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Highly resolved spatial data of complex systems encode rich and nonlinear information. Quantification of heterogeneous and noisy data—often with outliers, artifacts, and mislabeled points—such as those from tissues, remains a challenge. The mathematical field that extracts information from the shape of data, topological data analysis (TDA), has expanded its capability for analyzing real-world datasets in recent years by extending theory, statistics, and computation. An extension to the standard theory to handle heterogeneous data is multiparameter persistent homology (MPH). Here we provide an application of MPH landscapes, a statistical tool with theoretical underpinnings. MPH landscapes, computed for (noisy) data from agent-based model simulations of immune cells infiltrating into a spheroid, are shown to surpass existing spatial statistics and one-parameter persistent homology. We then apply MPH landscapes to study immune cell location in digital histology images from head and neck cancer. We quantify intratumoral immune cells and find that infiltrating regulatory T cells have more prominent voids in their spatial patterns than macrophages. Finally, we consider how TDA can integrate and interrogate data of different types and scales, e.g., immune cell locations and regions with differing levels of oxygenation. This work highlights the power of MPH landscapes for quantifying, characterizing, and comparing features within the tumor microenvironment in synthetic and real datasets.

Understanding how knotted proteins fold is a challenging problem in biology. Researchers have proposed several models for their folding pathways, based on theory, simulations and experiments. The geometry of proteins with the same knot type can vary substantially and recent simulations reveal different folding behaviour for deeply and shallow knotted proteins. We analyse proteins forming open-ended trefoil knots by introducing a topologically inspired statistical metric that measures their entanglement. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for such proteins. In particular, the folding pathway of shallow knotted carbonic anhydrases involves the creation of a double-looped structure, contrary to what has been observed for other knotted trefoil proteins. We validate this with Molecular Dynamics simulations. By leveraging the geometry and local symmetries of knotted proteins’ native states, we provide the first numerical evidence of a double-loop folding mechanism in trefoil proteins.

An ideal invariant for multiparameter persistence would be discriminative, computable and stable. In this work we analyse the discriminative power of a stable, computable invariant of multiparameter persistence modules: the fibered bar code. The fibered bar code is equivalent to the rank invariant and encodes the bar codes of the 1-parameter submodules of a multiparameter module. This invariant is well known to be globally incomplete. However in this work we show that the fibered bar code is locally complete for finitely presented modules by showing a local equivalence of metrics between the interleaving distance (which is complete on finitely-presented modules) and the matching distance on fibered bar codes. More precisely, we show that: for a finitely-presented multiparameter module \\(M\\) there is a neighbourhood of \\(M\\), in the interleaving distance \\(d_I\\), for which the matching distance, \\(d_0\\), satisfies the following bi-Lipschitz inequalities \\(\\frac{1}{34}d_I(M,N) \\leq d_0(M,N) \\leq d_I(M,N)\\) for all \\(N\\) in this neighbourhood about \\(M\\). As a consequence no other module in this neighbourhood has the same fibered bar code as \\(M\\).

An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable representation of persistence diagrams amenable to statistical analysis and machine learning tools. In this paper we generalise the persistence landscape to multiparameter persistence modules providing a stable representation of the rank invariant. We show that multiparameter landscapes are stable with respect to the interleaving distance and persistence weighted Wasserstein distance, and that the collection of multiparameter landscapes faithfully represents the rank invariant. Finally we provide example calculations and statistical tests to demonstrate a range of potential applications and how one can interpret the landscapes associated to a multiparameter module.

Let \\(M\\) be a compact, unit volume, Riemannian manifold with boundary. In this paper we study the homology of a random Čech-complex generated by a homogeneous Poisson process in \\(M\\). Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the \\(k\\)-th homology of \\(M\\) with high probability, and a lower threshold below which it almost certainly does not. These thresholds are close together in the sense that they have the same leading term. Here \\(k\\) is positive and strictly less than the dimension \\(d\\) of the manifold. This extends work of Bobrowski and Weinberger in [BW17] and Bobrowski and Oliveira [BO19] who establish similar formulas when \\(M\\) is a torus and, more generally, is closed and has no boundary. We note that the cases with and without boundary lead to different answers: The corresponding common leading terms for the upper and lower thresholds differ being \\(\\log (n) \\) when \\(M\\) is closed and \\((2-2/d)\\log (n)\\) when \\(M\\) has boundary; here \\(n\\) is the expected number of sample points. Our analysis identifies a special type of homological cycle, which we call a \\(\\Theta\\)-like-cycle, which occur close to the boundary and establish that the first order term of the lower threshold is \\((2-2/d)\\log (n)\\).

Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of a protein's planar projections using topological objects called knotoids. We approach the analysis of proteins with the same topology by introducing a topologically inspired statistical metric between their knotoid distributions. We detect geometric differences between trefoil proteins by characterising their entanglement and we recover a clustering by sequence similarity. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for proteins forming open-ended trefoil knots. Interestingly, our pipeline reveals that the folding pathway of shallow knotted Carbonic Anhydrases involves the creation of a double-looped structure, differently from what was previously observed for deeply knotted trefoil proteins. We validate this with Molecular Dynamics simulations.

Background We are now in the fifth year of an ongoing pandemic, and Canada continues to experience significant surges of COVID-19 infections. In addition to the acute impacts of deaths and hospitalizations, there is growing awareness of an accumulation of organ damage and disability which is building a “health debt” that will affect Canadians for decades to come. Calls in 2023 for an inquiry into the handling of the COVID-19 pandemic went unheeded, despite relevant precedent. Canada urgently needs a comprehensive review of its successes and failures to chart a better response in the near- and long-term. Main body While Canada fared better than many comparators in the early years of the COVID-19 pandemic, it is clearly still in a public health crisis. Infections are not only affecting Canadians’ daily lives but also eroding healthcare capacity. Post-COVID condition is having accumulating and profound individual, social, and economic consequences. An inquiry is needed to understand the current evidence underlying policy choices, identify a better course of action on various fronts, and build resilience. More must be done to reduce transmission, including a serious public education campaign to better inform Canadians about COVID and effective mitigations, especially the benefits of respirator masks. We need a national standard for indoor air quality to make indoor public spaces safer, particularly schools. Data collection must be more robust, especially to understand and mitigate the disproportionate impacts on under-served communities and high-risk populations. General confidence in public health must be rebuilt, with a focus on communication and transparency. In particular, the wide variation in provincial policies has sown mistrust: evidence-based policy should be consistent. Finally, Canada’s early success in vaccination has collapsed, and this development needs a careful post-mortem. Conclusions A complete investigation of Canada’s response to the pandemic is not yet possible because that response is still ongoing and, while we have learned much, there remain areas of dispute and uncertainty. However, an inquiry is needed to conduct a rapid assessment of the current evidence and policies and provide recommendations on how to improve in 2025 and beyond as well as guidance for future pandemics.

trish.greenhalgh@phc.ox.ac.uk We congratulate the authors on successfully completing a randomised controlled trial of advice to mask in a community setting.1 This topic is notoriously difficult to study experimentally and few previous randomised controlled trials have been published. The main finding of this new study—that people advised to mask are significantly less likely to develop symptoms of a new respiratory infection over the next 17 days—is consistent with our recently published comprehensive review of existing evidence.2 The short follow-up period and lack of virologically confirmed outcomes and data on compliance are a limitation. While this study showed a significant advantage of surgical masks compared with no masks, it also demonstrated a failure rate of up to 70% (depending on the unknown non-compliance rate), which is not surprising given the airborne nature of respiratory viruses3 and the known propensity of surgical masks to leak.

The mutation profile of the SARS-CoV-2 Omicron (lineage BA.1) variant posed a concern for naturally acquired and vaccine-induced immunity. We investigated the ability of prior infection with an early SARS-CoV-2 ancestral isolate (Australia/VIC01/2020, VIC01) to protect against disease caused by BA.1. We established that BA.1 infection in naïve Syrian hamsters resulted in a less severe disease than a comparable dose of the ancestral virus, with fewer clinical signs including less weight loss. We present data to show that these clinical observations were almost absent in convalescent hamsters challenged with the same dose of BA.1 50 days after an initial infection with ancestral virus. These data provide evidence that convalescent immunity against ancestral SARS-CoV-2 is protective against BA.1 in the Syrian hamster model of infection. Comparison with published pre-clinical and clinical data supports consistency of the model and its predictive value for the outcome in humans. Further, the ability to detect protection against the less severe disease caused by BA.1 demonstrates continued value of the Syrian hamster model for evaluation of BA.1-specific countermeasures.