CMOR Publications

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Strategy investments in zero-sum games
(Springer Nature, 2024) Garcia, Raul; Hosseinian, Seyedmohammadhossein; Pai, Mallesh; Schaefer, Andrew J.
We propose an extension of two-player zero-sum games, where one player may select available actions for themselves and the opponent, subject to a budget constraint. We present a mixed-integer linear programming (MILP) formulation for the problem, provide analytical results regarding its solution, and discuss applications in the security and advertising domains. Our computational experiments demonstrate that heuristic approaches, on average, yield suboptimal solutions with at least a 20% relative gap with those obtained by the MILP formulation.
An ultraweak-local discontinuous Galerkin method for nonlinear biharmonic Schrödinger equations
(edp Sciences, 2024) Wang, Qi; Zhang, Lu; Ken Kennedy Institute
This paper proposes and analyzes a fully discrete scheme for nonlinear biharmonic Schrödinger equations. We first write the single equation into a system of problems with second-order spatial derivatives and then discretize the space variable with an ultraweak discontinuous Galerkin scheme and the time variable with the Crank–Nicolson method. The proposed scheme proves to be computationally more efficient compared to the local discontinuous Galerkin method in terms of the number of equations needed to be solved at each single time step, and it is unconditionally stable without imposing any penalty terms. It also achieves optimal error convergence in L2 norm both in the solution and in the auxiliary variable with general nonlinear terms. We also prove several physically relevant properties of the discrete schemes, such as the conservation of mass and the Hamiltonian for the nonlinear biharmonic Schrödinger equations. Several numerical studies demonstrate and support our theoretical results.
Stable Recovery of Coefficients in an Inverse Fault Friction Problem
(Springer Nature, 2024) de Hoop, Maarten V.; Lassas, Matti; Lu, Jinpeng; Oksanen, Lauri
We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.
A new diagonalization based method for parallel-in-time solution of linear-quadratic optimal control problems
(EDP Sciences, 2024) Heinkenschloss, Matthias; Kroeger, Nathaniel J.
A new diagonalization technique for the parallel-in-time solution of linear-quadratic optimal control problems with time-invariant system matrices is introduced. The target problems are often derived from a semi-discretization of a Partial Differential Equation (PDE)-constrained optimization problem. The solution of large-scale time dependent optimal control problems is computationally challenging as the states, controls, and adjoints are coupled to each other throughout the whole time domain. This computational difficulty motivates the use of parallel-in-time methods. For time-periodic problems our diagonalization efficiently transforms the discretized optimality system into nt (=number of time steps) decoupled complex valued 2ny × 2ny systems, where ny is the dimension of the state space. These systems resemble optimality systems corresponding to a steady-state version of the optimal control problem and they can be solved in parallel across the time steps, but are complex valued. For optimal control problems with initial value state equations a direct solution via diagonalization is not possible, but an efficient preconditioner can be constructed from the corresponding time periodic optimal control problem. The preconditioner can be efficiently applied parallel-in-time using the diagonalization technique. The observed number of preconditioned GMRES iterations is small and insensitive to the size of the problem discretization.
Benchmarking lung cancer screening programmes with adaptive screening frequency against the optimal screening schedules derived from the ENGAGE framework: a comparative microsimulation study
(Elsevier, 2024) Hemmati, Mehdi; Ishizawa, Sayaka; Meza, Rafael; Ostrin, Edwin; Hanash, Samir M.; Antonoff, Mara; Schaefer, Andrew J.; Tammemägi, Martin C.; Toumazis, Iakovos
Background Lung cancer screening recommendations employ annual frequency for eligible individuals, despite evidence that it may not be universally optimal. The impact of imposing a structure on the screening frequency remains unknown. The ENGAGE framework, a validated framework that offers fully dynamic, analytically optimal, personalised lung cancer screening recommendations, could be used to assess the impact of screening structure on the effectiveness and efficiency of lung cancer screening. Methods In this comparative microsimulation study, we benchmarked alternative clinically relevant structured lung cancer screening programmes employing a fixed (annual or biennial) or adaptive (start with annual/biennial screening and then switch to biennial/annual at ages 60- or 65-years) screening frequency, against the ENGAGE framework. Individuals were eligible for screening according to the 2021 US Preventive Services Task Force recommendation on lung cancer screening. We assessed programmes' efficiency based on the number of screenings per death avoided (LDCT/DA) and the number of screenings per ever-screened individual (LDCT/ESI), and programmes’ effectiveness using quality-adjusted life years (QALY) gained from screening, lung cancer-specific mortality reduction (MR), and number of screen-detected lung cancer cases. We used validated natural history, smoking history generator, and risk prediction models to inform our analysis. Sensitivity analysis of key inputs was conducted. Findings ENGAGE was the best performing strategy. Among the structured policies, adaptive biennial-to-annual at age 65 was the best strategy requiring 24% less LDCT/DA and 60% less LDCT/ESI compared to TF2021, but yielded 105 more deaths per 100,000 screen-eligible individuals (10.2% vs. 11.8% MR for TF2021, p = 0.28). Fixed annual screening was the most effective strategy but the least efficient and was ranked as the fifth best strategy. All strategies yielded similar QALYs gained. Adherence levels did not affect the rankings. Interpretation Adaptive lung cancer screening strategies that start with biennial and switch to annual screening at a prespecified age perform well and warrant further consideration, especially in settings with limited availability of CT scanners and radiologists. Funding National Cancer Institute.
