In the context of cellular signaling and gene regulatory networks, the concept of molecular memory emerges as a crucial determinant of molecular mechanisms. This study introduces a novel memory quantifier designed to comprehensively capture and quantify a system's memory in response to transient stimuli. We propose and validate this quantifier using simplified models, demonstrating its effectiveness in systems with positive feedback loops and bistability.

Additionally, we develop an algorithm to assess long-term memory in circuits, leading to the identification of minimal motifs that play pivotal roles in conferring memory. The research explores the comparative impact of positive and negative feedback loops on memory, revealing that positive feedback enhances memory, while certain negative feedback loops may diminish it. An intriguing finding is that oscillating circuits, even in the absence of positive feedback, exhibit memory, with the phase of oscillations storing information about the stimulus duration.

Finally, we experimentally validate the quantifier using mouse embryonic stem cells (mESCs) subjected to transient differentiation stimuli. The proposed memory quantifier is applied to gene expression dynamics, revealing varying degrees of memory retention among different genes. The vectorial nature of the quantifier proves advantageous in capturing the holistic memory dynamics of the system.