4) The gene network receives as input and generates as output. The kinetic parameters of the gene network () are found by fitting the gene network to the optimal input-output relationship.

Figure 2. Modular representation of the galactose network and it's regulatory interactions in yeast.

Shown are the inputs and outputs of the galactose metabolism and galactose regulatory network, using a similar representation as in Figure 1A. Galactose metabolism (shown in blue) consists of four metabolic enzymes (gal2p, gal1p, gal7pd, gal10pd, shown in red). External galactose (Gal_out, green), is imported by gal2p, resulting in intracellular galactose (Gal_in, orange), which is further metabolised into glucose-1-phosphate (Glc-1P) by the enzymes gal7pd and gal10pd. Gal_in is needed for activation of the galactose regulatory network by binding to gal3p. Within this network, a distinction can be made between the regulatory proteins, gal3p, gal80p, gal4p (brown) and structural proteins (metabolic enzymes; red). Transcription of all genes is dependent on the concentration of gal4p dimer (gal4pd) and the number of gal4dp binding sites that the upstream activating sequences (UAS's) possess. The resulting mRNA's are shown in yellow. Degradation of every mRNA and protein is the net effect of intrinsic degradation and the growth rate dependent dilution.

Figure 3. Optimal gene network input-output relationship for the galactose network in yeast.

(A) Relationship between external galactose concentration (mM) and the galactose steady state flux (mM/min). The green line corresponds to the galactose flux as obtained by the Monod-equation. The red line shows the metabolic steady state flux that is calculated using the entire galactose model with the fitted gene network parameters. (B) Relationship between environmental dynamics and intracellular signalling metabolite. For a range of external galactose concentrations the corresponding range of intracellular galactose concentrations (the signalling metabolite for the gene network) range between 0 and 0.87 mM. (C-F) Input-output relationship for the galactose gene network. The blue lines correspond to the relationship between intracellular galactose (mM) and the metabolic enzyme concentrations (μM), as obtained by optimising the –isolated– metabolic network. The red solid lines represents the gene network behaviour with the gene kinetic parameters obtained by fitting the gene network to the input-output data. Panels correspond to: C gal2p; D gal1p; E gal7pd; F gal10dp.

Figure 4. Illustration of optimal tracking of the environment by the optimal regulatory gene network.

Shown is the response in the dynamic metabolic flux profile for the model with the optimised gene network parameters with a time interval of two hours between the perturbations. The external galactose concentrations are perturbed as shown in the upper part of the figure and the corresponding response of the metabolic flux is plotted relative to the optimal flux for the indicated galactose concentration.

Figure 5. Optimal tracking by the regulatory gene network fails for short switch times.

Shown is the metabolic flux profile over time based on metabolic enzyme expression of the gene regulatory network with the fitted gene parameters. The system starts at a steady state with an external galactose concentration of 0.05 mM. External galactose is perturbed in similar steps and using similar concentrations as shown in upper part of Figure 4 at time intervals as indicated in each plot. We plot the metabolic steady state flux relative to the optimal flux at that galactose concentration. The red dashed line corresponds to the optimal metabolic flux for the galactose concentration corresponding to that perturbation. For sake of comparison, we have normalised the time to each perturbation interval, giving rise to the equal space between the perturbations in the different plots.

Figure 6. Influence of parameters for optimal metabolic gene regulation by the gene network for other objective functions.

Scaled parameter sensitivities corresponding to the fold change in a system property upon 2-fold increase (green arrow) and decrease (red arrow) relative to the unperturbed value. The parameter sensitivities per objective were scaled between -1 and 1 and coloured as indicated by the colourbar. The upper two rows, indicated by the red-dashed box, corresponds to the parameter sensitivities of the optimal metabolic flux and the gene network parameters are sorted according to their influence on this system function. The remaining rows report the effects of the gene network parameters on: the steady state concentration of (potential toxic) metabolic intermediate Gal-1P, the response time of the steady state flux after a shift in the external galactose concentration from 0.5 to 5 mM, and the noise (quantified by the coefficient of variation) calculated from the linear noise approximation of some key regulators within the gene network: gal3p*, gal4pd, gal7pd and the complex gal80pgal3p*. The numbers above each columns correspond to the gene network parameters as listed in Supplementary Table 1.