Cluster-Based Toxicity Estimation of Osteoradionecrosis Via Unsupervised Machine Learning: Moving Beyond Single Dose-Parameter Normal Tissue Complication Probability by Using Whole Dose-Volume Histograms for Cohort Risk Stratification
(Elsevier, 2024) Hosseinian, Seyedmohammadhossein; Hemmati, Mehdi; Dede, Cem; Salzillo, Travis C.; van Dijk, Lisanne V.; Mohamed, Abdallah S. R.; Lai, Stephen Y.; Schaefer, Andrew J.; Fuller, Clifton D.
Purpose Given the limitations of extant models for normal tissue complication probability estimation for osteoradionecrosis (ORN) of the mandible, the purpose of this study was to enrich statistical inference by exploiting structural properties of data and provide a clinically reliable model for ORN risk evaluation through an unsupervised-learning analysis that incorporates the whole radiation dose distribution on the mandible. Methods and Materials The analysis was conducted on retrospective data of 1259 patients with head and neck cancer treated at The University of Texas MD Anderson Cancer Center between 2005 and 2015. During a minimum 12-month posttherapy follow-up period, 173 patients in this cohort (13.7%) developed ORN (grades I to IV). The (structural) clusters of mandibular dose-volume histograms (DVHs) for these patients were identified using the K-means clustering method. A soft-margin support vector machine was used to determine the cluster borders and partition the dose-volume space. The risk of ORN for each dose-volume region was calculated based on incidence rates and other clinical risk factors. Results The K-means clustering method identified 6 clusters among the DVHs. Based on the first 5 clusters, the dose-volume space was partitioned by the soft-margin support vector machine into distinct regions with different risk indices. The sixth cluster entirely overlapped with the others; the region of this cluster was determined by its envelopes. For each region, the ORN incidence rate per preradiation dental extraction status (a statistically significant, nondose related risk factor for ORN) was reported as the corresponding risk index. Conclusions This study presents an unsupervised-learning analysis of a large-scale data set to evaluate the risk of mandibular ORN among patients with head and neck cancer. The results provide a visual risk-assessment tool for ORN (based on the whole DVH and preradiation dental extraction status) as well as a range of constraints for dose optimization under different risk levels.
Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation
(Elsevier, 2024) Anderson, Thomas G.; Bonnet, Marc; Faria, Luiz M.; Pérez-Arancibia, Carlos
This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented.
Depletion of lamins B1 and B2 promotes chromatin mobility and induces differential gene expression by a mesoscale-motion-dependent mechanism
(Springer Nature, 2024) Pujadas Liwag, Emily M.; Wei, Xiaolong; Acosta, Nicolas; Carter, Lucas M.; Yang, Jiekun; Almassalha, Luay M.; Jain, Surbhi; Daneshkhah, Ali; Rao, Suhas S. P.; Seker-Polat, Fidan; MacQuarrie, Kyle L.; Ibarra, Joe; Agrawal, Vasundhara; Aiden, Erez Lieberman; Kanemaki, Masato T.; Backman, Vadim; Adli, Mazhar; Center for Theoretical Biological Physics
B-type lamins are critical nuclear envelope proteins that interact with the three-dimensional genomic architecture. However, identifying the direct roles of B-lamins on dynamic genome organization has been challenging as their joint depletion severely impacts cell viability. To overcome this, we engineered mammalian cells to rapidly and completely degrade endogenous B-type lamins using Auxin-inducible degron technology.
