The Dynamics of the Neuropeptide Y Receptor Type 1 Investigated by Solid-State NMR and Molecular Dynamics Simulation

We report data on the structural dynamics of the neuropeptide Y (NPY) G-protein-coupled receptor (GPCR) type 1 (Y1R), a typical representative of class A peptide ligand GPCRs, using a combination of solid-state NMR and molecular dynamics (MD) simulation. First, the equilibrium dynamics of Y1R were studied using 15N-NMR and quantitative determination of 1H-13C order parameters through the measurement of dipolar couplings in separated-local-field NMR experiments. Order parameters reporting the amplitudes of the molecular motions of the C-H bond vectors of Y1R in DMPC membranes are 0.57 for the Cα sites and lower in the side chains (0.37 for the CH2 and 0.18 for the CH3 groups). Different NMR excitation schemes identify relatively rigid and also dynamic segments of the molecule. In monounsaturated membranes composed of longer lipid chains, Y1R is more rigid, attributed to a higher hydrophobic thickness of the lipid membrane. The presence of an antagonist or NPY has little influence on the amplitude of motions, whereas the addition of agonist and arrestin led to a pronounced rigidization. To investigate Y1R dynamics with site resolution, we conducted extensive all-atom MD simulations of the apo and antagonist-bound state. In each state, three replicas with a length of 20 μs (with one exception, where the trajectory length was 10 μs) were conducted. In these simulations, order parameters of each residue were determined and showed high values in the transmembrane helices, whereas the loops and termini exhibit much lower order. The extracellular helix segments undergo larger amplitude motions than their intracellular counterparts, whereas the opposite is observed for the loops, Helix 8, and termini. Only minor differences in order were observed between the apo and antagonist-bound state, whereas the time scale of the motions is shorter for the apo state. Although these relatively fast motions occurring with correlation times of ns up to a few µs have no direct relevance for receptor activation, it is believed that they represent the prerequisite for larger conformational transitions in proteins

Expected Length of the Longest Chain in Linear Hashing

Hash table with chaining is a data structure that chains objects with identical hash values together with an entry or a memory address. It works by calculating a hash value from an input then placing the input in the hash table entry. When we place two inputs in the same entry, they chain together in a linear linked list. We are interested in the expected length of the longest chain in linear hashing and methods to reduce the length because the worst-case look-up time is directly proportional to it. The linear hash function used to calculate hash value is defined by ax+b mod p mod m, for any x ∈ {0,1, . . . , p−1} and a, b chosen uniformly at random from the set {0,1, . . . , p−1}, where p is a prime and p≥m. This class of hash functions is a 2-wise independent hash function family. For any 2-wise independent hash functions, the expected length of the longest chain is O(n1/2). Additionally, Alon et al. (JACM 1999) proved that when using a similar class of 2-wise independent hash function, the expected length of the longest chain has a tight lower bound of Ω(n1/2). Recently, in 2016, Knudsen (FOCS 2016) showed that the upper bound of the expected length of the longest chain of the linear hashing function is surprisingly n1/3+o(1). This bound is strictly better than O(n1/2), which, due to Alon et al.’s result, is already known to be tight for 2-wise independent hash functions. Consequently, there are exclusive properties of the linear hashing function, in addition to being 2-wise independent, that results in this phenomenon. Even though Knudsen’s upper bound on the expected length of the longest chain is remarkable, it is still unknown whether it is tight. In other words, does there exist a set of n inputs such that, when hashed using the linear hash function, the expected length of the longest chain is roughly n1/3. If Knudsen’s bound is not tight, then there is an additional motivation to study further and tighten the upper bound. Another focus of our research is to reduce the expected length of the longest chain by using the load balancing power of “two choices.” The idea is, instead of choosing one bin (hash table entry) for a ball (input), to choose two or more bins and put the ball in the bin with the least load at that moment. Mitzenmacher et al. proved that the power of two choices exponentially improves the expected max-load (from Θ(log n/log log n)) to Θ(log log n)) for the hash table that uses two truly random hash functions. We shall conduct an empirical study by simulation with SageMath (System for Algebra and Geometry Experimentation) to verify whether similar improvements are observed for the linear hash function as well. We anticipate that the length of the longest chain of our linear hash table can be significantly improved when used with two linear hash functions

Internal Structure of Addition Chains: Well-Ordering

An addition chain for $n$ is defined to be a sequence $(a_0,a_1,\ldots,a_r)$ such that $a_0=1$, $a_r=n$, and, for any $1\le k\le r$, there exist $0\le i, j<k$ such that $a_k = a_i + a_j$; the number $r$ is called the length of the addition chain. The shortest length among addition chains for $n$, called the addition chain length of $n$, is denoted $\ell(n)$. The number $\ell(n)$ is always at least $\log_2 n$; in this paper we consider the difference $\delta^\ell(n):=\ell(n)-\log_2 n$, which we call the addition chain defect. First we use this notion to show that for any $n$, there exists $K$ such that for any $k\ge K$, we have $\ell(2^k n)=\ell(2^K n)+(k-K)$. The main result is that the set of values of $\delta^\ell$ is a well-ordered subset of $[0,\infty)$, with order type $\omega^\omega$. The results obtained here are analogous to the results for integer complexity obtained in [1] and [3]. We also prove similar well-ordering results for restricted forms of addition chain length, such as star chain length and Hansen chain length.Comment: 19 page

