On the Round Complexity of Randomized Byzantine Agreement

We prove lower bounds on the round complexity of randomized Byzantine agreement (BA) protocols, bounding the halting probability of such protocols after one and two rounds. In particular, we prove that: 1) BA protocols resilient against n/3 [resp., n/4] corruptions terminate (under attack) at the end of the first round with probability at most o(1) [resp., 1/2+ o(1)]. 2) BA protocols resilient against n/4 corruptions terminate at the end of the second round with probability at most 1-Theta(1). 3) For a large class of protocols (including all BA protocols used in practice) and under a plausible combinatorial conjecture, BA protocols resilient against n/3 [resp., n/4] corruptions terminate at the end of the second round with probability at most o(1) [resp., 1/2 + o(1)]. The above bounds hold even when the parties use a trusted setup phase, e.g., a public-key infrastructure (PKI). The third bound essentially matches the recent protocol of Micali (ITCS\u2717) that tolerates up to n/3 corruptions and terminates at the end of the third round with constant probability

On the Round Complexity of Randomized Byzantine Agreement

We prove lower bounds on the round complexity of randomized Byzantine agreement (BA) protocols, bounding the halting probability of such protocols after one and two rounds. In particular, we prove that: (1) BA protocols resilient against $n/3$ [resp., $n/4$] corruptions terminate (under attack) at the end of the first round with probability at most $o(1)$ [resp., $1/2+ o(1)$]. (2) BA protocols resilient against $n/4$ corruptions terminate at the end of the second round with probability at most $1-\Theta(1)$. (3) For a large class of protocols (including all BA protocols used in practice) and under a plausible combinatorial conjecture, BA protocols resilient against $n/3$ [resp., $n/4$] corruptions terminate at the end of the second round with probability at most $o(1)$ [resp., $1/2 + o(1)$]. The above bounds hold even when the parties use a trusted setup phase, e.g., a public-key infrastructure (PKI). The third bound essentially matches the recent protocol of Micali (ITCS'17) that tolerates up to $n/3$ corruptions and terminates at the end of the third round with constant probability.Comment: DISC 201

On the Round Complexity of Randomized Byzantine Agreement

We prove lower bounds on the round complexity of randomized Byzantine agreement (BA) protocols, bounding the halting probability of such protocols after one and two rounds. In particular, we prove that: (1) BA protocols resilient against $n/3$ [resp., $n/4$] corruptions terminate (under attack) at the end of the first round with probability at most $o(1)$ [resp., $1/2+ o(1)$]. (2) BA protocols resilient against a fraction of corruptions greater than $1/4$ terminate at the end of the second round with probability at most $1-\Theta(1)$. (3) For a large class of protocols (including all BA protocols used in practice) and under a plausible combinatorial conjecture, BA protocols resilient against a fraction of corruptions greater than $1/3$ [resp., $1/4$] terminate at the end of the second round with probability at most $o(1)$ [resp., $1/2 + o(1)$]. The above bounds hold even when the parties use a trusted setup phase, e.g., a public-key infrastructure (PKI). The third bound essentially matches the recent protocol of Micali (ITCS\u2717) that tolerates up to $n/3$ corruptions and terminates at the end of the third round with constant probability

Controlled-Deactivation CB1 Receptor Ligands as a Novel Strategy to Lower Intraocular Pressure

Nearly half a century has passed since the demonstration that cannabis and its chief psychoactive component Δ9-THC lowers intraocular pressure (IOP). Elevated IOP remains the chief hallmark and therapeutic target for glaucoma, a condition that places millions at risk of blindness. It is likely that Δ9-THC exerts much of its IOP-lowering effects via the activation of CB1 cannabinoid receptors. However, the initial promise of CB1 as a target for treating glaucoma has not thus far translated into a credible therapeutic strategy. We have recently shown that blocking monoacylglycerol lipase (MAGL), an enzyme that breaks the endocannabinoid 2-arachidonoyl glycerol (2-AG), substantially lowers IOP. Another strategy is to develop cannabinoid CB1 receptor agonists that are optimized for topical application to the eye. Recently we have reported on a controlled-deactivation approach where the “soft” drug concept of enzymatic deactivation was combined with a “depot effect” that is commonly observed with Δ9-THC and other lipophilic cannabinoids. This approach allowed us to develop novel cannabinoids with a predictable duration of action and is particularly attractive for the design of CB1 activators for ophthalmic use with limited or no psychoactive effects. We have tested a novel class of compounds using a combination of electrophysiology in autaptic hippocampal neurons, a well-characterized model of endogenous cannabinoid signaling, and measurements of IOP in a mouse model. We now report that AM7410 is a reasonably potent and efficacious agonist at CB1 in neurons and that it substantially (30%) lowers IOP for as long as 5 h after a single topical treatment. This effect is absent in CB1 knockout mice. Our results indicate that the direct targeting of CB1 receptors with controlled-deactivation ligands is a viable approach to lower IOP in a murine model and merits further study in other model systems

Controlled-Deactivation CB1 Receptor Ligands as a Novel Strategy to Lower Intraocular Pressure

