Testing and learning Boolean functions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 203-207).Given a function f on n inputs, we consider the problem of testing whether f belongs to a concept class C, or is far from every member of C. An algorithm that achieves this goal for a particular C is called a property testing algorithm, and can be viewed as relaxation of a proper learning algorithm, which must also return an approximation to f if it is in C. We give property testing algorithms for many different classes C, with a focus on those that are fundamental to machine learning, such as halfspaces, decision trees, DNF formulas, and sparse polynomials. In almost all cases, the property testing algorithm has query complexity independent of n, better than the best possible learning algorithm.by Kevin Michael Matulef.Ph.D

Testing (subclasses of) halfspaces

We address the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w . x − θ). We consider halfspaces over the continuous domain R n (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube { − 1,1} n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are ε-far from any halfspace using only poly(1) queries, independent of the dimension n. In contrast to the case of general halfspaces, we show that testing natural subclasses of halfspaces can be markedly harder; for the class of { − 1,1}-weight halfspaces, we show that a tester must make at least Ω(logn) queries. We complement this lower bound with an upper bound showing that O(√n) queries suffice.National Basic Research Program of China (grant 2007CB807900)National Basic Research Program of China (grant 2007CB807901)National Natural Science Foundation (China) (grant 60553001

The analysis of desensitizing CNGA1 channels reveals molecular interactions essential for normal gating

The pore region of cyclic nucleotide–gated (CNG) channels acts as the channel gate. Therefore, events occurring in the cyclic nucleotide–binding (CNB) domain must be coupled to the movements of the pore walls. When Glu363 in the pore region, Leu356 and Thr355 in the P helix, and Phe380 in the upper portion of the S6 helix are mutated into an alanine, gating is impaired: mutant channels E363A, L356A, T355A, and F380A desensitize in the presence of a constant cGMP concentration, contrary to what can be observed in wild-type (WT) CNGA1 channels. Similarly to C-type inactivation of K+ channels, desensitization in these mutant channels is associated with rearrangements of residues in the outer vestibule. In the desensitized state, Thr364 residues in different subunits become closer and Pro366 becomes more accessible to extracellular reagents. Desensitization is also observed in the mutant channel L356C, but not in the double-mutant channel L356C+F380C. Mutant channels L356F and F380K did not express, but cGMP-gated currents with a normal gating were observed in the double-mutant channels L356F+F380L and L356D+F380K. Experiments with tandem constructs with L356C, F380C, and L356C+F380C and WT channels indicate that the interaction between Leu356 and Phe380 is within the same subunit. These results show that Leu356 forms a hydrophobic interaction with Phe380, coupling the P helix with S6, whereas Glu363 could interact with Thr355, coupling the pore wall to the P helix. These interactions are essential for normal gating and underlie the transduction between the CNB domain and the pore

Ion-binding properties of a K+ channel selectivity filter in different conformations

K(+) channels are membrane proteins that selectively conduct K(+) ions across lipid bilayers. Many voltage-gated K(+) (K(V)) channels contain two gates, one at the bundle crossing on the intracellular side of the membrane and another in the selectivity filter. The gate at the bundle crossing is responsible for channel opening in response to a voltage stimulus, whereas the gate at the selectivity filter is responsible for C-type inactivation. Together, these regions determine when the channel conducts ions. The K(+) channel from Streptomyces lividians (KcsA) undergoes an inactivation process that is functionally similar to K(V) channels, which has led to its use as a practical system to study inactivation. Crystal structures of KcsA channels with an open intracellular gate revealed a selectivity filter in a constricted conformation similar to the structure observed in closed KcsA containing only Na(+) or low [K(+)]. However, recent work using a semisynthetic channel that is unable to adopt a constricted filter but inactivates like WT channels challenges this idea. In this study, we measured the equilibrium ion-binding properties of channels with conductive, inactivated, and constricted filters using isothermal titration calorimetry (ITC). EPR spectroscopy was used to determine the state of the intracellular gate of the channel, which we found can depend on the presence or absence of a lipid bilayer. Overall, we discovered that K(+) ion binding to channels with an inactivated or conductive selectivity filter is different from K(+) ion binding to channels with a constricted filter, suggesting that the structures of these channels are different

