Functional consequences of leucine and tyrosine mutations in the dual pore motifs of the yeast K(+) channel, Tok1p.

Tandem pore-loop potassium channels differ from the majority of K(+) channels in that a single polypeptide chain carries two K(+)-specific segments (P) each sandwiched between two transmembrane helices (M) to form an MP(1)M-MP(2)M series. Two of these peptide molecules assemble to form one functional potassium channel, which is expected to have biaxial symmetry (commonly described as asymmetric) due to independent mutation in the two MPM units. The resulting intrinsic asymmetry is exaggerated in fungal 2P channels, especially in Tok1p of Saccharomyces, by the N-terminal presence of four more transmembrane helices. Functional implications of such structural asymmetry have been investigated via mutagenesis of residues (L290 in P(1) and Y424 in P(2)) that are believed to provide the outermost ring of carbonyl oxygen atoms for coordination with potassium ions. Both complementary mutations (L290Y and Y424L) yield functional potassium channels having quasi-normal conductance when expressed in Saccharomyces itself, but the P(1) mutation (only) accelerates channel opening about threefold in response to depolarizing voltage shifts. The more pronounced effect at P(1) than at P(2) appears paradoxical in relation to evolution, because a comparison of fungal Tok1p sequences (from 28 ascomycetes) shows the filter sequence of P(2) (overwhelmingly TIGYGD) to be much stabler than that of P(1) (mostly TIGLGD). Profound functional asymmetry is revealed by the fact that combining mutations (L290Y + Y424L)-which inverts the order of residues from the wild-type channel-reduces the expressed channel conductance by a large factor (20-fold, cf. <twofold for the single mutants)

In the yeast potassium channel, Tok1p, the external ring of aspartate residues modulates both gating and conductance.

The yeast plasma-membrane potassium channel, Tok1p, is a voltage-dependent outward rectifier, the gating and steady-state conductance of which are conspicuously modulated by extracellular [K(+)] ([K(+)](o)). Activation is slow at high [K(+)](o), showing time constants (tau(a)) of approximately 90 ms when [K(+)](o) is 150 mM (depolarizing step to +100 mV), and inactivation is weak (<30%) during sustained depolarization. Lowering [K(+)](o) accelerates activation, increases peak current, and enhances inactivation, so that at 15 mM [K(+)](o) tau(a) is less than 50 ms and inactivation suppresses approximately 60% of peak current. Two negative residues, Asp292 and Asp426, near the mouth of the assembled channel, modulate both kinetics and conductance of the channel. Charge neutralization in the mutant Asp292Asn allows fast activation (tau(a) approximately 20 ms) at high [K(+)](o), peak currents diminishing with decreasing [K(+)](o), and fast, nearly complete, inactivation. The voltage dependence of tau(a) persists in the mutant, but the [K(+)](o) dependence almost disappears. Similar but smaller changes are seen in the Asp426Asn mutant, implying that pore geometry in the functional channel has twofold, not fourfold, symmetry

Interior point methods are not worse than Simplex

Whereas interior point methods provide polynomial-time linear programming algorithms, the running time bounds depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. We introduce a new polynomial-time path-following interior point method where the number of iterations also admits a combinatorial upper bound O(2nn1.5 log n) for an n-variable linear program in standard form. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations. The number of iterations of our algorithm is at most O(n1.5 log n) times the number of segments of any piecewise linear curve in the wide neighborhood of the central path. In particular, it matches the number of iterations of any path following interior point method up to this polynomial factor. The overall exponential upper bound derives from studying the max central path', a piecewise-linear curve with the number of pieces bounded by the total length of 2n shadow vertex simplex paths. From the existence of a line segment in the wide neighborhood we derive strong implications on the structure of the corresponding segment of the central path. Our algorithm is able to detect this structure from the local geometry at the current iterate, and constructs a step direction that descends along this segment. The bound O(n1.5 log n) that applies for arbitrarily long line segments is derived from a combinatorial progress measure. Our algorithm falls into the family of layered least squares interior point methods introduced by Vavasis and Ye (Math. Prog. 1996). In contrast to previous layered least squares methods that partition the kernel of the constraint matrix into coordinate subspaces, our method creates layers based on a general subspace providing more flexibility. Our result also implies the same bound on the number of iterations of the trust region interior point method by Lan, Monteiro, and Tsuchiya (SIOPT 2009)