Nearly half a century has passed since the demonstration that cannabis and its chief psychoactive component $\Delta^9$-THC lowers intraocular pressure (IOP). Elevated IOP remains the chief hallmark and therapeutic target for glaucoma, a condition that places millions at risk of blindness. It is likely that $\Delta^9$-THC exerts much of its IOP-lowering effects via the activation of CB1 cannabinoid receptors. However, the initial promise of CB1 as a target for treating glaucoma has not thus far translated into a credible therapeutic strategy. We have recently shown that blocking monoacylglycerol lipase (MAGL), an enzyme that breaks the endocannabinoid 2-arachidonoyl glycerol (2-AG), substantially lowers IOP. Another strategy is to develop cannabinoid CB1 receptor agonists that are optimized for topical application to the eye. Recently we have reported on a controlled-deactivation approach where the “soft” drug concept of enzymatic deactivation was combined with a “depot effect” that is commonly observed with $\Delta^9$-THC and other lipophilic cannabinoids. This approach allowed us to develop novel cannabinoids with a predictable duration of action and is particularly attractive for the design of CB1 activators for ophthalmic use with limited or no psychoactive effects. We have tested a novel class of compounds using a combination of electrophysiology in autaptic hippocampal neurons, a well-characterized model of endogenous cannabinoid signaling, and measurements of IOP in a mouse model. We now report that AM7410 is a reasonably potent and efficacious agonist at CB1 in neurons and that it substantially (30%) lowers IOP for as long as 5 h after a single topical treatment. This effect is absent in CB1 knockout mice. Our results indicate that the direct targeting of CB1 receptors with controlled-deactivation ligands is a viable approach to lower IOP in a murine model and merits further study in other model systems

Conformational Memories and the Endocannabinoid Binding Site at the Cannabinoid CB1 Receptor

Endocannabinoid structure-activity relationships (SAR) indicate that the CB1 receptor recognizes ethanolamides whose fatty acid acyl chains have 20 or 22 carbons, with at least three homoallylic double bonds and satn. in at least the last five carbons of the acyl chain. To probe the mol. basis for these acyl chain requirements, the method of conformational memories (CM) was used to study the conformations available to an n-6 series of ethanolamide fatty acid acyl chain congeners: 22:4, n-6 (Ki = 34.4±3.2 nM); 20:4, n-6 (Ki = 39.2±5.7 nM); 20:3, n-6 (Ki = 53.4±5.5 nM); and 20:2, n-6 (Ki \u3e 1500 nM). CM studies indicated that each analog could form both extended and U/J-shaped families of conformers. However, for the low affinity 20:2, n-6 ethanolamide, the higher populated family was the extended conformer family, while for the other analogs in the series, the U/J-shaped family had the higher population. In addn., the 20:2, n-6 ethanolamide U-shaped family was not as tightly curved as were those of the other analogs studied. To quantitate this variation in curvature, the radius of curvature (in the C-3 to C-17 region) of each member of each U/J-shaped family was measured. The av. radii of curvature (with their 95% confidence intervals) were found to be 5.8 Å (5.3-6.2) for 20:2, n-6; 4.4 Å (4.1-4.7) for 20:3, n-6; 4.0 Å (3.7-4.2) for 20:4, n-6; and 4.0 Å (3.6-4.5) for 22:4, n-6. These results suggest that higher CB1 affinity is assocd. with endocannabinoids that can form tightly curved structures. Endocannabinoid SAR also indicate that the CB1 receptor does not tolerate large endocannabinoid headgroups; however, it does recognize both polar and nonpolar moieties in the headgroup region. To identify a headgroup orientation that results in high CB1 affinity, a series of di-Me anandamide analogs (R)-N-(1-methyl-2-hydroxyethyl)-2-(R)-methyl-arachidonamide (Ki = 7.42±0.86 nM), (R)-N-(1-methyl-2-hydroxyethyl)-2-(S)-methyl-arachidonamide (Ki = 185±12 nM), (S)-N-(1-methyl-2-hydroxyethyl)-2-(S)-methyl-arachidonamide (Ki = 389±72 nM), and (S)-N-(1-methyl-2-hydroxyethyl)-2-(R)-methyl-arachidonamide (Ki = 233±69 nM) were then studied using CM and computer receptor docking studies in an active state (R*) model of CB1. These studies suggested that the high CB1 affinity of the R,R stereoisomer is due to the ability of the headgroup to form an intramol. hydrogen bond between the carboxamide oxygen and the headgroup hydroxyl that orients the C2 and C1\u27 Me groups to have hydrophobic interactions with valine 3.32(196), while the carboxamide oxygen forms a hydrogen bond with lysine 3.28(192) at CB1. In this position in the CB1 binding pocket, the acyl chain has hydrophobic and C-H···π interactions with residues in the transmembrane helix (TMH) 2-3-7 region. Taken together, the studies reported here suggest that anandamide and its congeners adopt tightly curved U/J-shaped conformations at CB1 and suggest that the TMH 2-3-7 region is the endocannabinoid binding region at CB1