Magnetic induction inspires a schematic theory for crosstalk-driven relaxation dynamics in cells

Establishing formal mathematical analogies between disparate physical systems can be a powerful tool, allowing for the well studied behavior of one system to be directly translated into predictions about the behavior of another that may be harder to probe. In this paper we lay the foundation for such an analogy between the macroscale electrodynamics of simple magnetic circuits and the microscale chemical kinetics of transcriptional regulation in cells. By artificially allowing the inductor coils of the former to elastically expand under the action of their Lorentz pressure, we introduce nonlinearities into the system that we interpret through the lens of our analogy as a schematic model for the impact of crosstalk on the rates of gene expression near steady state. Synthetic plasmids introduced into a cell must compete for a finite pool of metabolic and enzymatic resources against a maelstrom of crisscrossing biological processes, and our theory makes sensible predictions about how this noisy background might impact the expression profiles of synthetic constructs without explicitly modeling the kinetics of numerous interconnected regulatory interactions. We conclude the paper with a discussion of how our theory might be expanded to a broader class of plasmid circuits and how our predictions might be tested experimentally.

Figure 1

Analogous circuits. (a) A schematic representation of a simple plasmid circuit consisting of a single, constitutively regulated gene and (b) a schematic diagram of its equivalent magnetic circuit, drawn using standard linear circuit notation for its various components.

Figure 2

Mutually regulating circuits. (a) A schematic representation of a plasmid circuit in which two genes act as transcriptional regulators for each other. The arrows indicate regulatory interactions, with the pointed arrowhead representing positive, promotional regulation, and the flat arrowhead representing negative, repressive regulation. (b) A schematic diagram of the equivalent magnetic circuit, which is a simple transformer. As drawn, the circuit is analogous to the case of mutual repression; the case of mutual promotion can be obtained by reversing the transformer polarity.

Figure 3

Conceptual illustration of the elastic inductor. As current flows through the solenoid, a magnetic field is produced that exerts an outward Lorentz pressure on the coil. This causes it to expand radially, increasing the cross-sectional area of its turns and the total magnetic flux passing through the coil. This additional flux induces an even larger electromotive force to oppose any changes to the current, making the elastic inductor more slowly relaxing than a normal, rigid inductor—especially for larger currents.

Figure 4

The time-dependent current profile of the elastified $RL$ circuit for two different initial conditions, one above the steady-state current $V/R$ (in red) and one below it (in blue). All parameter values ($V$, $R$, $L$, etc.) were set to unity for simplicity. Dashed lines indicate the behavior of the standard $RL$ circuit initialized from the same two initial conditions (in matching colors). The inset plots the same curves over a shorter timescale to better illustrate the exponential decay of the standard circuit.

Figure 5

Phase portraits of the elastic transformer. (a) A dynamical phase portrait for the elastic transformer circuit in the case where the two loop currents rotate in the same direction. All parameters were set equal to unity for convenience, except for the coupling parameter $k$, which was set equal to $1/2$. The red dot at $(I_1,I_2)=(1,1)$ marks the single, stable fixed point of the system. (b) For the same set of parameter values, the phase portrait for the oppositely polarized elastic transformer is plotted. The same stable fixed point exists, but now there are two additional saddle points that lie along the curves defined by the sets of current values that reduce one or the other of the bracketed terms in Eqs. (25) to zero. Two phase trajectories that illustrate the behavior of the dynamics near these curves are plotted in green and blue, illustrating, respectively, how these saddle points either deflect the currents away towards a trivial fixed point at infinity or back towards the stable fixed point at (1,1).

Figure 6

Phase portrait of the mutually regulating plasmid circuit with crosstalk. The dynamical phase portrait defined by Eqs. (29) and (30) is plotted for $U_{\text{max}}=U_{\text{max}}^{\prime }=1/2$ and all other parameters equal to unity (an equivalent parameter set to that used for the phase portraits in Fig. 5). Once again, the red dot marks the fixed point at (1,1).