Non-Kramers State Transitions in a Synthetic Toggle Switch Biosystem

TL;DR

This study directly images state transitions in a synthetic bacterial toggle switch at the single-cell level using a mother-machine device, revealing that transitions do not conform to small-noise Kramers-type single-rate dynamics. By reconstructing an effective one-dimensional stochastic process along the readout coordinate $r$, the authors show a barrier–trap landscape with significant, position-dependent noise $D(r)$, leading to strong initial-condition effects on first-passage times. The work integrates first-passage analysis, landscape reconstruction, and Langevin simulations to demonstrate that multiplicative noise and non-separable timescales govern transitions, challenging the universality of discrete-state, single-rate paradigms. These findings highlight the need for theoretical frameworks beyond the small-noise assumption to describe biological state transitions in developmental and synthetic circuits, with implications for how cellular states are defined and controlled in noisy, living systems.

Abstract

State transitions are fundamental in biological systems but challenging to observe directly. Here, we present the first single-cell observation of state transitions in a synthetic bacterial genetic circuit. Using a mother machine, we tracked over 1007 cells for 27 hours. First-passage analysis and dynamical reconstruction reveal that transitions occur outside the small-noise regime, challenging the applicability of classical Kramers' theory. The process lacks a single characteristic rate, questioning the paradigm of transitions between discrete cell states. We observe significant multiplicative noise that distorts the effective potential landscape yet increases transition times. These findings necessitate theoretical frameworks for biological state transitions beyond the small-noise assumption.

Figures (18)

• Figure 1: Temporal evolution of the synthetic toggle switch circuit (inset) is tracked on the single-cell level. Three typical trajectories of the RFP ($r$) and GFP ($g$) fluorecsence intensities are shown on the $r-g$ plane. The contour lines indicate the frequency of the $(r,g)$ readings over the whole observation, according to the logarithm of the counts. The red solid line marks the $r=g$ line, separating the two states. To be noted, the contour map should not be interpreted as a distribution or an effective landscape, since ergodicity is not achieved yet.
• Figure 2: (a) Reconstructed landscape $U$ (solid line) and the noise strength $D$ (dashed line) from the experiment data. Both quantities share the same dimensions with the unit $\text{FI}^2/\text{hr}$. The unit of $r$ is FI, the fluorescent intensity. $U$ and $D$ are estimated for the $R$-state regime with the boundary $r_a=58$ (red dashed dot line). The peak of the landscape locates at $r_b=86$ (blue dashed dot line). The center of the trap locates at $r_c=174$ (green dashed dot line). (b) Rescaled landscape $U_{\text{rsc}}$, defined in Eq. (\ref{['rsc']}) (blue dashed line), and the modified one $U_{\text{st}}=\ln D+U_{\text{rsc}}$ (black solid line). The red dashed-dot line shows $U_{\text{st}}$ from a case of homogeneous noise as comparison. (See Eq.(\ref{['dgamma']}) for the definition of $\gamma$. )
• Figure 3: Mean first passage time (MFPT) $\tau$ to the boundary at $r_a=58$ for the initial value $r(t=0)=r_0$, simulated by Langevin dynamics using the reconstructed $f(r)$ and $D(r)$ (black solid line). The time unit is hours. Colored dashed lines illustrate the contributions from the landscape and the multiplicative noise. The parameter $\beta=4,1,0$ controls the barrier scale (see Eq.(\ref{['fbeta']})), while $\gamma=1,0$ controls the presence of the multiplicative noise (see Eq. (\ref{['dgamma']})). The MFPT of the small noise case with $\beta=7$ (red dashed line) is significantly larger ( $\tau(r_0=250)=1.6\times10^4\text{hr}$) and has been rescaled here for better visualization.
• Figure S1: Long-term single-cell tracking using the “mother machine” microfluidic device. The lower panel shows time-course of RFP fluorescence intensity from 100 randomly selected mother cells growing under steady-state conditions in rich defined medium (RDM). Three representative cell lineages are highlighted in red, blue, and green. The upper panel shows xy–t montages of raw fluorescence images from the same three channels, with each growth channel imaged every 3 frames (about $9.3$ minutes per frame).
• Figure S2: Cell morphology statistics. Distribution of cell area (a) and inferred diameter (b) for all cells across all time frames.