Cell Division Changes Fate Decisions in a Genetic Toggle Switch

Technical Summary: Cell Division Changes Fate Decisions in a Genetic Toggle Switch

Problem Statement
Gene regulatory networks (GRNs) govern cellular fate decisions through multistable dynamics, with the genetic toggle switch serving as a canonical model for such behavior. While classical models of gene expression often incorporate the effects of cell division via an "effective dilution" term added to the degradation rate, these models treat division as a continuous process rather than an explicit event. In contrast, explicit division mechanisms remove proteins discretely at the end of a cell cycle. Although previous studies have noted that fully accounting for division can reduce the region of bistability, it remains unclear whether, within parameter regimes where both models exhibit bistability, identical initial conditions lead to the same stable cellular state. This paper addresses the gap in understanding how explicit modeling of cell division impacts the dynamics and fate predictions of multistable regulatory networks.

Methodology
To analytically investigate the impact of division, the authors propose a simplified, analytically tractable version of the toggle switch: the Boolean Toggle Switch (BTS). This model is derived by taking the limit of infinite Hill coefficients in the standard coupled ordinary differential equations (ODEs), replacing Hill functions with Heaviside step functions. This binarization of protein synthesis (ON/OFF based on concentration thresholds) preserves the fundamental dynamical features of the toggle switch while allowing for the derivation of closed-form expressions.

The study compares two distinct versions of the BTS:

1. Standard BTS: Follows classical conventions where cell volume is fixed ($V=1$). Division effects are incorporated solely through an effective dilution rate ($\delta = \ln 2 / T$) added to the active degradation rates.
2. BTS with Division: Explicitly models exponential cell growth ($V(t) = V_0 e^{\delta t}$) and discrete cell division. In this version, protein counts are halved at the end of each cell cycle ($t=T$), and there is no effective dilution term in the differential equations.

The authors derive analytical expressions for the separatrices (the stable manifolds of saddle equilibria that divide the phase plane into basins of attraction) for both models. They assume zero active protein degradation ($\beta_A = \beta_B = 0$) to focus on the effects of division, noting that protein half-lives in mammalian cells often exceed the cell cycle length. Additionally, stochastic simulations (500 runs per initial condition) are performed to validate the deterministic analysis against intrinsic noise, particularly in low-count regimes.

Key Contributions and Results
The primary contribution of the paper is the derivation of analytical separatrices for both the standard and division-inclusive BTS models, revealing that the two models possess distinct boundaries for cellular fate decisions.

• Divergent Trajectories: The analysis demonstrates that cell division can redirect trajectories with identical initial conditions to opposing stable states. Specifically, the separatrix for the BTS with division does not coincide with that of the standard BTS.
• Region of Disagreement (ROD): The authors define a "Region of Disagreement" (ROD) in the phase plane where the two models predict different stable states for the same initial protein counts.
  • When the synthesis and threshold parameters satisfy $\alpha_A \lambda_B > \alpha_B \lambda_A$, the ROD lies below the standard separatrix.
  • When $\alpha_A \lambda_B < \alpha_B \lambda_A$, the ROD lies to the left of the standard separatrix.
  • The area of the ROD is strictly positive (except when synthesis and threshold ratios are perfectly balanced) and is maximized when the ratios between protein synthesis rates or repression thresholds are large.
• Stochastic Validation: Stochastic simulations confirm that the deterministic ROD is robust to intrinsic noise. Initial conditions falling within the deterministic ROD result in a high proportion of stochastic simulations where the final states of the two models disagree.

Significance and Claims
The paper claims that cell division can fundamentally reshape the fate boundaries in multistable regulatory networks. By neglecting the explicit mechanism of division and relying solely on effective dilution, models may incorrectly predict cellular fates for a significant subset of initial conditions (the ROD). The authors conclude that these discrepancies are not merely artifacts of specific parameter choices but represent a fundamental difference in how the two modeling approaches handle the dynamics of cell growth and division. Consequently, the results suggest that similar discrepancies may arise in more complex multistable gene regulatory networks, implying that the explicit modeling of division is crucial for accurate predictions of cellular decision-making.