Simplexes Characterize the Coupling in Gene Expression Due to Competition for Rnap and Ribosomes

The limited availability of shared cellular resources couples the expressions of any two genes. This, among other factors, renders the behavior of gene circuits context-dependent. As a result, our ability of predicting the emergent behavior of a complex gene network from that of the composing modules remains limited, posing a challenge in systems and synthetic biology. To overcome this limitation, here we characterize the coupling among protein concentrations due to competition for the shared transcriptional/translational machinery in gene networks with arbitrary topology.

In particular, we show that the set of protein concentrations that are simultaneously realizable is given by the intersection of simplexes. Exploiting these simplexes, we further characterize how changing the concentration of one protein affects the concentration of other proteins, even in the absence of regulatory linkages among them. The fact that the vertices of these simplexes are straightforward to measure experimentally renders this approach easy to employ in practice. 

This simplex-based method aids the rational and predictable design of complex biocircuits as follows. First, it provides a simple tool to check whether the desired behavior of a large gene circuit is realizable. If the transcriptional/translational resources are insufficient, the simplexes aid the partitioning of circuits into smaller components (subnetworks), each realizable within a single cell. Second, we can predict how tuning various biochemical parameters affect the strength of interdependence among protein concentrations. Therefore, we can mitigate the coupling among components, facilitating the modular design of complex gene circuits. Finally, standard pathway optimization problems, such as maximizing the concentration of protein complexes or maximizing the rate at which a reaction occurs, can be formulated as convex optimization problems. These problems can be naturally interpreted in the context of economics, giving rise to concepts such as the price and utility of proteins.