Modelling co-translational dimerization for programmable nonlinearity in synthetic biology

Abstract

Nonlinearity plays a fundamental role in the performance of both natural and synthetic biological networks. Key functional motifs in living microbial systems, such as the emergence of bistability or oscillations, rely on nonlinear molecular dynamics. Despite its core importance, the rational design of nonlinearity remains an unmet challenge. This is largely due to a lack of mathematical modelling that accounts for the mechanistic basis of nonlinearity. We introduce a model for gene regulatory circuits that explicitly simulates protein dimerization-a well-known source of nonlinear dynamics. Specifically, our approach focuses on modelling co-translational dimerization: the formation of protein dimers during-and not after-translation. This is in contrast to the prevailing assumption that dimer generation is only viable between freely diffusing monomers (i.e. post-translational dimerization). We provide a method for fine-tuning nonlinearity on demand by balancing the impact of co- versus post-translational dimerization. Furthermore, we suggest design rules, such as protein length or physical separation between genes, that may be used to adjust dimerization dynamics in vivo. The design, build and test of genetic circuits with on-demand nonlinear dynamics will greatly improve the programmability of synthetic biological systems.

Keywords: genetic circuits; mathematical modelling; nonlinearity; protein dimerization; synthetic biology; systems biology.

Figure 1.

Co-translational dimerization. (a) Upon DNA transcription by RNA polymerase, ribosomes bind the resulting RNA to translate it into proteins. There is more than one translation process at a given time, and ribosomes go along the RNA at different speeds, leading to the appearance of traffic jams. Our model simulates the process by which, when distance among ribosomes is short, partially formed monomers (represented in the figure by chains of yellow circles) dimerize with other partially formed monomers as they are being translated. (b) Detail of the translation-mediated dimerization area. The extension of this region depends on several physical features, such as the length of the protein to be translated or the distance between ribosomes; these constraints will affect nonlinearities due to protein dimerization.

Figure 2.

Mechanistic principles of co- versus post-translational dimerization dynamics. (a) Diagram showing the modelling parameters affecting dimerization types: ribosome binding (λ), the rate at which two partially formed monomers interact (k_D) and protein length. (b) At a rate where monomers interact weakly (k_D = 1, dimensionless) the fraction of co-translational dimerization (i.e. the total dimerization minus the dimerization in the cellular cytosol, divided by the total), α, as described by equation (2.3), responds primarily to ribosome binding: if binding increases (x-axis), α also increases (colour map). (c) The pattern in α when monomers strongly interact (k_D = 10) is similar, but with a (much) sharper transition from lower to higher values.

Figure 3.

Impact of co-translational dimerization (α) on a genetic oscillator. (a) Diagram of a three-component genetic oscillator (as in [45])—each repressor protein (R) inhibits the expression of its successor. As a result, the concentration of each repressor oscillates in time. (b) Eigenvalue analysis, which shows a bifurcation point around α = 0.022. This suggests that the emergence of oscillations is highly dependent on the balance between co- and post-translational dimerization. Solid and dotted lines are the real and imaginary values of the eigenvalues in the Jacobian at equilibrium, respectively. SS, steady state. (c) Time-course simulations of the genetic oscillator at different values of α. As shown, reliable oscillations are lost for values indicated in (b).

Figure 4.

Impact of co-translational dimerization and intergenic distance on a genetic toggle switch. (a) The two circuit modules are located in proximity (i.e. sharing the same chromosomal position). In this scenario, only in the case of α being low is there enough nonlinearity to achieve bistability. (b) Genetic components are placed at a distance (i.e. different chromosomal location). The physical separation forces proteins to travel from their source gene to their target promoter—a nonlinear process which counteracts the effect of high α values. (a,b) The value of α = 0.1 (tagged as predicted) is our theoretical approximation; up to now, and to the best of our knowledge, this value has not been experimentally obtained.

Figure 5.

Calculation of the spherical overlap between partially formed monomers. At the centre of each sphere, there is a ribosome bound to the RNA and that is actively translating. The radii (r and r + Δx) define the length of a monomer. Note that the dark grey overlapping section is composed of two spherical domes; for visualization purposes, here we show the two-dimensional analogous circular segments. The value of h is the height of the dome and a is the radius of its disc.