Mass and spring dimer Fermi-Pasta-Ulam-Tsingou nanopterons with exponentially small, nonvanishing ripples

We study traveling waves in mass and spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattices in the long wave limit. Such lattices are known to possess nanopteron traveling waves in relative displacement coordinates. These nanopteron profiles consist of the superposition of an exponentially localized "core," which is close to a KdV solitary wave, and a periodic "ripple," whose amplitude is small beyond all algebraic orders of the long wave parameter, although a zero amplitude is not precluded. Here we deploy techniques of spatial dynamics, inspired by results of Iooss and Kirchg\"{a}ssner, Iooss and James, and Venney and Zimmer, to construct mass and spring dimer nanopterons whose ripples are both exponentially small and also nonvanishing. We first obtain "growing front" traveling waves in the original position coordinates and then pass to relative displacement. To study position, we recast its traveling wave problem as a first-order equation on an infinite-dimensional Banach space; then we develop hypotheses that, when met, allow us to reduce such a first-order problem to one solved by Lombardi. A key part of our analysis is then the passage back from the reduced problem to the original one. Our hypotheses free us from working strictly with lattices but are easily checked for FPUT mass and spring dimers. We also give a detailed exposition and reinterpretation of Lombardi's methods, to illustrate how our hypotheses work in concert with his techniques, and we provide a dialogue with prior methods of constructing FPUT nanopterons, to expose similarities and differences with the present approach

Micropterons, Nanopterons and Solitary Wave Solutions to the Diatomic Fermi-Pasta-Ulam-Tsingou Problem

We use a specialized boundary-value problem solver for mixed-type functional differential equations to numerically examine the landscape of traveling wave solutions to the diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) problem. By using a continuation approach, we are able to uncover the relationship between the branches of micropterons and nanopterons that have been rigorously constructed recently in various limiting regimes. We show that the associated surfaces are connected together in a nontrivial fashion and illustrate the key role that solitary waves play in the branch points. Finally, we numerically show that the diatomic solitary waves are stable under the full dynamics of the FPUT system

Nanopteron-Stegoton Traveling Waves in Mass and Spring Dimer Fermi-Pasta-Ulam-Tsingou Lattices

We study the existence of traveling waves in mass and spring dimer Fermi-Pasta- Ulam-Tsingou (FPUT) lattices. These are infinite, one-dimensional lattices of particles connected by nonlinear springs, in which either the masses alternate (the mass dimer or diatomic lattice) or the spring forces alternate (the spring dimer). Under the classical "long wave" scaling, the lattice equations of motion turn out to be sin- gularly perturbed. In response to this complication, we apply a method of Beale to produce nanopteron traveling wave solutions with wave speed slightly greater than the lattice's speed of sound. The nanopteron wave profiles are the superposition of an exponentially decaying term (which itself is a small perturbation of a KdV soliton) and a periodic term of very small amplitude. This dissertation builds on the previous work of Faver and Wright on mass dimer lattices to treat spring dimer lattices. Further generalizing the spring forces from the mass dimer case, we allow the springs' nonlinearity to contain higher order terms be- yond the quadratic. This necessitates the use of composition operators to phrase the long wave problem, and these operators require delicate estimates due to the characteristic superposition of different function types from Beale's ansatz. Additionally, the value of the leading order term in the spring dimer traveling wave profiles alternates between particle sites, so that, unlike in the mass dimer, the spring dimer traveling waves are also "stegotons."Ph.D., Mathematics -- Drexel University, 201

DNA-encoded chemistry technology yields expedient access to SARS-CoV-2 Mpro inhibitors.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has killed more than 4 million humans globally, but there is no bona fide Food and Drug Administration-approved drug-like molecule to impede the COVID-19 pandemic. The sluggish pace of traditional therapeutic discovery is poorly suited to producing targeted treatments against rapidly evolving viruses. Here, we used an affinity-based screen of 4 billion DNA-encoded molecules en masse to identify a potent class of virus-specific inhibitors of the SARS-CoV-2 main protease (Mpro) without extensive and time-consuming medicinal chemistry. CDD-1714, the initial three-building-block screening hit (molecular weight [MW] = 542.5 g/mol), was a potent inhibitor (inhibition constant [K i] = 20 nM). CDD-1713, a smaller two-building-block analog (MW = 353.3 g/mol) of CDD-1714, is a reversible covalent inhibitor of Mpro (K i = 45 nM) that binds in the protease pocket, has specificity over human proteases, and shows in vitro efficacy in a SARS-CoV-2 infectivity model. Subsequently, key regions of CDD-1713 that were necessary for inhibitory activity were identified and a potent (K i = 37 nM), smaller (MW = 323.4 g/mol), and metabolically more stable analog (CDD-1976) was generated. Thus, screening of DNA-encoded chemical libraries can accelerate the discovery of efficacious drug-like inhibitors of emerging viral disease targets

DNA-encoded chemistry technology yields expedient access to SARS-CoV-2 Mpro inhibitors.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has killed more than 4 million humans globally, but there is no bona fide Food and Drug Administration-approved drug-like molecule to impede the COVID-19 pandemic. The sluggish pace of traditional therapeutic discovery is poorly suited to producing targeted treatments against rapidly evolving viruses. Here, we used an affinity-based screen of 4 billion DNA-encoded molecules en masse to identify a potent class of virus-specific inhibitors of the SARS-CoV-2 main protease (Mpro) without extensive and time-consuming medicinal chemistry. CDD-1714, the initial three-building-block screening hit (molecular weight [MW] = 542.5 g/mol), was a potent inhibitor (inhibition constant [Ki] = 20 nM). CDD-1713, a smaller two-building-block analog (MW = 353.3 g/mol) of CDD-1714, is a reversible covalent inhibitor of Mpro (Ki = 45 nM) that binds in the protease pocket, has specificity over human proteases, and shows in vitro efficacy in a SARS-CoV-2 infectivity model. Subsequently, key regions of CDD-1713 that were necessary for inhibitory activity were identified and a potent (Ki = 37 nM), smaller (MW = 323.4 g/mol), and metabolically more stable analog (CDD-1976) was generated. Thus, screening of DNA-encoded chemical libraries can accelerate the discovery of efficacious drug-like inhibitors of emerging viral disease targets