Fig. 1.

Thermodynamic model of dCas12a binding. (A) Thermodynamic states, energies, and Boltzmann weights of a dCas12a ($\beta =k_{B}T$, where $k_B$ = Boltzmann constant, T = temperature) for the PAM attachment, crRNA–DNA inspection, and reconfiguration steps. The fold change (FC), PAM occupancy ${\theta }_{PAM}$, and CRISPR-Cas occupancy ${\theta }_c$ depend on the effective PAM energy ${ϵ}_{PAM}$, the CRISPR-Cas binding energy ${ϵ}_c$, and the DNA replication rate $\mathrm{\Lambda }$. All expressions assume that binding occurs in the weak promoter (${\lambda }_p{e}^{-\beta {ϵ}_p}\ll 1$) and weak PAM binding (${\lambda }_c{e}^{-\beta {ϵ}_{PAM}}\ll 1$) limits, where ${\lambda }_c=e^{\beta {\mu }_c}$ is the fugacity of a dCas12a molecule with chemical potential ${\mu }_c$. (B) Internal base-dependent energies ${ϵ}_b^i$ define a PAM-specific binding energy ${ϵ}_{PAM}={\sum }_i{ϵ}_b^i$. In this model, the PAM-specific binding energy between two targets with energy that differs by $\mathrm{\Delta }{ϵ}_{PAM}$ scales their relative PAM attachment efficiency by $e^{-\beta \mathrm{\Delta }{ϵ}_{PAM}}$. The presence of crRNA–target DNA mismatches increases the effective activation energy $E_a$ by a mismatch energy $\mathrm{\Delta }{ϵ}^{*}$ and scales the effective reconfiguration rate $\nu$ by a multiplicative factor equal to $e^{-\beta \mathrm{\Delta }{ϵ}^{*}}$.