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We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way. 

Fibrinolysis, the plasmin-mediated degradation of the fibrin mesh that stabilizes blood clots, is an important physiological process, and understanding mechanisms underlying lysis is critical for improved stroke treatment. Experimentalists are now able to study lysis on the scale of single fibrin fibers, but mathematical models of lysis continue to focus mostly on fibrin network degradation. Experiments have shown that while some degradation occurs along the length of a fiber, ultimately the fiber is cleaved at a single location. We built a 2-dimensional stochastic model of a fibrin fiber cross-section that uses the Gillespie algorithm to study single fiber lysis initiated by plasmin. We simulated the model over a range of parameter values to learn about patterns and rates of single fiber lysis in various physiological conditions. We also used epifluorescent microscopy to measure the cleavage times of fibrin fibers with different apparent diameters. By comparing our model results to the laboratory experiments, we were able to: 1) suggest value ranges for unknown rate constants(namely that the degradation rate of fibrin by plasmin should be ≤ 10 s⁻¹and that if plasmin crawls, the rate of crawling should be between 10 s⁻¹and 60 s⁻¹); 2) estimate the fraction of fibrin within a fiber cross-section that must be degraded for the fiber to cleave in two; and 3) propose that that fraction is higher in thinner fibers and lower in thicker fibers. Collectively, this information provides more details about how fibrin fibers degrade, which can be leveraged in the future for a better understanding of why fibrinolysis is impaired in certain disease states, and could inform intervention strategies. 

Abstract This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on $${\mathbb{R}}^{+}$$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain ‘weak Harris theorems’. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting stochastic partial differential equation examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier–Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on $${\mathbb{R}}^{+}$$. To develop these numerical analysis results, we provide a refinement of $$L^{2}_{x}$$ accuracy bounds in comparison to the existing literature, which are results of independent interest. 

Abstract Parallel Markov Chain Monte Carlo (pMCMC) algorithms generate clouds of proposals at each step to efficiently resolve a target probability distribution $$\mu $$. We build a rigorous foundational framework for pMCMC algorithms that situates these methods within a unified ‘extended phase space’ measure-theoretic formalism. Drawing on our recent work that provides a comprehensive theory for reversible single-proposal methods, we herein derive general criteria for multiproposal acceptance mechanisms that yield ergodic chains on general state spaces. Our formulation encompasses a variety of methodologies, including proposal cloud resampling and Hamiltonian methods, while providing a basis for the derivation of novel algorithms. In particular, we obtain a top-down picture for a class of methods arising from ‘conditionally independent’ proposal structures. As an immediate application of this formalism, we identify several new algorithms including a multiproposal version of the popular preconditioned Crank–Nicolson (pCN) sampler suitable for high- and infinite-dimensional target measures that are absolutely continuous with respect to a Gaussian base measure. To supplement the aforementioned theoretical results, we carry out a selection of numerical case studies that evaluate the efficacy of these novel algorithms. First, noting that the true potential of pMCMC algorithms arises from their natural parallelizability and the ease with which they map to modern high-performance computing architectures, we provide a limited parallelization study using TensorFlow and a graphics processing unit to scale pMCMC algorithms that leverage as many as 100k proposals at each step. Second, we use our multiproposal pCN algorithm (mpCN) to resolve a selection of problems in Bayesian statistical inversion for partial differential equations motivated by fluid measurement. These examples provide preliminary evidence of the efficacy of mpCN for high-dimensional target distributions featuring complex geometries and multimodal structures. 

Abstract BackgroundPathogenicLeptospiraspecies are globally important zoonotic pathogens capable of infecting a wide range of host species. In marine mammals, reports ofLeptospirahave predominantly been in pinnipeds, with isolated reports of infections in cetaceans. Case presentationOn 28 June 2021, a 150.5 cm long female, short-beaked common dolphin (Delphinus delphis delphis) stranded alive on the coast of southern California and subsequently died. Gross necropsy revealed multifocal cortical pallor within the reniculi of the kidney, and lymphoplasmacytic tubulointerstitial nephritis was observed histologically. Immunohistochemistry confirmedLeptospirainfection, and PCR followed bylfb1gene amplicon sequencing suggested that the infecting organism wasL.kirschneri. LeptospiraDNA capture and enrichment allowed for whole-genome sequencing to be conducted. Phylogenetic analyses confirmed the causative agent was a previously undescribed, divergent lineage ofL.kirschneri. ConclusionsWe report the first detection of pathogenicLeptospirain a short-beaked common dolphin, and the first detection in any cetacean in the northeastern Pacific Ocean. Renal lesions were consistent with leptospirosis in other host species, including marine mammals, and were the most significant lesions detected overall, suggesting leptospirosis as the likely cause of death. We identified the cause of the infection asL.kirschneri, a species detected only once before in a marine mammal – a northern elephant seal (Mirounga angustirostris) of the northeastern Pacific. These findings raise questions about the mechanism of transmission, given the obligate marine lifestyle of cetaceans (in contrast to pinnipeds, which spend time on land) and the commonly accepted view thatLeptospiraare quickly killed by salt water. They also raise important questions regarding the source of infection, and whether it arose from transmission among marine mammals or from terrestrial-to-marine spillover. Moving forward, surveillance and sampling must be expanded to better understand the extent to whichLeptospirainfections occur in the marine ecosystem and possible epidemiological linkages between and among marine and terrestrial host species. GeneratingLeptospiragenomes from different host species will yield crucial information about possible transmission links, and our study highlights the power of new techniques such as DNA enrichment to illuminate the complex ecology of this important zoonotic pathogen. 

The continuous-time Markov chain (CTMC) is the mathematical workhorse of evolutionary biology. Learning CTMC model parameters using modern, gradient-based methods requires the derivative of the matrix exponential evaluated at the CTMC’s infinitesimal generator (rate) matrix. Motivated by the derivative’s extreme computational complexity as a function of state space cardinality, recent work demonstrates the surprising effectiveness of a naive, first-order approximation for a host of problems in computational biology. In response to this empirical success, we obtain rigorous deterministic and probabilistic bounds for the error accrued by the naive approximation and establish a “blessing of dimensionality” result that is universal for a large class of rate matrices with random entries. Finally, we apply the first-order approximation within surrogate-trajectory Hamiltonian Monte Carlo for the analysis of the early spread of Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) across 44 geographic regions that comprise a state space of unprecedented dimensionality for unstructured (flexible) CTMC models within evolutionary biology. 

As the era of omics continues to expand with increasing ubiquity and success in both academia and industry, omics-based experiments are becoming commonplace in industrial biotechnology, including efforts to develop novel solutions in bioprocess optimization and cell line development. Omic technologies provide particularly valuable ‘observational’ insights for discovery science, especially in academic research and industrial R&D; however, biomanufacturing requires a different paradigm to unlock ‘actionable’ insights from omics. Here, we argue the value of omic experiments in biotechnology can be maximized with deliberate selection of omic approaches and forethought about analysis techniques. We describe important considerations when designing and implementing omic-based experiments and discuss how systems biology analysis strategies can enhance efforts to obtain actionable insights in mammalian-based biologics production.