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An analogue of Furstenberg–Sárközy’s theorem and an alternative solution to Waring’s problem over finite fields
No full textIn this paper, we use Cayley digraphs to obtain some new self-contained proofs for Waring’s problem over finite fields, proving that any element of a finite field F_q can be written as a sum of m many kth powers as long as q > k^(2m/m-1); and we also compute the smallest positive integers such that every element of can be written as a sum of many powers for all too small to be covered by the above mentioned results when 2 ⩽ k ⩽ 37.
In the process of developing the proofs mentioned above, we arrive at another result (providing a finite field analogue of Furstenberg–Sárközy’s Theorem) showing that any subset
of a finite field F_q for which |E| > qk/√q̅-̅1̅ must contain at least two distinct elements whose difference is a power
Predicting DNA kinetics with a truncated continuous-time Markov chain method
No full textPredicting the kinetics of reactions involving nucleic acid strands is a fundamental task in biology and biotechnology. Reaction kinetics can be modeled as an elementary step continuous-time Markov chain, where states correspond to secondary structures and transitions correspond to base pair formation and breakage. Since the number of states in the Markov chain could be large, rates are determined by estimating the mean first passage time from sampled trajectories. As a result, the cost of kinetic predictions becomes prohibitively expensive for rare events with extremely long trajectories. Also problematic are scenarios where multiple predictions are needed for the same reaction, e.g., under different environmental conditions, or when calibrating model parameters, because a new set of trajectories is needed multiple times. We propose a new method, called pathway elaboration, to handle these scenarios. Pathway elaboration builds a truncated continuous-time Markov chain through both biased and unbiased sampling. The resulting Markov chain has moderate state space size, so matrix methods can efficiently compute reaction rates, even for rare events. Also the transition rates of the truncated Markov chain can easily be adapted when model or environmental parameters are perturbed, making model calibration feasible. We illustrate the utility of pathway elaboration on toehold-mediated strand displacement reactions, show that it well-approximates trajectory-based predictions of unbiased elementary step models on a wide range of reaction types for which such predictions are feasible, and demonstrate that it performs better than alternative truncation-based approaches that are applicable for mean first passage time estimation. Finally, in a small study, we use pathway elaboration to optimize the Metropolis kinetic model of Multistrand, an elementary step simulator, showing that the optimized parameters greatly improve reaction rate predictions. Our framework and dataset are available at https://github.com/DNA-and-Natural-Algorithms-Group/PathwayElaboration
Isotopic Constraints on the Nature of Primary Precipitates in Archean–Early Paleoproterozoic Iron Formations from Determinations of the Iron Phonon Density of States of Greenalite and 2L- and 6L-Ferrihydrite
No full textIron formations (IFs) are chemical sedimentary rocks that were widely deposited before the Great Oxidation Event (GOE) around 2.4–2.2 Ga. It is generally thought that IFs precipitated as hydrated Fe3+ oxides (HFOs) such as ferrihydrite following surface oxidation of Fe²⁺-rich, anoxic deep waters. This model often implicates biological oxidation and underpins reconstructions of marine nutrient concentrations. However, nanoscale petrography indicates that an Fe²⁺ silicate, greenalite, is a common primary mineral in well-preserved IFs, motivating an alternative depositional model of anoxic ferrous silicate precipitation. It is unclear, however, if Fe²⁺-rich silicates can produce the Fe isotopic variations in IFs that are well explained by Fe²⁺ oxidation. To address this question, we constrain the equilibrium Fe isotopic (⁵⁶Fe/⁵⁴Fe) fractionation of greenalite and ferrihydrite by determining the iron phonon densities of states for those minerals. We use ab initio density functional theory (DFT + U) calculations and nuclear resonant inelastic X-ray scattering spectroscopy to show that ferrous greenalite should be isotopically lighter than ferrihydrite by ∼1–1.2‰ at equilibrium, and fractionation should scale linearly with increasing Fe³⁺ content in greenalite. By anchoring ferrihydrite–greenalite mineral pair fractionations to published experimental Fe isotopic fractionations between HFOs and aqueous Fe²⁺, we show that ferrous greenalite may produce all but the heaviest pre-GOE Fe isotopic compositions and mixed valence greenalites can produce the entire record. Our results suggest that heavy Fe isotope enrichments alone are not diagnostic of primary IF mineralogies, and ferrihydrite and partially oxidized or even purely ferrous greenalite are all viable primary IF mineralogies
