Solve the MFPT equation for non-constant energy landscapes

Solve the second-order inhomogeneous ordinary differential equation d^2 T(x)/dx^2 − (k/D) T(x) + e^{−E(x)}/D · q(x) = 0 for the mean first passage time T(x) within a single 1D round, under general non-constant binding energy landscapes E(x), with boundary conditions T(0) = 0 and T(∞) = 0, where q(x) = exp(−√(k/D) · |x|) and k, D are the detachment and diffusion rate constants defined by k(x) = k · exp(E(x)) and D(x) = D · exp(E(x)).

Background

In the facilitated diffusion model, T(x) denotes the mean first passage time to the origin (central PAM) during a single 1D sliding round, starting from position x on the DNA. The authors derive the governing equation for T(x) in the continuous limit as d^2 T(x)/dx^2 − (k/D) T(x) + e^{−E(x)}/D * q(x) = 0, where E(x) is the sequence-dependent binding energy, k and D are baseline detachment and diffusion rates, and q(x) = exp(−√(k/D) * |x|) is the probability to reach the origin before detaching.

While they provide solutions for the constant-energy case and proceed by ensemble averaging to obtain expressions for ⟨T(x)⟩ when E(x) is random, they explicitly state that they cannot directly solve the MFPT equation for general non-constant E(x). A direct solution would characterize T(x) for specific heterogeneous energy landscapes without resorting to averaging assumptions.