Integrating quantitative knowledge into a qualitative gene regulatory network

Abstract

Despite recent improvements in molecular techniques, biological knowledge remains incomplete. Any theorizing about living systems is therefore necessarily based on the use of heterogeneous and partial information. Much current research has focused successfully on the qualitative behaviors of macromolecular networks. Nonetheless, it is not capable of taking into account available quantitative information such as time-series protein concentration variations. The present work proposes a probabilistic modeling framework that integrates both kinds of information. Average case analysis methods are used in combination with Markov chains to link qualitative information about transcriptional regulations to quantitative information about protein concentrations. The approach is illustrated by modeling the carbon starvation response in Escherichia coli. It accurately predicts the quantitative time-series evolution of several protein concentrations using only knowledge of discrete gene interactions and a small number of quantitative observations on a single protein concentration. From this, the modeling technique also derives a ranking of interactions with respect to their importance during the experiment considered. Such a classification is confirmed by the literature. Therefore, our method is principally novel in that it allows (i) a hybrid model that integrates both qualitative discrete model and quantities to be built, even using a small amount of quantitative information, (ii) new quantitative predictions to be derived, (iii) the robustness and relevance of interactions with respect to phenotypic criteria to be precisely quantified, and (iv) the key features of the model to be extracted that can be used as a guidance to design future experiments.

Figure 1. Flowchart of the Event Transition Markov chain modeling protocol.

Figure 2. Event Transition Graph composed of 2 genes (left) and its corresponding probability transition matrix (right), that includes two unknowns and .

Figure 3. Set of probabilities that satisfy the constraints for the Event Transition Graph depicted in Figure 2 .

Figure 4. Comparison of the protein Y growth ratio in two different situations.

(A) Distribution of the Y protein growth rate estimated from probabilities randomly picked; (B) Difference of the distribution described in (A), and the distribution of protein Y growth rate estimated from 10 000 combinations of probabilities that satisfies the constraints of the ETG model that depict the interactions of two genes.

Figure 5. Biological information concerning Escherichia coli carbon starvation system.

(A) represents interactions between genes involved in the regulatory network (adapted from [29]). (B) shows quantitative variations of macromolecules of interest (based on [30]). Note the linear relationship between fis mRNA and Fis protein productions that allows to infer protein product behaviors based on the gene regulatory network.

Figure 6. Even transition graph of the genes regulatory network of carbon starvation response in E. coli.

Each component represents an active event that concerns a gene product (), either its increase () or its decrease (). Arrows between events depict the active effect of one event on another. Two transitions are absent when the system is subject to carbon starvation.

Figure 7. Simulations of changes in bacterial protein concentration during both stationary and exponential growth phases.

The corresponding probability matrix is estimated in the stationary growth condition based on three experimental data for the protein Fis. After 80 minutes, the signal of carbon starvation manually switches from 1 to 0, emphasizing a switch from starvation to non-starvation conditions, which leads respectively to a stationary and an exponential growth phase of the bacterial population. Experimental data are marked with dashed lines, whereas computation results are depicted using plain lines for the five proteins of interest (Fis, Cya, Topa, GyrAB and Crp).