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

Projected and Hidden Markov Models for calculating kinetics and metastable states of complex molecules

Markov state models (MSMs) have been successful in computing metastable states, slow relaxation timescales and associated structural changes, and stationary or kinetic experimental observables of complex molecules from large amounts of molecular dynamics simulation data. However, MSMs approximate the true dynamics by assuming a Markov chain on a clusters discretization of the state space. This approximation is difficult to make for high-dimensional biomolecular systems, and the quality and reproducibility of MSMs has therefore been limited. Here, we discard the assumption that dynamics are Markovian on the discrete clusters. Instead, we only assume that the full phase- space molecular dynamics is Markovian, and a projection of this full dynamics is observed on the discrete states, leading to the concept of Projected Markov Models (PMMs). Robust estimation methods for PMMs are not yet available, but we derive a practically feasible approximation via Hidden Markov Models (HMMs). It is shown how various molecular observables of interest that are often computed from MSMs can be computed from HMMs / PMMs. The new framework is applicable to both, simulation and single-molecule experimental data. We demonstrate its versatility by applications to educative model systems, an 1 ms Anton MD simulation of the BPTI protein, and an optical tweezer force probe trajectory of an RNA hairpin

Tehokas strategia biokemiallisten verkkojen päättelyyn mekanististen mallien tilastollisen sovittamisen avulla

Various fields of science employ systems of ordinary differential equations (ODEs) to model the behaviour of dynamical systems, such as gene regulatory networks. However, the system model often contains uncertainty in both its structure and the model parameters. When experimental data are available, the model parameters can be calibrated using well-established statistical techniques and also different model structures can be compared in the light of their statistical evidence. If the set of alternative model structures is small enough, it is possible to evaluate the validity of each individual model separately. However, for biochemical networks, the number of viable model configurations is often enormous, which renders it computationally impossible to draw inferences about the network structure using such an exhaustive strategy. This thesis introduces a novel computationally efficient approach to obtain probabilistic structure inferences for general ODE models. The proposed approach relies on exploring the discrete set of alternative models using Markov chain Monte Carlo methods. Inference problems involving simulated data are used to demonstrate that the method is suitable for efficiently extracting information about the characteristics of the likely models. Furthermore, the method is applied to infer the structure of the transiently evolving core regulatory network that steers the T helper 17 (Th17) cell differentiation. The obtained results are in agreement with earlier studies that suggest that the Th17 differentiation program involves three sequential phases.Differentiaaliyhtälösysteemejä käytetään monilla tieteenaloilla mallintamaan dynaamisia systeemejä, kuten geenisäätelyverkkoja. Systeemiä kuvaavassa mallissa on kuitenkin usein epävarmuutta sekä sen rakenteen että mallin parametrien osalta. Kun kokeellista dataa on saatavilla, mallien parametrit voidaan sovittaa käyttäen vakiintuneita tilastollisia menetelmiä, ja myös erilaisia malleja voidaan vertailla niiden tilastollisen todennäköisyyden avulla. Jos vaihtoehtoisia malleja on vain vähän, voidaan jokainen yksittäinen malli validoida erikseen. Biokemiallisten verkkojen tapauksessa mahdollisia mallikonfiguraatioita on usein lukemattomia, minkä takia yllä kuvattu tapa verkkojen rakenteen päättelyyn on laskennallisesti mahdotonta. Tässä työssä esitellään uusi laskennallisesti tehokas lähestymistapa tehdä probabilistisia päätelmiä differentiaaliyhtälömallien rakenteesta. Ehdotettu lähestymistapa perustuu diskreetin mallijoukon tutkimiseen Markov Chain Monte Carlo -menetelmillä. Työssä muotoillaan simuloituun dataan liittyviä ongelmia, joilla näytetään, että menetelmällä voi tehokkaasti saada tietoa todennäköisimmistä mallirakenteista. Menetelmää sovelletaan myös erään auttaja-T-solujen alityypin (Th17) erilaistumista ajavan aikariippuvan ydinverkon rakenteen päättelyyn. Saadut tulokset ovat linjassa aiempien tutkimusten kanssa, joiden mukaan Th17-solujen erilaistuminen tapahtuu kolmessa peräkkäisessä vaiheessa

Decoding Single Molecule Time Traces with Dynamic Disorder

Single molecule time trajectories of biomolecules provide glimpses into complex folding landscapes that are difficult to visualize using conventional ensemble measurements. Recent experiments and theoretical analyses have highlighted dynamic disorder in certain classes of biomolecules, whose dynamic pattern of conformational transitions is affected by slower transition dynamics of internal state hidden in a low dimensional projection. A systematic means to analyze such data is, however, currently not well developed. Here we report a new algorithm - Variational Bayes-double chain Markov model (VB-DCMM) - to analyze single molecule time trajectories that display dynamic disorder. The proposed analysis employing VB-DCMM allows us to detect the presence of dynamic disorder, if any, in each trajectory, identify the number of internal states, and estimate transition rates between the internal states as well as the rates of conformational transition within each internal state. Applying VB-DCMM algorithm to single molecule FRET data of H-DNA in 100 mM-Na$^+$ solution, followed by data clustering, we show that at least 6 kinetic paths linking 4 distinct internal states are required to correctly interpret the duplex-triplex transitions of H-DNA

Approximation of event probabilities in noisy cellular processes

Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discrete-state continuous-time stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive. We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required

Accelerated Sensitivity Analysis in High-Dimensional Stochastic Reaction Networks

In this paper, a two-step strategy for parametric sensitivity analysis for such systems is proposed, exploiting advantages and synergies between two recently proposed sensitivity analysis methodologies for stochastic dynamics. The first method performs sensitivity analysis of the stochastic dynamics by means of the Fisher Information Matrix on the underlying distribution of the trajectories; the second method is a reduced-variance, finite-difference, gradient-type sensitivity approach relying on stochastic coupling techniques for variance reduction. Here we demonstrate that these two methods can be combined and deployed together by means of a new sensitivity bound which incorporates the variance of the quantity of interest as well as the Fisher Information Matrix estimated from the first method. The first step of the proposed strategy labels sensitivities using the bound and screens out the insensitive parameters in a controlled manner based also on the new sensitivity bound. In the second step of the proposed strategy, the finite-difference method is applied only for the sensitivity estimation of the (potentially) sensitive parameters that have not been screened out in the first step. Results on an epidermal growth factor network with fifty parameters and on a protein homeostasis with eighty parameters demonstrate that the proposed strategy is able to quickly discover and discard the insensitive parameters and in the remaining potentially sensitive parameters it accurately estimates the sensitivities. The new sensitivity strategy can be several times faster than current state-of-the-art approaches that test all parameters, especially in "sloppy" systems. In particular, the computational acceleration is quantified by the ratio between the total number of parameters over the number of the sensitive parameters