9 research outputs found
Trajectory ensemble methods for understanding complex stochastic systems
This thesis investigates the equilibrium and dynamic properties of stochastic systems of varying complexity. The dynamic properties of lattice models -- the 1-d Ising model and a 3-d protein model -- and equilibrium properties of continuous models -- particles in various potentials -- are presented. Dynamics are studied according to a large deviation formalism, by looking at non-equilibrium ensembles of trajectories, classified according to a dynamical order parameter. The phase structure of the ensembles of trajectories is deduced from the properties of large-deviation functions, representing dynamical free-energies.
The 1-d Ising model is studied with Glauber dynamics uncovering the dynamical second-order transition at critical values of the counting field 's', confirming the analytical predictions by Jack and Solich. Next, the dynamics in an external magnetic field are studied, allowing the construction of a dynamic phase diagram in the space of temperature, s-field and magnetic field. The dynamic phase diagram is reminiscent of that of the 2-d Ising model. In contrast, Kawasaki dynamics give rise to a dynamical phase structure similar to the one observed in kinetically constrained models.
The dynamics of a lattice protein model, represented by a self avoiding walk with three different Hamiltonians, are studied. For the uniform Go Hamiltonian all dynamics occurs between non-native and native trajectories, whereas for heterogeneous Hamiltonians and Full interaction Hamiltonians a first-order dynamical transition to sets of trapping trajectories is observed in the s-ensemble. The model is studied exhaustively for a particular sequence, constructing a qualitative phase diagram, from which a more general dynamic behaviour is extrapolated.
Lastly, an estimator for equilibrium expectations, represented by a transition matrix in an extended space between temperatures and a set of discrete states obtained through the discretisation of a continuous space, is proposed. It is then demonstrated that this estimator outperforms conventional multi-temperature ensemble estimates by up to three orders of magnitude, by considering three models of increasing complexity: diffusive particles in a double-well potential, a multidimensional folding potential and a molecular dynamics simulations of alanine dipeptide
Trajectory ensemble methods for understanding complex stochastic systems
This thesis investigates the equilibrium and dynamic properties of stochastic systems of varying complexity. The dynamic properties of lattice models -- the 1-d Ising model and a 3-d protein model -- and equilibrium properties of continuous models -- particles in various potentials -- are presented. Dynamics are studied according to a large deviation formalism, by looking at non-equilibrium ensembles of trajectories, classified according to a dynamical order parameter. The phase structure of the ensembles of trajectories is deduced from the properties of large-deviation functions, representing dynamical free-energies.
The 1-d Ising model is studied with Glauber dynamics uncovering the dynamical second-order transition at critical values of the counting field 's', confirming the analytical predictions by Jack and Solich. Next, the dynamics in an external magnetic field are studied, allowing the construction of a dynamic phase diagram in the space of temperature, s-field and magnetic field. The dynamic phase diagram is reminiscent of that of the 2-d Ising model. In contrast, Kawasaki dynamics give rise to a dynamical phase structure similar to the one observed in kinetically constrained models.
The dynamics of a lattice protein model, represented by a self avoiding walk with three different Hamiltonians, are studied. For the uniform Go Hamiltonian all dynamics occurs between non-native and native trajectories, whereas for heterogeneous Hamiltonians and Full interaction Hamiltonians a first-order dynamical transition to sets of trapping trajectories is observed in the s-ensemble. The model is studied exhaustively for a particular sequence, constructing a qualitative phase diagram, from which a more general dynamic behaviour is extrapolated.
Lastly, an estimator for equilibrium expectations, represented by a transition matrix in an extended space between temperatures and a set of discrete states obtained through the discretisation of a continuous space, is proposed. It is then demonstrated that this estimator outperforms conventional multi-temperature ensemble estimates by up to three orders of magnitude, by considering three models of increasing complexity: diffusive particles in a double-well potential, a multidimensional folding potential and a molecular dynamics simulations of alanine dipeptide
xTRAM: Estimating equilibrium expectations from time-correlated simulation data at multiple thermodynamic states
Computing the equilibrium properties of complex systems, such as free energy
differences, is often hampered by rare events in the dynamics. Enhanced
sampling methods may be used in order to speed up sampling by, for example,
using high temperatures, as in parallel tempering, or simulating with a biasing
potential such as in the case of umbrella sampling. The equilibrium properties
of the thermodynamic state of interest (e.g., lowest temperature or unbiased
potential) can be computed using reweighting estimators such as the weighted
histogram analysis method or the multistate Bennett acceptance ratio (MBAR).
weighted histogram analysis method and MBAR produce unbiased estimates, the
simulation samples from the global equilibria at their respective thermodynamic
state--a requirement that can be prohibitively expensive for some simulations
such as a large parallel tempering ensemble of an explicitly solvated
biomolecule. Here, we introduce the transition-based reweighting analysis
method (TRAM)--a class of estimators that exploit ideas from Markov modeling
and only require the simulation data to be in local equilibrium within subsets
of the configuration space. We formulate the expanded TRAM (xTRAM) estimator
that is shown to be asymptotically unbiased and a generalization of MBAR. Using
four exemplary systems of varying complexity, we demonstrate the improved
convergence (ranging from a twofold improvement to several orders of magnitude)
of xTRAM in comparison to a direct counting estimator and MBAR, with respect to
the invested simulation effort. Lastly, we introduce a random-swapping
simulation protocol that can be used with xTRAM, gaining orders-of-magnitude
advantages over simulation protocols that require the constraint of sampling
from a global equilibrium.Comment: 23 pages with appendices, 5 figure
Dynamic Profiling of β-Coronavirus 3CL M<sup>pro</sup>Protease Ligand-Binding Sites
Data availability statement: The trajectories of Mpro simulations and models of the metastable states can be downloaded from 10.5281/zenodo.4782284.β-coronavirus (CoVs) alone has been responsible for three major global outbreaks in the 21st century. The current crisis has led to an urgent requirement to develop therapeutics. Even though a number of vaccines are available, alternative strategies targeting essential viral components are required as a backup against the emergence of lethal viral variants. One such target is the main protease (Mpro) that plays an indispensable role in viral replication. The availability of over 270 Mpro X-ray structures in complex with inhibitors provides unique insights into ligand–protein interactions. Herein, we provide a comprehensive comparison of all nonredundant ligand-binding sites available for SARS-CoV2, SARS-CoV, and MERS-CoV Mpro. Extensive adaptive sampling has been used to investigate structural conservation of ligand-binding sites using Markov state models (MSMs) and compare conformational dynamics employing convolutional variational auto-encoder-based deep learning. Our results indicate that not all ligand-binding sites are dynamically conserved despite high sequence and structural conservation across β-CoV homologs. This highlights the complexity in targeting all three Mpro enzymes with a single pan inhibitor.There was no funding for this wor