Revisiting Hybridization Kinetics with Improved Elementary Step Simulation

Nucleic acid strands, which react by forming and breaking Watson-Crick base pairs, can be designed to form complex nanoscale structures or devices. Controlling such systems requires accurate predictions of the reaction rate and of the folding pathways of interacting strands. Simulators such as Multistrand model these kinetic properties using continuous-time Markov chains (CTMCs), whose states and transitions correspond to secondary structures and elementary base pair changes, respectively. The transient dynamics of a CTMC are determined by a kinetic model, which assigns transition rates to pairs of states, and the rate of a reaction can be estimated using the mean first passage time (MFPT) of its CTMC. However, use of Multistrand is limited by its slow runtime, particularly on rare events, and the quality of its rate predictions is compromised by a poorly-calibrated and simplistic kinetic model. The former limitation can be addressed by constructing truncated CTMCs, which only include a small subset of states and transitions, selected either manually or through simulation. As a first step to address the latter limitation, Bayesian posterior inference in an Arrhenius-type kinetic model was performed in earlier work, using a small experimental dataset of DNA reaction rates and a fixed set of manually truncated CTMCs, which we refer to as Assumed Pathway (AP) state spaces. In this work we extend this approach, by introducing a new prior model that is directly motivated by the physical meaning of the parameters and that is compatible with experimental measurements of elementary rates, and by using a larger dataset of 1105 reactions as well as larger truncated state spaces obtained from the recently introduced stochastic Pathway Elaboration (PE) method. We assess the quality of the resulting posterior distribution over kinetic parameters, as well as the quality of the posterior reaction rates predicted using AP and PE state spaces. Finally, we use the newly parameterised PE state spaces and Multistrand simulations to investigate the strong variation of helix hybridization reaction rates in a dataset of Hata et al. While we find strong evidence for the nucleation-zippering model of hybridization, in the classical sense that the rate-limiting phase is composed of elementary steps reaching a small "nucleus" of critical stability, the strongly sequence-dependent structure of the trajectory ensemble up to nucleation appears to be much richer than assumed in the model by Hata et al. In particular, rather than being dominated by the collision probability of nucleation sites, the trajectory segment between first binding and nucleation tends to visit numerous secondary structures involving misnucleation and hairpins, and has a sizeable effect on the probability of overcoming the nucleation barrier

Bayesian modelling of DNA secondary structure kinetics : revisiting path space approximations and posterior inference in exponentially large state spaces

Nucleic acid strands, which react by forming and breaking Watson-Crick base pairs, can be designed to form complex nanoscale structures or devices. Controlling such systems requires accurate predictions of the reaction rate and folding pathways of the interacting strands. These kinetic properties can be modelled using continuous-time Markov chains (CTMCs), whose states and transitions correspond to secondary structures and elementary base pair changes, respectively. The transient dynamics of a CTMC are determined by a kinetic model, which assigns transition rates to pairs of states. The rate of a reaction can be estimated using its CTMC's mean first passage time (MFPT), which can be computed exactly by solving a linear system, or approximated via Monte Carlo simulation. However, both approaches may be computationally infeasible for rare event reactions in larger systems. This limitation can be addressed by constructing truncated CTMCs, which only include a small subset of states and transitions, selected either manually or through simulation. In recent work, posterior inference in an Arrhenius-type kinetic model was performed using a fixed set of manually truncated CTMCs and a small experimental dataset of DNA reaction rates. We extend this Bayesian approach, using a larger dataset of 1105 reactions, a new prior model that is directly motivated by the physical meaning of the parameters and is compatible with experimental measurements of elementary rates, and larger truncated state spaces, constructed stochastically using the recently introduced pathway elaboration method. Despite a significantly higher computational cost, we find that the larger state spaces do not necessarily lead to more accurate rate predictions than the small, manually designed state spaces. For posterior approximation, we apply the standard random walk Metropolis algorithm and the gradient-based no-u-turn sampler. Our posterior approximations, which are often multimodal, recover an expected correlation structure among the kinetic parameters. However, we also uncover severe numerical instability in the MPFT computations. Due to numerous design limitations in the legacy software, a significant refactoring effort was required to implement the above extensions, resulting also in improvements in performance and reproducibility.Science, Faculty ofComputer Science, Department ofGraduat

Computing Four-Center Two-Electron Coulomb Integrals Using Exponential Transformations and Trapezoidal Rule

The numerical evaluations of the four-center two-electron Coulomb integrals are among the most time-consuming computations involved in molecular electronic structure calculations. In the present paper we extend the double exponential (DE) transform method, previously developed for the numerical evaluation of threecenter one-electron molecular integrals [J. Lovrod, H. Safouhi, Molecular Physics (2019) DOI:10.1030/0026867.2019.1619854], to four-center two-electron integrals. The fast convergence properties analyzed in the numerical section illustrate the advantages of the new approach

Computing Four-Center Two-Electron Coulomb Integrals Using Exponential Transformations and Trapezoidal Rule

The numerical evaluations of the four-center two-electron Coulomb integrals are among the most time-consuming computations involved in molecular electronic structure calculations. In the present paper we extend the double exponential (DE) transform method, previously developed for the numerical evaluation of threecenter one-electron molecular integrals [J. Lovrod, H. Safouhi, Molecular Physics (2019) DOI:10.1030/0026867.2019.161985