Explicit solution for unevenly spaced measurement times in QWENDY

Derive an explicit solution that generalizes QWENDY to measurements at four non-equidistant time points t0 < t1 < t2 < t3, where the intervals Δ1 = t1 − t0, Δ2 = t2 − t1, and Δ3 = t3 − t2 are unequal. Specifically, given single-cell gene expression covariance matrices K0, K1, K2, and K3 at these times under the linearized dynamics K_{i+1} ≈ B(Δ_{i+1})^T K_i B(Δ_{i+1}) with B(Δ) = I + Δ A for an unknown gene regulatory network matrix A, construct a procedure to determine A without relying on the heuristic approximation I + (t2 − t1) A ≈ [(t2 − t1)/(t1 − t0)] [I + (t1 − t0) A].

Background

QWENDY infers gene regulatory networks by leveraging a linear approximation of covariance dynamics using single-cell expression data collected at four evenly spaced time points. In this equidistant setting, the transformation between consecutive covariance matrices can be expressed with a single matrix B = I + t A, enabling a unique recovery of the regulatory matrix A.

When the measurement intervals are unequal, the covariance evolution is more complex and the current QWENDY derivation does not yield a closed-form procedure. A scaling-based approximation is suggested to reuse one transformation across different intervals, but a rigorous explicit solution for the non-equidistant case is not available, leaving this generalization unresolved.