Markov state models based on milestoning

Markov state models (MSMs) have become the tool of choice to analyze large amounts of molecular dynamics data by approximating them as a Markov jump process between suitably predefined states. Here we investigate "Core Set MSMs," a new type of MSMs that build on metastable core sets acting as milestones for tracing the rare event kinetics. We present a thorough analysis of Core Set MSMs based on the existing milestoning framework, Bayesian estimation methods and Transition Path Theory (TPT). We show that Core Set MSMs can be used to extract phenomenological rate constants between the metastable sets of the system and to approximate the evolution of certain key observables. The performance of Core Set MSMs in comparison to standard MSMs is analyzed and illustrated on a toy example and in the context of the torsion angle dynamics of alanine dipeptide.     

Validation of Markov state models using Shannon's entropy

Markov state models are kinetic models built from the dynamics of molecular simulation trajectories by grouping similar configurations into states and examining the transition probabilities between states. Here we present a procedure for validating the underlying Markov assumption in Markov state models based on information theory using Shannon's entropy. This entropy method is applied to a simple system and is compared with the previous eigenvalue method. The entropy method also provides a way to identify states that are least Markovian, which can then be divided into finer states to improve the model.     

Building Markov state models along pathways to determine free energies and rates of transitions

An efficient method is proposed for building Markov models with discrete states able to accurately describe the slow relaxation of a complex system with two stable conformations. First, the reaction pathway described by a set of collective variables between the two stable states is determined using the string method with swarms of trajectories. Then, short trajectories are initiated at different points along this pathway to build the state-to-state transition probability matrix. It is shown, using a model system, how this strategy makes it possible to use trajectories that are significantly shorter than the slowest relaxation time to efficiently build a reliable and accurate Markov model. Extensions of the method to multiple pathways, as well as some common pitfalls arising from poorly relaxed paths or an inappropriate choice of collective variables, are illustrated and discussed.     

Foldamer dynamics expressed via Markov state models. II. State space decomposition

The structural landscape of poly-phenylacetylene (pPA), otherwise known as m-phenylene ethynylene oligomers, has been shown to consist of a very diverse set of conformations, including helices, turns, and knots. Defining a state space decomposition to classify these conformations into easily identifiable states is an important step in understanding the dynamics in relation to Markov state models. We define the state decomposition of pPA oligomers in terms of the sequence of discretized dihedral angles between adjacent phenyl rings along the oligomer backbone. Furthermore, we derive in mathematical detail an approach to further reduce the number of states by grouping symmetrically equivalent states into a single parent state. A more challenging problem requires a formal definition for knotted states in the structural landscape. Assuming that the oligomer chain can only cross the ideal helix path once, we propose a technique to define a knotted state derived from a helical state determined by the position along the helical nucleus where the chain crosses the ideal helix path. Several examples of helical states and knotted states from the pPA 12-mer illustrate the principles outlined in this article.     

The p53 network: p53 and its downstream genes

The tumor-suppressor gene p53 and its downstream genes consist of a complicated gene network. p53 is a key molecular node in the network, which is activated in response to several cellular signals resulting in the maintenance of genetic stability. Several cellular signals may activate the p53 network. When the expression of P53 is elevated, P53-MDM2 module and the ubiquitin system can accurately regulate the expression level of P53. P53 can bind to specific DNA sequence, activate its downstream genes expression, and control cell-cycle arrest, DNA repair, and apoptosis. Elucidating the function of p53 gene network will help understand the interaction mechanisms of p53 and its downstream genes.     

A Markov model for relaxation and exchange in NMR spectroscopy

A two-state Markov noise process for lattice fluctuations and chemical exchange dynamics is used to derive a stochastic Liouville equation describing the evolution of the spin-density operator in nuclear magnetic resonance spectroscopy. Relaxation through lattice fluctuations and chemical exchange processes is incorporated into the theory at the same fundamental level, and the results are valid for all time scales provided that lattice fluctuations are much faster than chemical exchange kinetics. Time-scale separation emerges as an essential feature from the lowest-order perturbation expansion of the average resolvent in the Laplace domain.     

