Time-dependent saddle-node bifurcation: Breaking time and the point of no return in a non-autonomous model of critical transitions.

There is a growing awareness that catastrophic phenomena in biology and medicine can be mathematically represented in terms of saddle-node bifurcations. In particular, the term "tipping", or critical transition has in recent years entered the discourse of the general public in relation to ecology, medicine, and public health. The saddle-node bifurcation and its associated theory of catastrophe as put forth by Thom and Zeeman has seen applications in a wide range of fields including molecular biophysics, mesoscopic physics, and climate science. In this paper, we investigate a simple model of a non-autonomous system with a time-dependent parameter p(τ) and its corresponding "dynamic" (time-dependent) saddle-node bifurcation by the modern theory of non-autonomous dynamical systems. We show that the actual point of no return for a system undergoing tipping can be significantly delayed in comparison to the breaking time τ ^ at which the corresponding autonomous system with a time-independent parameter p a = p ( τ ^ ) undergoes a bifurcation. A dimensionless parameter α = λ p 0 3 V - 2 is introduced, in which λ is the curvature of the autonomous saddle-node bifurcation according to parameter p(τ), which has an initial value of p 0 and a constant rate of change V. We find that the breaking time τ ^ is always less than the actual point of no return τ ∗ after which the critical transition is irreversible; specifically, the relation τ * - τ ^ ≃ 2.338 ( λ V ) - 1 3 is analytically obtained. For a system with a small λV, there exists a significant window of opportunity ( τ ^ , τ ∗) during which rapid reversal of the environment can save the system from catastrophe.

Quantifying information accumulation encoded in the dynamics of biochemical signaling.

Cellular responses to environmental changes are encoded in the complex temporal patterns of signaling proteins. However, quantifying the accumulation of information over time to direct cellular decision-making remains an unsolved challenge. This is, in part, due to the combinatorial explosion of possible configurations that need to be evaluated for information in time-course measurements. Here, we develop a quantitative framework, based on inferred trajectory probabilities, to calculate the mutual information encoded in signaling dynamics while accounting for cell-cell variability. We use it to understand NFκB transcriptional dynamics in response to different immune threats, and reveal that some threats are distinguished faster than others. Our analyses also suggest specific temporal phases during which information distinguishing threats becomes available to immune response genes; one specific phase could be mapped to the functionality of the IκBα negative feedback circuit. The framework is generally applicable to single-cell time series measurements, and enables understanding how temporal regulatory codes transmit information over time.