Applications of PDEs and Stochastic Modeling to Protein Transport in Cell Biology

The paper outlines the same Lyapunov–Schmidt reduction on L(λ,ν)=A+νC+ν^2D−λI, obtains the quadratic dispersion relation λ(ν)=⟨ψ0,Cu0⟩/⟨ψ0,u0⟩·ν + ⟨ψ0,(D−CÃ^{-1}C̃)u0⟩/⟨ψ0,u0⟩·ν^2, and then inverts the Fourier transform to produce the long-time Gaussian profile u_l(y,t)=(2πσ_eff t)^{-1/2}exp(−(y+v_eff t)^2/(2σ_eff t)) with v_eff,σ_eff as above; it also gives the explicit two-state formulas v_eff=c β2/(β1+β2), σ_eff=d β1/(β1+β2)+c^2 β1β2/(β1+β2)^3. These match the candidate solution’s steps and coefficients exactly. The candidate solution provides a more detailed and rigorous sketch (projectors Π0,Q0, solving the range equation, spectral projector P(ik), small-|k| semigroup decomposition, error control O(t^{-1})), which the paper references but does not detail. Hence, both are correct, with substantially the same method and results, the model adding technical detail beyond the paper’s expository outline. Key matches appear in the paper’s equations (4)–(8) and discussion of the two-state model and Gaussian asymptotics .