Hidden Markov models and optimized sequence alignments

We present a formulation of the Needleman-Wunsch type algorithm for sequence alignment in which the mutation matrix is allowed to vary under the control of a hidden Markov process. The fully trainable model is applied to two problems in bioinformatics: the recognition of related gene/protein names and the alignment and scoring of homologous proteins.     

Decreased in vitro interaction between p53 and nuclear stress proteins in the p53-deficient mouse

In a previous study, the strength of the interaction between the nuclear stress proteins (sps) 25a, 70i, 72c, and 90 and the tumor suppressor protein p53 was determined by an in vitro fluorescence binding assay. The relative binding of the individual sps with p53, derived from the bone marrow of transgenic mice heterozygous at the p53 locus (p53+/-), was reduced compared to the interaction of sps and p53 derived from wild-type (p53+/+) mice. In order to determine if the genotype of the p53 donor or the genotype of the sp donor determined the binding efficiency, p53 expression was induced by retinoic acid and sp synthesis by bleomycin. P53 derived from either wild-type or heterozygous animals was cross-reacted with nuclear sps obtained from either wild-type or heterozygous animals. Each of the sps, 25a, 70i, 72c, and 90, bound to wild-type p53 with a similar efficiency, irrespective of the genotype of the sp donor mouse (p53+/+ or p53+/-). In contrast, when the sp interaction with p53 obtained from the heterozygous mouse was measured, the relative value of the fluorescence complex was significantly reduced. The data suggest that the strength of the interaction between p53 and nuclear sps is related to the genotype of the p53 donor, and not to the genotype of the animals from which the sps are derived.     

Involvement of JNK-initiated p53 accumulation and phosphorylation of p53 in pseudolaric acid B induced cell death

A terminal deoxynucleotidyl transferase-mediated dUTP nick end-labeling (TUNEL) assay was used to determine that apoptosis causes HeLa cell death induced by pseudolaric acid B. The c-Jun N-terminal kinase (JNK) inhibitor SP600125 decreased p53 protein expression during exposure to pseudolaric acid B. SP600125 decreased the phosphorylation of p53 during pseudolaric acid B exposure, indicating that JNK mediates phosphorylation of p53 during the response to pseudolaric acid B. SP600125 reversed pseudolaric acid B-induced down-regulation of phosphorylated extracellular signal-regulated protein kinase (ERK), and protein kinase C (PKC) was activated by pseudolaric acid B, whereas staurosporine, calphostin C, and H7 partly blocked this effect. These results indicate that p53 is partially regulated by JNK in pseudolaric acid B-induced HeLa cell death and that PKC participates in pseudolaric acid B-induced HeLa cell death.     

Markov models of molecular kinetics: generation and validation

Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years. This approach has many appealing characteristics compared to straightforward molecular dynamics simulation and analysis, including the potential to mitigate the sampling problem by extracting long-time kinetic information from short trajectories and the ability to straightforwardly calculate expectation values and statistical uncertainties of various stationary and dynamical molecular observables. In this paper, we summarize the current state of the art in generation and validation of MSMs and give some important new results. We describe an upper bound for the approximation error made by modeling molecular dynamics with a MSM and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast to previous practice, it becomes clear that the best MSM is not obtained by the most metastable discretization, but the MSM can be much improved if non-metastable states are introduced near the transition states. Moreover, we show that it is not necessary to resolve all slow processes by the state space partitioning, but individual dynamical processes of interest can be resolved separately. We also present an efficient estimator for reversible transition matrices and a robust test to validate that a MSM reproduces the kinetics of the molecular dynamics data.     