Biological nitrogen fixation in theory, practice, and reality: a perspective on the molybdenum nitrogenase system
No full textNitrogenase is the sole enzyme responsible for the ATP-dependent conversion of atmospheric dinitrogen into the bioavailable form of ammonia (NH₃), making this protein essential for the maintenance of the nitrogen cycle and thus life itself. Despite the widespread use of the Haber–Bosch process to industrially produce NH₃, biological nitrogen fixation still accounts for half of the bioavailable nitrogen on Earth. An important feature of nitrogenase is that it operates under physiological conditions, where the equilibrium strongly favours ammonia production. This biological, multielectron reduction is a complex catalytic reaction that has perplexed scientists for decades. In this review, we explore the current understanding of the molybdenum nitrogenase system based on experimental and computational research, as well as the limitations of the crystallographic, spectroscopic, and computational techniques employed. Finally, essential outstanding questions regarding the nitrogenase system will be highlighted alongside suggestions for future experimental and computational work to elucidate this essential yet elusive process
Competitive Activation Experiments Reveal Significantly Different Mechanochemical Reactivity of Furan-Maleimide and Anthracene-Maleimide Mechanophores
No full textDuring the past two decades, our understanding of mechanochemical reactivity has advanced considerably. Nevertheless, an incomplete knowledge of structure–activity relationships and the principles that govern mechanochemical transformations limits molecular design. The experimental development of mechanophores has thus benefited from simple computational tools like CoGEF, from which quantitative metrics like rupture force can be extracted to estimate reactivity. Furan–maleimide (FM) and anthracene–maleimide (AM) Diels–Alder adducts are widely studied mechanophores that undergo retro-Diels–Alder reactions upon mechanical activation in polymers. Despite possessing significantly different thermal stability, similar rupture forces predicted by CoGEF calculations suggest that these compounds exhibit similar mechanochemical reactivity. Here, we directly probe the relative mechanochemical reactivity of FM and AM adducts through competitive activation experiments. Ultrasound-induced mechanochemical activation of bis-adduct mechanophores comprising covalently tethered FM and AM subunits reveals pronounced selectivity─as high as ∼13:1─for reaction of the FM adduct compared to the AM adduct. Computational models provide insight into the greater reactivity of the FM mechanophore, indicating a more efficient mechanochemical coupling for the FM adduct compared to the AM adduct. The methodology employed here to directly interrogate the relative reactivity of two different mechanophores using a tethered bis-adduct configuration may be useful for other systems where more common sonication-based approaches are limited by poor sensitivity
Detection of cesium in the atmosphere of the hot He-rich white dwarf HD 149499B
No full textWe report the first detection of cesium (Z = 55) in the atmosphere of a white dwarf. Around a dozen absorption lines of Cs IV, Cs V, and Cs VI have been identified in the Far Ultraviolet Spectroscopic Explorer spectrum of the He-rich white dwarf HD 149499B (T_(eff) = 49 500 K, log g = 7.97). The lines have equivalent widths ranging from 2.3 to 26.9 mÅ. We performed a spectral synthesis analysis to determine the cesium content in the atmosphere. Non-local thermodynamic equilibrium atmosphere models were computed by considering cesium explicitly in the calculations. For this purpose we calculated oscillator strengths for the bound–bound transitions of Cꜱ ɪᴠ–Cꜱ ᴠɪ with both AUTOSTRUCTURE (multiconfiguration Breit–Pauli) and GRASP2K (multiconfiguration Dirac–Fock) atomic structure codes as neither measured nor theoretical values are reported in the literature. We determined a cesium abundance of log N(Cs)/N(He) = −5.45 ± 0.35, which can also be expressed in terms of the mass fraction logX_(Cꜱ) = −3.95 ± 0.35
Arithmetic circuit tensor networks, multivariable function representation, and high-dimensional integration