Predicting A/B compartments from histone modifications using deep learning
(Elsevier, 2024) Zheng, Suchen; Thakkar, Nitya; Harris, Hannah L.; Liu, Susanna; Zhang, Megan; Gerstein, Mark; Aiden, Erez Lieberman; Rowley, M. Jordan; Noble, William Stafford; Gürsoy, Gamze; Singh, Ritambhara
The three-dimensional organization of genomes plays a crucial role in essential biological processes. The segregation of chromatin into A and B compartments highlights regions of activity and inactivity, providing a window into the genomic activities specific to each cell type. Yet, the steep costs associated with acquiring Hi-C data, necessary for studying this compartmentalization across various cell types, pose a significant barrier in studying cell type specific genome organization. To address this, we present a prediction tool called compartment prediction using recurrent neural networks (CoRNN), which predicts compartmentalization of 3D genome using histone modification enrichment. CoRNN demonstrates robust cross-cell-type prediction of A/B compartments with an average AuROC of 90.9%. Cell-type-specific predictions align well with known functional elements, with H3K27ac and H3K36me3 identified as highly predictive histone marks. We further investigate our mispredictions and found that they are located in regions with ambiguous compartmental status. Furthermore, our model’s generalizability is validated by predicting compartments in independent tissue samples, which underscores its broad applicability.
Stable reconstruction of simple Riemannian manifolds from unknown interior sources
(IOP Publishing Ltd, 2023) Hoop, Maarten V. de; Ilmavirta, Joonas; Lassas, Matti; Saksala, Teemu
Consider the geometric inverse problem: there is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a Riemannian manifold with boundary? With a finite set of sources we can only hope to get an approximate reconstruction, and we indeed provide a discrete metric approximation to the manifold with explicit data-driven error bounds when the manifold is simple. This is the geometrization of a seismological inverse problem where we measure the arrival times on the surface of waves from an unknown number of unknown interior microseismic events at unknown times. The closeness of two metric spaces with a marked boundary is measured by a labeled Gromov–Hausdorff distance. If measurements are done for infinite time and spatially dense sources, our construction produces the true Riemannian manifold and the finite-time approximations converge to it in the metric sense
A rapid, low-cost, and highly sensitive SARS-CoV-2 diagnostic based on whole-genome sequencing
(Public Library of Science, 2023) Adastra, Per A.; Durand, Neva C.; Mitra, Namita; Pulido, Saul Godinez; Mahajan, Ragini; Blackburn, Alyssa; Colaric, Zane L.; Theisen, Joshua W. M.; Weisz, David; Dudchenko, Olga; Gnirke, Andreas; Rao, Suhas S. P.; Kaur, Parwinder; Aiden, Erez Lieberman; Aiden, Aviva Presser; Center for Theoretical Biological Physics
Early detection of SARS-CoV-2 infection is key to managing the current global pandemic, as evidence shows the virus is most contagious on or before symptom onset. Here, we introduce a low-cost, high-throughput method for diagnosing and studying SARS-CoV-2 infection. Dubbed Pathogen-Oriented Low-Cost Assembly & Re-Sequencing (POLAR), this method amplifies the entirety of the SARS-CoV-2 genome. This contrasts with typical RT-PCR-based diagnostic tests, which amplify only a few loci. To achieve this goal, we combine a SARS-CoV-2 enrichment method developed by the ARTIC Network (https://artic.network/) with short-read DNA sequencing and de novo genome assembly. Using this method, we can reliably (>95% accuracy) detect SARS-CoV-2 at a concentration of 84 genome equivalents per milliliter (GE/mL). The vast majority of diagnostic methods meeting our analytical criteria that are currently authorized for use by the United States Food and Drug Administration with the Coronavirus Disease 2019 (COVID-19) Emergency Use Authorization require higher concentrations of the virus to achieve this degree of sensitivity and specificity. In addition, we can reliably assemble the SARS-CoV-2 genome in the sample, often with no gaps and perfect accuracy given sufficient viral load. The genotypic data in these genome assemblies enable the more effective analysis of disease spread than is possible with an ordinary binary diagnostic. These data can also help identify vaccine and drug targets. Finally, we show that the diagnoses obtained using POLAR of positive and negative clinical nasal mid-turbinate swab samples 100% match those obtained in a clinical diagnostic lab using the Center for Disease Control’s 2019-Novel Coronavirus test. Using POLAR, a single person can manually process 192 samples over an 8-hour experiment at the cost of ~$36 per patient (as of December 7th, 2022), enabling a 24-hour turnaround with sequencing and data analysis time. We anticipate that further testing and refinement will allow greater sensitivity using this approach.
The contagion number: How fast can a disease spread?