Topology-based models and NMR structures in protein folding simulations

Topology-based interaction potentials are simplified models that use the native contacts in the folded structure of a protein to define an energetically unfrustrated folding funnel. They have been widely used to analyze the folding transition and pathways of different proteins through computer simulations. Obviously, they need a reliable, experimentally determined folded structure to define the model interactions. In structures elucidated through NMR spectroscopy, a complex treatment of the raw experimental data usually provides a series of models, a set of different conformations compatible with the available experimental data. Here, we use an efficient coarse-grained simulation technique to independently consider the contact maps from every different NMR model in a protein whose structure has been resolved by the use of NMR spectroscopy. For lambda-Cro repressor, a homodimeric protein, we have analyzed its folding characteristics with a topology-based model. We have focused on the competition between the folding of the individual chains and their binding to form the final quaternary structure. From 20 different NMR models, we find a predominant three-state folding behavior, in agreement with experimental data on the folding pathway for this protein. Individual NMR models, however, show distinct characteristics, which are analyzed both at the level of the interplay between tertiary/quaternary structure formation and also regarding the thermal stability of the tertiary structure of every individual chain.     

Quadrupolar coupling selective cross-polarization in solid state NMR

A new strategy is presented for achieving selective heteronuclear polarization transfers from half-integer quadrupolar spins in magic-angle spinning (MAS) NMR. By combining cross-polarization with a recently introduced RAPT pulse sequence that selectively excites the signal of a half-integer quadrupolar nucleus based on its quadrupolar coupling constant magnitude, we demonstrate that hetero-nuclei in its close proximity may be selectively excited. Selective 23Na --> 1H polarization transfers are demonstrated in Na2MoO4 x 2 H2O, Na2HPO4 x 2 H2O and a mixture of NaHCO3 and Na2HPO4 x 2 H2O.     

Frequency-selective homonuclear dipolar recoupling in solid state NMR

We introduce a new approach to frequency-selective homonuclear dipolar recoupling in solid state nuclear magnetic resonance (NMR) with magic-angle spinning (MAS). This approach, to which we give the acronym SEASHORE, employs alternating periods of double-quantum recoupling and chemical shift evolution to produce phase modulations of the recoupled dipole-dipole interactions that average out undesired couplings, leaving only dipole-dipole couplings between nuclear spins with a selected pair of NMR frequencies. In principle, SEASHORE is applicable to systems with arbitrary coupling strengths and arbitrary sets of NMR frequencies. Arbitrary MAS frequencies are also possible, subject only to restrictions imposed by the pulse sequence chosen for double-quantum recoupling. We demonstrate the efficacy of SEASHORE in experimental (13)C NMR measurements of frequency-selective polarization transfer in uniformly (15)N, (13)C-labeled L-valine powder and frequency-selective intermolecular polarization transfer in amyloid fibrils formed by a synthetic decapeptide containing uniformly (15)N, (13)C-labeled residues.     

Broadband rotational resonance in solid state NMR spectroscopy

A new technique for restoring nuclear magnetic dipole-dipole couplings under magic-angle spinning (MAS) in solid state nuclear magnetic resonance (NMR) spectroscopy is described and demonstrated. In this technique, called broadband rotational resonance (BroBaRR), the coupling between a pair of nuclear spins with NMR frequency difference close (but not necessarily equal) to the MAS frequency is restored by the application of a train of weak radio-frequency pulses at a carrier frequency close to the average of the two NMR frequencies. Phase or amplitude modulation of the pulse train at half the MAS frequency splits the carrier into sidebands close to the two NMR frequencies. The pulse train then removes offsets from the exact rotational resonance condition, leading to dipolar recoupling over a bandwidth controlled by the amplitude of the pulse train. (13)C NMR experiments on uniformly (15)N,(13)C-labeled L-valineHClH(2)O powder validate the theoretical analysis. BroBaRR will be useful in studies of molecular structures by solid state NMR, for example in the detection of long-range couplings between carbons in uniformly labeled organic and biological materials.     

New solid state NMR techniques in coal analysis

Nuclear magnetic resonance (NMR) spectroscopy has become a valuable technique for the determination of the aromaticity of fossil fuels. Now, by exploiting ¹³C relaxation, intimate details of the structure of coal may be determined. This article briefly describes these new applications.