Inspired by recent experiments on the spontaneous assembly of virus-like particles from a solution containing a synthetic coat protein and double-stranded DNA [Carlos Calcines-Cruz, Ilya J. Finkelstein, and Armando Hernandez-Garcia, Nano Lett. 21 (2021), 2752-2757], we put forward a kinetic model that has as main ingredients a stochastic nucleation and a deterministic growth process. The efficiency and rate of the packaging of the DNA turn out to strongly increase by introducing proteins onto the DNA template that are modified using CRISPR-Cas techniques to bind specifically at predesignated locations, mimicking assembly signals in viruses. Our model shows that treating these proteins as nucleation-inducing diffusion barriers is sufficient to explain experimentally observed increase in encapsulation efficiency, but only if the nucleation rate is sufficiently high. We find an optimum in the encapsulation kinetics for conditions where the number of packaging signals is equal to the number of nucleation events that can occur during time required to fully encapsulate the DNA template, presuming that the nucleation events can only take place adjacent to a packaging signal. Our theory is in satisfactory agreement with the available experimental data.