Negative Feedback and Transcriptional Overshooting in a Regulatory Network for Horizontal Gene Transfer

Horizontal gene transfer (HGT) is a major force driving bacterial evolution. Because of their ability to cross inter-species barriers, bacterial plasmids are essential agents for HGT. This ability, however, poses specific requisites on plasmid physiology, in particular the need to overcome a multilevel selection process with opposing demands. We analyzed the transcriptional network of plasmid R388, one of the most promiscuous plasmids in Proteobacteria. Transcriptional analysis by fluorescence expression profiling and quantitative PCR revealed a regulatory network controlled by six transcriptional repressors. The regulatory network relied on strong promoters, which were tightly repressed in negative feedback loops. Computational simulations and theoretical analysis indicated that this architecture would show a transcriptional burst after plasmid conjugation, linking the magnitude of the feedback gain with the intensity of the transcriptional burst. Experimental analysis showed that transcriptional overshooting occurred when the plasmid invaded a new population of susceptible cells. We propose that transcriptional overshooting allows genome rebooting after horizontal gene transfer, and might have an adaptive role in overcoming the opposing demands of multilevel selection.

The dynamics of the negative feedback follow dx dt Λ 1 k n k n y t n Β 1 x t dy dt Λ 2 x t Β 2 y t Eq 1.
Where x is the RNA concentration, Y is the protein concentration , Λ stands for production rates, Β represents degradation/dilution rates, k is the feedback constant, and n is the cooperativity of the system (non-linearity of the feedback). In these condition the steady-state of the system is dx dt Λ 1 k n k n y t n Β 1 x t 0 dy dt Λ 2 x t Β 2 y t 0 In order to simplify the calculations we replace k n K and the steady-state for strong repression ( K y ss y ss for X follows Eq 1. Where x is the RNA concentration, Y is the protein concentration , Λ stands for production rates, Β represents degradation/dilution rates, k is the feedback constant, and n is the cooperativity of the system (non-linearity of the feedback). In these condition the steady-state of the system is dx dt In order to simplify the calculations we replace k n K and the steady-state for strong repression ( K y ss y ss for X follows X fb Eq 2. Similarly the equations for the open loop dx dt Therefore the feedback gain equals Eq 4. This expressions yields the following limits: This indicates that while decreasing k (increasing the affinity of the repressor for its cognate site) increases indefinitely the feedback gain, increasing n reaches a limit gain that equals the intrinsic promoter strength ( times the inverse of the half-repression constant (k).
The gain in y is calculated in the same way, yielding Which indicates that the feedback gain is equivalent for values of x and y

Feedback Overshoot existence
System of Ec. 1 produces a transient overshoot whenever x and y reach a maximum that is higher than the steady state value. A max. in x is reached iif Which indicates that the feedback gain is equivalent for values of x and y

Feedback Overshoot existence
System of Ec. 1 produces a transient overshoot whenever x and y reach a maximum that is higher than the steady state value. A max. in x is reached iif x " Λ 1 K n y n 1 y ' 0 n 0 since there cannot be negative cooperativity y n 1 0 since y cannot take negative values Λ 1 K are always positive then x " 0 y ' 0 From this follows that y at x x max y is always smaller than its steady state value Since y ss Equivalent reasoning yields that x ' 0 at y y max and x ss x y max

Gain-Overshoot relationship
The overshoot (O) is the ratio between the maximal value of X or Y and its steady-state value. In the case of an RNA Eq.7 The higly non-linear nature of the ODE system prevents the calculation of Xmax, but in the case of strong selfrepression (k+y y) and high cooperativity we can linearize the system, approximating Xmax to the equivalent value for the open loop when t= t x max dx dt Eq.7 The higly non-linear nature of the ODE system prevents the calculation of Xmax, but in the case of strong selfrepression (k+y y) and high cooperativity we can linearize the system, approximating Xmax to the equivalent value for the open loop when t= t x max dx dt This approximation holds for higly-nolinear systems, where the high cooperativity index (n) acts as an effective delay between the accumulation of the repressor (y) and the onset of repression. Therefore To calculate the overshoot for y we follow the same reasoning. Linearizing x until the onset of the negative feedback, we can approximate the value of Ymax Simulations indicate that these approximation holds for highly-nonlinear systems (Figure 4, main text). This condition can be met by systems where the repressor exhibits high cooperativity to its cognate binding site or where the repressor dimer/multimerization is required for binding.

Effect of multimerization
The following system includes a step of repressor dimerization Supplementary Material 2-Calculations.nb