(National Library of Serbia, 2023) Blessley, Misty; Davila, Randy; Hale, Trevor; Pepper, Ryan
The burning number of a graph models the rate at which a disease, information, or other externality can propagate across a network. The burning number is known to be NP-hard even for a tree. Herein, we define a relative of the burning number that we coin the contagion number (CN). We aver that the CN is a better metric to model disease spread than the burning number as it only counts first time infections (i.e., constrains a node from getting the same disease/same variant/same alarm more than once). This is important because the Centers for Disease Control and Prevention report that COVID-19 reinfections are rare. This paper delineates a method to solve for the contagion number of any tree, in polynomial time, which addresses how fast a disease could spread (i.e., a worst-cast analysis) and then employs simulation to determine the average contagion number (ACN) (i.e., a most-likely analysis) of how fast a disease would spread. The latter is analyzed on scale-free graphs, which are used to model human social networks generated through a preferential attachment mechanism. With CN differing across network structures and almost identical to ACN, our findings advance disease spread understanding and reveal the importance of network structure. In a borderless world without replete resources, understanding disease spread can do much to inform public policy and managerial decision makers’ allocation decisions. Furthermore, our direct interactions with supply chain executives at two COVID-19 vaccine developers provided practical grounding on what the results suggest for achieving social welfare objectives.
Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source
(EDP Sciences, 2023) Masri, Rami; Shen, Boqian; Riviere, Beatrice
The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L^{2^{norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L^{2^{norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L^{2^{norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L^{2^{in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.}}}}}}}}
Quantitative unique continuation for the elasticity system with application to the kinematic inverse rupture problem
(Taylor & Francis, 2023) de Hoop, Maarten V.; Lassas, Matti; Lu, Jinpeng; Oksanen, Lauri
We obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular, our result applies to the elasticity system and also the Maxwell system. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the stable unique continuation up to the rupture surface.
Multi-Patch Epidemic Models with General Exposed and Infectious Periods
(EDP Sciences, 2023) Pang, Guodong; Pardoux, Étienne
We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as “from distance” from all the other patches. The migration processes among the patches in either of the four states are assumed to be Markovian, and independent of the exposed and infectious periods. These periods have general distributions, and are not affected by the possible migrations of the individuals. The infection “from distance” aspect introduces a new formulation of the infection process, which, together with the migration processes, brings technical challenges in proving the functional limit theorems. Generalizing the methods in Pang and Pardoux [Ann. Appl. Probab. 32 (2022) 1615–1665], we establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.
Conditional Injective Flows for Bayesian Imaging
(IEEE, 2023) Khorashadizadeh, AmirEhsan; Kothari, Konik; Salsi, Leonardo; Harandi, Ali Aghababaei; de Hoop, Maarten; Dokmanić, Ivan
Most deep learning models for computational imaging regress a single reconstructed image. In practice, however, ill-posedness, nonlinearity, model mismatch, and noise often conspire to make such point estimates misleading or insufficient. The Bayesian approach models images and (noisy) measurements as jointly distributed random vectors and aims to approximate the posterior distribution of unknowns. Recent variational inference methods based on conditional normalizing flows are a promising alternative to traditional MCMC methods, but they come with drawbacks: excessive memory and compute demands for moderate to high resolution images and underwhelming performance on hard nonlinear problems. In this work, we propose C-Trumpets—conditional injective flows specifically designed for imaging problems, which greatly diminish these challenges. Injectivity reduces memory footprint and training time while low-dimensional latent space together with architectural innovations like fixed-volume-change layers and skip-connection revnet layers, C-Trumpets outperform regular conditional flow models on a variety of imaging and image restoration tasks, including limited-view CT and nonlinear inverse scattering, with a lower compute and memory budget. C-Trumpets enable fast approximation of point estimates like MMSE or MAP as well as physically-meaningful uncertainty quantification.
Optimized decision support for selection of transoral robotic surgery or (chemo)radiation therapy based on posttreatment swallowing toxicity
(Wiley, 2023) Hemmati, Mehdi; Barbon, Carly; Mohamed, Abdallah S.R.; van Dijk, Lisanne V.; Moreno, Amy C.; Gross, Neil D.; Goepfert, Ryan P.; Lai, Stephen Y.; Hutcheson, Katherine A.; Schaefer, Andrew J.; Fuller, Clifton D.