Extracellular protons titrate voltage gating of a ligand-gated ion channel

Cyclic nucleotide–gated channels mediate transduction of light into electric signals in vertebrate photoreceptors. These channels are primarily controlled by the binding of intracellular cyclic GMP (cGMP). Glutamate residue 363 near the extracellular end of the ion selectivity filter interacts with the pore helix and helps anchor the filter to the helix. Disruption of this interaction by mutations renders the channels essentially fully voltage gated in the presence of saturating concentrations of cGMP. Here, we find that lowering extracellular pH makes the channels conduct in an extremely outwardly rectifying manner, as does a neutral glutamine substitution at E363. A pair of cysteine mutations, E363C and L356C (the latter located midway the pore helix), largely eliminates current rectification at low pH. Therefore, this low pH-induced rectification primarily reflects voltage-dependent gating involving the ion selectivity filter rather than altered electrostatics around the external opening of the ion pore and thus ion conduction. It then follows that protonation of E363, like the E363Q mutation, disrupts the attachment of the selectivity filter to the pore helix. Loosening the selectivity filter from its surrounding structure shifts the gating equilibrium toward closed states. At low extracellular pH, significant channel opening occurs only when positive voltages drive the pore from a low probability open conformation to a second open conformation. Consequently, at low extracellular pH the channels become practically fully voltage gated, even in the presence of a saturating concentration of cGMP

Multiple mechanisms underlying rectification in retinal cyclic nucleotide-gated (CNGA1) channels

In cyclic nucleotide-gated (CNGA1) channels, in the presence of symmetrical ionic conditions, current-voltage (I-V) relationship depends, in a complex way, on the radius of permeating ion. It has been suggested that both the pore and S4 helix contribute to the observed rectification. In the present manuscript, using tail and gating current measurements from homotetrameric CNGA1 channels expressed in Xenopus oocytes, we clarify and quantify the role of the pore and of the S4 helix. We show that in symmetrical Rb(+) and Cs(+) single-channel current rectification dominates macroscopic currents while voltage-dependent gating becomes larger in symmetrical ethylammonium and dimethylammonium, where the open probability strongly depends on voltage. Isochronal tail currents analysis in dimethylammonium shows that at least two voltage-dependent transitions underlie the observed rectification. Only the first voltage-dependent transition is sensible to mutation of charge residues in the S4 helix. Moreover, analysis of tail and gating currents indicates that the number of elementary charges per channel moving across the membrane is less than 2, when they are about 12 in K(+) channels. These results indicate the existence of distinct mechanisms underlying rectification in CNG channels. A restricted motion of the S4 helix together with an inefficient coupling to the channel gate render CNGA1 channels poorly sensitive to voltage in the presence of physiological Na(+) and K(+)

Sensory Transduction Channel Subunits, tax-4 and tax-2, Modify Presynaptic Molecular Architecture in C. elegans

During development, neural activity is important for forming proper connections in neural networks. The effect of activity on the gross morphology and synaptic strength of neurons has been well documented, but little is known about how activity affects different molecular components during development. Here, we examine the localization of four fluorescently-tagged presynaptic proteins, RAB-3, SNG-1/synaptogyrin, SYD-2/Liprin-α, and SAD-1/SAD kinase, in the C. elegans thermosensory neuron AFD. We show that tax-4 and tax-2, two genes that encode the cyclic nucleotide-gated channel necessary for sensory transduction in AFD, disrupt the localization of all four proteins. In wild-type animals, the synaptic vesicle (SV) markers RAB-3 and SNG-1 and the active zone markers SYD-2 and SAD-1 localize in a stereotyped, punctate pattern in the AFD axon. In tax-4 and tax-2 mutants, SV and SYD-2 puncta are more numerous and less intense. Interestingly, SAD-1 puncta are also less intense but do not increase in number. The change in puncta number can be rescued cell-autonomously in AFD. These results suggest that sensory transduction genes tax-4 and tax-2 are necessary for the proper assembly of presynapses