Modeling of Open, Closed, and Open-Inactivated States of the hERG1 Channel: Structural Mechanisms of the State-Dependent Drug Binding

The human ether-a-go-go related gene 1 (hERG1) K ion channel is a key element for the rapid component of the delayed rectified potassium current in cardiac myocytes. Since there are no crystal structures for hERG channels, creation and validation of its reliable atomistic models have been key targets in molecular cardiology for the past decade. In this study, we developed and vigorously validated models for open, closed, and open-inactivated states of hERG1 using a multistep protocol. The conserved elements were derived using multiple-template homology modeling utilizing available structures for Kv1.2, Kv1.2/2.1 chimera, and KcsA channels. Then missing elements were modeled with the ROSETTA <i>De Novo</i> protein-designing suite and further refined with all-atom molecular dynamics simulations. The final ensemble of models was evaluated for consistency to the reported experimental data from biochemical, biophysical, and electrophysiological studies. The closed state models were cross-validated against available experimental data on toxin footprinting with protein–protein docking using hERG state-selective toxin BeKm-1. Poisson–Boltzmann calculations were performed to determine gating charge and compare it to electrophysiological measurements. The validated structures offered us a unique chance to assess molecular mechanisms of state-dependent drug binding in three different states of the channel

Motion Planning of Multi-Agent Systems under Temporal Logic Specifications

In this thesis, a top-down hierarchical framework is proposed to deal with multiagentsymbolic motion planning and control. Agents are assigned a distributable globalmission, and are partitioned into leaders and followers. The global mission is distributedamong the followers and each one synthesizes a discrete motion plan. By exploitingthe concept of network controllability, an open-loop optimal control law is synthesizedcentrally and executed in a distributed manner. This control law guarantees the concurrentexecution and fulfillment of the followers’ discrete motion plans. Simulations areperformed to verify the proposed control law both for single- and multi-leader, leaderfollowernetworks

Simulation system of the sodium channel Na_vAb embedded in a lipid bilayer.

<p>For clarity, the voltage sensor domains are not shown; only two monomers, a few ions, and a thin layer of water are shown. The right panel depicts the selectivity filter lined by the amino acids TLESW. The green dots represent z-coordinates along the central axis, whose arrow indicates the inward direction. The distance of 8 Å between the oxygen atoms of opposite hydroxyl groups of S178 is greater than the smallest width of the selectivity filter, which is 4.6 Å.</p

(a) Perturbed Helmholtz free-energy change (useful work) of three-ion pulling configurations. The value of <i>λ</i> indicates the center of the harmonic pulling potential (see Fig 1). (b) Perturbed Helmholtz free-energy change (useful work) of individual ions in the three-ion pulling simulations. The numbers indicate the ordered ions in the three-ion configurations. (c) Averaged z-coordinate of three ions.

<p>(a) Perturbed Helmholtz free-energy change (useful work) of three-ion pulling configurations. The value of <i>λ</i> indicates the center of the harmonic pulling potential (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004482#pcbi.1004482.g001" target="_blank">Fig 1</a>). (b) Perturbed Helmholtz free-energy change (useful work) of individual ions in the three-ion pulling simulations. The numbers indicate the ordered ions in the three-ion configurations. (c) Averaged z-coordinate of three ions.</p

(a) z-coordinate of K₁ (horizontal axis; z₁) versus z-coordinate of K₂ (vertical axis; z₂) during the pulling simulations. (b) Frequency of data points in (a).

<p>(a) z-coordinate of K₁ (horizontal axis; z₁) versus z-coordinate of K₂ (vertical axis; z₂) during the pulling simulations. (b) Frequency of data points in (a).</p

(a) Perturbed Helmholtz free-energy changes (useful work) of single K⁺ and Na⁺ ions pulled through the Na_vAb channel with relaxation times equal to 0.5 ns for each pulling step. The value of <i>λ</i> indicates the center of the harmonic pulling potential (see Fig 1). (b) Averaged z-coordinate of single K⁺ and Na⁺ ions. The z-coordinates more negative than –15 Å are on the extracellular side of the membrane, and z-coordinates greater than 10 Å are on the intracellular of the membrane.

<p>(a) Perturbed Helmholtz free-energy changes (useful work) of single K⁺ and Na⁺ ions pulled through the Na_vAb channel with relaxation times equal to 0.5 ns for each pulling step. The value of <i>λ</i> indicates the center of the harmonic pulling potential (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004482#pcbi.1004482.g001" target="_blank">Fig 1</a>). (b) Averaged z-coordinate of single K⁺ and Na⁺ ions. The z-coordinates more negative than –15 Å are on the extracellular side of the membrane, and z-coordinates greater than 10 Å are on the intracellular of the membrane.</p

(a) Snapshot of three Na⁺ ions aligned along one lateral corner of adjacent SEL amino acids at λ = –12Å. (b) Snapshot of all atoms in (a) showing water molecules that fill the rest of the selectivity filter.

<p>(a) Snapshot of three Na⁺ ions aligned along one lateral corner of adjacent SEL amino acids at λ = –12Å. (b) Snapshot of all atoms in (a) showing water molecules that fill the rest of the selectivity filter.</p

(a-b) Histogram of z-coordinate positions of all ions during the three-ion inward pulling protocol. (c) Averaged z-coordinate of each ion in the three-ion configurations during the pulling simulations.

<p>The numbers indicate the same ordered numbers in (a-b). The Na_vAb selectivity filter is shown on the right side of the Fig for reference.</p

Snapshots of two (a) and three (b) K⁺ ions binding to oxygen atoms of carboxylate and two hydroxyl groups of E177 and S178 at λ = –10 and –9 Å, respectively. The views are from the extracellular side of the membrane. The red numbers are distances (Å) between the side chain oxygen atoms of E177 and S178 and the ions. The black numbers are angles of O_S–O_E–O_S. (c) Averaged distance and number of water molecules between K₁ and K₂. The data of 2.5 ns in each pulling step are collected for the averages. The number of water molecules is counted in the overlap between the two spheres, which have instantaneous radius r_K1-K2 and are centered at the positions of the ions.

<p>Snapshots of two (a) and three (b) K⁺ ions binding to oxygen atoms of carboxylate and two hydroxyl groups of E177 and S178 at λ = –10 and –9 Å, respectively. The views are from the extracellular side of the membrane. The red numbers are distances (Å) between the side chain oxygen atoms of E177 and S178 and the ions. The black numbers are angles of O_S–O_E–O_S. (c) Averaged distance and number of water molecules between K₁ and K₂. The data of 2.5 ns in each pulling step are collected for the averages. The number of water molecules is counted in the overlap between the two spheres, which have instantaneous radius r_K1-K2 and are centered at the positions of the ions.</p

(a) Perturbed Helmholtz free-energy change (useful work) of three potassium and sodium ions pulled in the outward direction. (b) Average position of each individual ion in the outward pulling simulations. The ions move from the right to left of the horizontal axis. (c) The orientation and structure of the selectivity filter of Na_vAb is shown at the right side of the Fig.

<p>(a) Perturbed Helmholtz free-energy change (useful work) of three potassium and sodium ions pulled in the outward direction. (b) Average position of each individual ion in the outward pulling simulations. The ions move from the right to left of the horizontal axis. (c) The orientation and structure of the selectivity filter of Na_vAb is shown at the right side of the Fig.</p