Multiscale Polymer And Nanoparticle Dynamics In Polymer Nanocomposites

The addition of nanoparticles (NPs) to a polymer matrix, forming a polymer nanocomposite (PNC), can extend and control macroscopic material properties. Many macroscopic properties (e.g. mechanical strength and small molecule transport) are dictated by microscopic dynamic processes, including dynamics of the polymer segments, chains, and NPs. Because the NPs and polymers have overlapping characteristic length, time, and energy scales, the interactions within these materials are complex, the dynamics are interrelated, and both remain poorly understood. Developing a fundamental and mechanistic understanding of polymer and NP dynamics in PNCs will lead to new opportunities, new innovations, and improved manufacturability, all of which may accelerate their universal introduction to society. This dissertation aims to navigate the hierarchy of dynamics in model PNCs. At the smallest length scale, we show that polymer segmental dynamics are slowed by the addition of highly-attractive, immobile NPs, particularly at the NP-polymer interface, and depend only weakly on temperature and matrix molecular weight. Despite measurable reductions in the timescale of motion, we show that the segmental diffusion process is mechanistically similar in PNCs and bulk. At longer length and timescales, we use molecular dynamics simulations to study chain-scale conformations and diffusion near confining athermal NPs. We show polymer diffusion is perturbed at longer length-scales than conformations and identify slow diffusion through confining NPs but bulk-like diffusion away from them. Using model attractive PNCs, we develop and demonstrate ion scattering measurements to extract the fraction of chains bound to the immobile NPs. These measurements show that the slow segmental relaxations at the interface persist to the chain-scale and reveal slow bound polymer desorption that occurs more readily at higher temperatures, lower polymer molecular weight, and longer times. Finally, we sample multiple length and timescales in mixtures of entangled polymer and very small, attractive NPs. We present experimental support of vehicular diffusion of NPs, which produces anomalously fast NP motion and commensurate slowing of polymer segments and polymer chain diffusion. Finally, we present X-ray photon correlation spectroscopy measurements of NP dynamics, small-angle neutron scattering measurements of the bound polymer layer in solution, and protocols for NP surface functionalization

Construction of Polar Codes with Sublinear Complexity

Consider the problem of constructing a polar code of block length $N$ for the transmission over a given channel $W$. Typically this requires to compute the reliability of all the $N$ synthetic channels and then to include those that are sufficiently reliable. However, we know from [1], [2] that there is a partial order among the synthetic channels. Hence, it is natural to ask whether we can exploit it to reduce the computational burden of the construction problem. We show that, if we take advantage of the partial order [1], [2], we can construct a polar code by computing the reliability of roughly a fraction $1/\log^{3/2} N$ of the synthetic channels. In particular, we prove that $N/\log^{3/2} N$ is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor $\log\log N$. This set of roughly $N/\log^{3/2} N$ synthetic channels is universal, in the sense that it allows one to construct polar codes for any $W$, and it can be identified by solving a maximum matching problem on a bipartite graph. Our proof technique consists of reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set. As such, this method is general and it can be used to further improve the complexity of the construction problem in case a new partial order on the synthetic channels of polar codes is discovered.Comment: 9 pages, 3 figures, presented at ISIT'17 and submitted to IEEE Trans. Inform. Theor

Signatures of arithmetic simplicity in metabolic network architecture

Metabolic networks perform some of the most fundamental functions in living cells, including energy transduction and building block biosynthesis. While these are the best characterized networks in living systems, understanding their evolutionary history and complex wiring constitutes one of the most fascinating open questions in biology, intimately related to the enigma of life's origin itself. Is the evolution of metabolism subject to general principles, beyond the unpredictable accumulation of multiple historical accidents? Here we search for such principles by applying to an artificial chemical universe some of the methodologies developed for the study of genome scale models of cellular metabolism. In particular, we use metabolic flux constraint-based models to exhaustively search for artificial chemistry pathways that can optimally perform an array of elementary metabolic functions. Despite the simplicity of the model employed, we find that the ensuing pathways display a surprisingly rich set of properties, including the existence of autocatalytic cycles and hierarchical modules, the appearance of universally preferable metabolites and reactions, and a logarithmic trend of pathway length as a function of input/output molecule size. Some of these properties can be derived analytically, borrowing methods previously used in cryptography. In addition, by mapping biochemical networks onto a simplified carbon atom reaction backbone, we find that several of the properties predicted by the artificial chemistry model hold for real metabolic networks. These findings suggest that optimality principles and arithmetic simplicity might lie beneath some aspects of biochemical complexity