Background A primary goal in transoral robotic surgery (TORS) for oropharyngeal squamous cell cancer (OPSCC) survivors is to optimize swallowing function. However, the uncertainty in the outcomes of TORS including postoperative residual positive margin (PM) and extranodal extension (ENE), may necessitate adjuvant therapy, which may cause significant swallowing toxicity to survivors. Methods A secondary analysis was performed on a prospective registry data with low- to intermediate-risk human papillomavirus–related OPSCC possibly resectable by TORS. Decision trees were developed to model the uncertainties in TORS compared with definitive radiation therapy (RT) and chemoradiation therapy (CRT). Swallowing toxicities were measured by Dynamic Imaging Grade of Swallowing Toxicity (DIGEST), MD Anderson Dysphagia Inventory (MDADI), and the MD Anderson Symptom Inventory–Head and Neck (MDASI-HN) instruments. The likelihoods of PM/ENE were varied to determine the thresholds within which each therapy remains optimal. Results Compared with RT, TORS resulted in inferior swallowing function for moderate likelihoods of PM/ENE (>60% in short term for all instruments, >75% in long term for DIGEST and MDASI) leaving RT as the optimal treatment. Compared with CRT, TORS remained the optimal therapy based on MDADI and MDASI but showed inferior swallowing outcomes based on DIGEST for moderate-to-high likelihoods of PM/ENE (>75% for short-term and >40% for long-term outcomes). Conclusion In the absence of reliable estimation of postoperative PM/ENE concurrent with significant postoperative PM, the overall toxicity level in OPSCC patients undergoing TORS with adjuvant therapy may become more severe compared with patients receiving nonsurgical treatments thus advocating definitive (C)RT protocols.
On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows
(Frontiers Media S.A., 2022) Chan, Jesse; Ranocha, Hendrik; Rueda-Ramírez, Andrés M.; Gassner, Gregor; Warburton, Tim
High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.
MR-Guided Adaptive Radiotherapy for OAR Sparing in Head and Neck Cancers
(MDPI, 2022) Mulder, Samuel L.; Heukelom, Jolien; McDonald, Brigid A.; Van Dijk, Lisanne; Wahid, Kareem A.; Sanders, Keith; Salzillo, Travis C.; Hemmati, Mehdi; Schaefer, Andrew; Fuller, Clifton D.
MR-linac devices offer the potential for advancements in radiotherapy (RT) treatment of head and neck cancer (HNC) by using daily MR imaging performed at the time and setup of treatment delivery. This article aims to present a review of current adaptive RT (ART) methods on MR-Linac devices directed towards the sparing of organs at risk (OAR) and a view of future adaptive techniques seeking to improve the therapeutic ratio. This ratio expresses the relationship between the probability of tumor control and the probability of normal tissue damage and is thus an important conceptual metric of success in the sparing of OARs. Increasing spatial conformity of dose distributions to target volume and OARs is an initial step in achieving therapeutic improvements, followed by the use of imaging and clinical biomarkers to inform the clinical decision-making process in an ART paradigm. Pre-clinical and clinical findings support the incorporation of biomarkers into ART protocols and investment into further research to explore imaging biomarkers by taking advantage of the daily MR imaging workflow. A coherent understanding of this road map for RT in HNC is critical for directing future research efforts related to sparing OARs using image-guided radiotherapy (IGRT).
Chromosome size affects sequence divergence between species through the interplay of recombination and selection
(Wiley, 2022) Tigano, Anna; Khan, Ruqayya; Omer, Arina D.; Weisz, David; Dudchenko, Olga; Multani, Asha S.; Pathak, Sen; Behringer, Richard R.; Aiden, Erez L.; Fisher, Heidi; MacManes, Matthew D.; Center for Theoretical and Biological Physics
The structure of the genome shapes the distribution of genetic diversity and sequence divergence. To investigate how the relationship between chromosome size and recombination rate affects sequence divergence between species, we combined empirical analyses and evolutionary simulations. We estimated pairwise sequence divergence among 15 species from three different mammalian clades—Peromyscus rodents, Mus mice, and great apes—from chromosome-level genome assemblies. We found a strong significant negative correlation between chromosome size and sequence divergence in all species comparisons within the Peromyscus and great apes clades but not the Mus clade, suggesting that the dramatic chromosomal rearrangements among Mus species may have masked the ancestral genomic landscape of divergence in many comparisons. Our evolutionary simulations showed that the main factor determining differences in divergence among chromosomes of different sizes is the interplay of recombination rate and selection, with greater variation in larger populations than in smaller ones. In ancestral populations, shorter chromosomes harbor greater nucleotide diversity. As ancestral populations diverge, diversity present at the onset of the split contributes to greater sequence divergence in shorter chromosomes among daughter species. The combination of empirical data and evolutionary simulations revealed that chromosomal rearrangements, demography, and divergence times may also affect the relationship between chromosome size and divergence, thus deepening our understanding of the role of genome structure in the evolution of species divergence.

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