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      Probability Fluxes and Transition Paths in a Markovian Model Describing Complex Subunit Cooperativity in HCN2 Channels

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      1 , * , 1 , 2
      PLoS Computational Biology
      Public Library of Science

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          Abstract

          Hyperpolarization-activated cyclic nucleotide-modulated (HCN) channels are voltage-gated tetrameric cation channels that generate electrical rhythmicity in neurons and cardiomyocytes. Activation can be enhanced by the binding of adenosine-3′,5′-cyclic monophosphate (cAMP) to an intracellular cyclic nucleotide binding domain. Based on previously determined rate constants for a complex Markovian model describing the gating of homotetrameric HCN2 channels, we analyzed probability fluxes within this model, including unidirectional probability fluxes and the probability flux along transition paths. The time-dependent probability fluxes quantify the contributions of all 13 transitions of the model to channel activation. The binding of the first, third and fourth ligand evoked robust channel opening whereas the binding of the second ligand obstructed channel opening similar to the empty channel. Analysis of the net probability fluxes in terms of the transition path theory revealed pronounced hysteresis for channel activation and deactivation. These results provide quantitative insight into the complex interaction of the four structurally equal subunits, leading to non-equality in their function.

          Author Summary

          The activation of a receptor protein by a small molecule (ligand) can be quantified by Markovian models. These models, which are widely used in natural sciences, consist of distinct states and transitions between them. In nature receptor proteins are often formed by the assembly of more than one subunit and each subunit can bind a ligand on its own. In such a multimeric receptor protein the translation of the ligand binding into receptor activation is more complex because the subunits interact. This usually limits the application of Markovian models. HCN2 pacemaker channels are tetrameric ion channels that mediate electrical rhythmicity in multiple brain and peripheral neurons and in specialized heart cells. The channels are modulated by cAMP binding to each subunit. We were recently successful to quantify ligand-induced activation for these channels by a complex Markovian model and a full set of rate constants. Herein we applied the transition path theory to further analyze the identified Markovian model, and we quantified time-dependent probability fluxes within the model. Our results provide unprecedented insight into the complex interaction of the four structurally equal subunits of a presumably fourfold symmetric channel that leads to pronounced non-equality of the subunit function.

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          How fast-folding proteins fold.

          An outstanding challenge in the field of molecular biology has been to understand the process by which proteins fold into their characteristic three-dimensional structures. Here, we report the results of atomic-level molecular dynamics simulations, over periods ranging between 100 μs and 1 ms, that reveal a set of common principles underlying the folding of 12 structurally diverse proteins. In simulations conducted with a single physics-based energy function, the proteins, representing all three major structural classes, spontaneously and repeatedly fold to their experimentally determined native structures. Early in the folding process, the protein backbone adopts a nativelike topology while certain secondary structure elements and a small number of nonlocal contacts form. In most cases, folding follows a single dominant route in which elements of the native structure appear in an order highly correlated with their propensity to form in the unfolded state.
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            Transition-path theory and path-finding algorithms for the study of rare events.

            Transition-path theory is a theoretical framework for describing rare events in complex systems. It can also be used as a starting point for developing efficient numerical algorithms for analyzing such rare events. Here we review the basic components of transition-path theory and path-finding algorithms. We also discuss connections with the classical transition-state theory.
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              Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states.

              Molecular dynamics simulation generates large quantities of data that must be interpreted using physically meaningful analysis. A common approach is to describe the system dynamics in terms of transitions between coarse partitions of conformational space. In contrast to previous work that partitions the space according to geometric proximity, the authors examine here clustering based on kinetics, merging configurational microstates together so as to identify long-lived, i.e., dynamically metastable, states. As test systems microsecond molecular dynamics simulations of the polyalanines Ala(8) and Ala(12) are analyzed. Both systems clearly exhibit metastability, with some kinetically distinct metastable states being geometrically very similar. Using the backbone torsion rotamer pattern to define the microstates, a definition is obtained of metastable states whose lifetimes considerably exceed the memory associated with interstate dynamics, thus allowing the kinetics to be described by a Markov model. This model is shown to be valid by comparison of its predictions with the kinetics obtained directly from the molecular dynamics simulations. In contrast, clustering based on the hydrogen-bonding pattern fails to identify long-lived metastable states or a reliable Markov model. Finally, an approach is proposed to generate a hierarchical model of networks, each having a different number of metastable states. The model hierarchy yields a qualitative understanding of the multiple time and length scales in the dynamics of biomolecules.
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                Author and article information

                Contributors
                Role: Editor
                Journal
                PLoS Comput Biol
                PLoS Comput. Biol
                plos
                ploscomp
                PLoS Computational Biology
                Public Library of Science (San Francisco, USA )
                1553-734X
                1553-7358
                October 2012
                October 2012
                18 October 2012
                : 8
                : 10
                : e1002721
                Affiliations
                [1 ]Friedrich-Schiller-Universität Jena, Universitätsklinikum Jena, Institut für Physiologie II, Jena, Germany
                [2 ]Fachhochschule Schmalkalden, Fakultät Elektrotechnik, Blechhammer, Schmalkalden, Germany
                University of Notre Dame, United States of America
                Author notes

                The authors have declared that no competing interests exist.

                Conceived and designed the experiments: KB JK. Performed the experiments: JK. Analyzed the data: KB ES. Wrote the paper: KB.

                Article
                PCOMPBIOL-D-12-00816
                10.1371/journal.pcbi.1002721
                3475657
                23093920
                7f796e49-69ac-432c-96ec-d83870e28c00
                Copyright @ 2012

                This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

                History
                : 18 May 2012
                : 16 August 2012
                Page count
                Pages: 10
                Funding
                This work was supported by a grant of the Deutsche Forschungsgemeinschaft to KB. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
                Categories
                Research Article
                Biology
                Biochemistry
                Biophysics
                Computational Biology

                Quantitative & Systems biology
                Quantitative & Systems biology

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