Fig. 2: Illustration of the spline-DV method proposed in this study.

A Two spline-fit curves are created independently with scRNA-seq data of two conditions, blue for condition 1 and red for condition 2. These spline curves represent the “expected” expression profile for genes based on their statistical properties in each condition. B Two spline-fit curves are projected into the same 3D space for comparison. For a given gene such as Gene A, its position is compared with respect to the condition 1 spline-fit curve. The vector \({\vec{v}}_{1}\) extends from the closest point on the blue curve (spline-fit) to the blue point (Gene A in condition 1). Similarly, for condition 2 (red point), the vector \({\vec{v}}_{2}\) extends from the closest point on the red curve to the red point (Gene A in condition 2). The L2-norm of \({\vec{v}}_{1}\) (or \({\vec{v}}_{2})\) quantifies the deviation of the same Gene A from its expectation in condition 1 (or condition 2). C Given that the deviation is the shortest Euclidean distance from the blue (or red) point to the blue (or red) spline-fit curve, representing the level of expression variability of the gene in condition 1 (or condition 2), \(\vec{{dv}}={\vec{v}}_{2}-{\vec{v}}_{1}\), is used to measure the level of differential variability. The L2-norm of \(\vec{{dv}}\) is called DV score. Note that, in B, \({\vec{v}}_{1}\) and \({\vec{v}}_{2}\) have their own origin with respect to their respective conditions; in C, \({\vec{v}}_{1}\) and \({\vec{v}}_{2}\) share the same origin to compute \(\vec{{dv}}\).