No full textMany computational problems can be formulated in terms of high-dimensional functions. Simple representations of such functions and resulting computations with them typically suffer from the “curse of dimensionality,” an exponential cost dependence on dimension. Tensor networks provide a way to represent certain classes of high-dimensional functions with polynomial memory. This results in computations where the exponential cost is ameliorated or, in some cases, removed, if the tensor network representation can be obtained. Here, we introduce a direct mapping from the arithmetic circuit of a function to arithmetic circuit tensor networks, avoiding the need to perform any optimization or functional fit. We demonstrate the power of the circuit construction in examples of multivariable integration on the unit hypercube in up to 50 dimensions, where the complexity of integration can be understood from the circuit structure. We find very favorable cost scaling compared with quasi–Monte Carlo integration for these cases and further give an example where efficient quasi–Monte Carlo integration cannot be performed without knowledge of the underlying tensor network circuit structure
Boundaries of quantum supremacy via random circuit sampling
No full textGoogle’s quantum supremacy experiment heralded a transition point where quantum computers can evaluate a computational task, random circuit sampling, faster than classical supercomputers. We examine the constraints on the region of quantum advantage for quantum circuits with a larger number of qubits and gates than experimentally implemented. At near-term gate fidelities, we demonstrate that quantum supremacy is limited to circuits with a qubit count and circuit depth of a few hundred. Larger circuits encounter two distinct boundaries: a return of a classical advantage and practically infeasible quantum runtimes. Decreasing error rates cause the region of a quantum advantage to grow rapidly. At error rates required for early implementations of the surface code, the largest circuit size within the quantum supremacy regime coincides approximately with the smallest circuit size needed to implement error correction. Thus, the boundaries of quantum supremacy may fortuitously coincide with the advent of scalable, error-corrected quantum computing
Selective Serotonin Reuptake Inhibitors within Cells: Temporal Resolution in Cytoplasm, Endoplasmic Reticulum, and Membrane
No full textSelective serotonin reuptake inhibitors (SSRIs) are the most prescribed treatment for individuals experiencing major depressive disorder. The therapeutic mechanisms that take place before, during, or after SSRIs bind the serotonin transporter (SERT) are poorly understood, partially because no studies exist on the cellular and subcellular pharmacokinetic properties of SSRIs in living cells. We studied escitalopram and fluoxetine using new intensity-based, drug-sensing fluorescent reporters targeted to the plasma membrane, cytoplasm, or endoplasmic reticulum (ER) of cultured neurons and mammalian cell lines. We also used chemical detection of drug within cells and phospholipid membranes. The drugs attain equilibrium in neuronal cytoplasm and ER at approximately the same concentration as the externally applied solution, with time constants of a few s (escitalopram) or 200–300 s (fluoxetine). Simultaneously, the drugs accumulate within lipid membranes by ≥18-fold (escitalopram) or 180-fold (fluoxetine), and possibly by much larger factors. Both drugs leave cytoplasm, lumen, and membranes just as quickly during washout. We synthesized membrane-impermeant quaternary amine derivatives of the two SSRIs. The quaternary derivatives are substantially excluded from membrane, cytoplasm, and ER for >2.4 h. They inhibit SERT transport-associated currents sixfold or 11-fold less potently than the SSRIs (escitalopram or fluoxetine derivative, respectively), providing useful probes for distinguishing compartmentalized SSRI effects. Although our measurements are orders of magnitude faster than the therapeutic lag of SSRIs, these data suggest that SSRI–SERT interactions within organelles or membranes may play roles during either the therapeutic effects or the antidepressant discontinuation syndrome.
SIGNIFICANCE STATEMENT:
Selective serotonin reuptake inhibitors stabilize mood in several disorders. In general, these drugs bind to SERT, which clears serotonin from CNS and peripheral tissues. SERT ligands are effective and relatively safe; primary care practitioners often prescribe them. However, they have several side effects and require 2–6 weeks of continuous administration until they act effectively. How they work remains perplexing, contrasting with earlier assumptions that the therapeutic mechanism involves SERT inhibition followed by increased extracellular serotonin levels. This study establishes that two SERT ligands, fluoxetine and escitalopram, enter neurons within minutes, while simultaneously accumulating in many membranes. Such knowledge will motivate future research, hopefully revealing where and how SERT ligands engage their therapeutic target(s)
Advances in theoretical and computational nonlinear control — Dedicated to the 80th birthday of Professor Arthur J. Krener
No full textIt is with great pleasure that we introduce this special issue of Systems & Control Letters, on “Advances in Theoretical and Computational Nonlinear Control — Dedicated to the 80th Birthday of Professor Arthur J. Krener”