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Establishing the host range for novel viruses remains a challenge. Here, we address the challenge of identifying non-human animal coronaviruses that may infect humans by creating an artificial neural network model that learns from spike protein sequences of alpha and beta coronaviruses and their binding annotation to their host receptor. The proposed method produces a human-Binding Potential (h-BiP) score that distinguishes, with high accuracy, the binding potential among coronaviruses. Three viruses, previously unknown to bind human receptors, were identified: Bat coronavirus BtCoV/133/2005 and Pipistrellus abramus bat coronavirus HKU5-related (both MERS related viruses), and Rhinolophus affinis coronavirus isolate LYRa3 (a SARS related virus). We further analyze the binding properties of BtCoV/133/2005 and LYRa3 using molecular dynamics. To test whether this model can be used for surveillance of novel coronaviruses, we re-trained the model on a set that excludes SARS-CoV-2 and all viral sequences released after the SARS-CoV-2 was published. The results predict the binding of SARS-CoV-2 with a human receptor, indicating that machine learning methods are an excellent tool for the prediction of host expansion events.

Mice are the most commonly used model animals for itch research and for development of antiitch drugs. Most labs manually quantify mouse scratching behavior to assess itch intensity. This process is labor-intensive and limits large-scale genetic or drug screenings. In this study, we developed a new system, Scratch-AID (Automatic Itch Detection), which could automatically identify and quantify mouse scratching behavior with high accuracy. Our system included a custom-designed videotaping box to ensure high-quality and replicable mouse behavior recording and a convolutional recurrent neural network (CRNN) trained with frame-labeled mouse scratching behavior videos, induced by nape injection of chloroquine (CQ). The best trained network achieved 97.6% recall and 96.9% precision on previously unseen test videos. Remarkably, Scratch-AID could reliably identify scratching behavior in other major mouse itch models, including the acute cheek model, the histaminergic model, and a chronic itch model. Moreover, our system detected significant differences in scratching behavior between control and mice treated with an anti-itch drug. Taken together, we have established a novel deep learning-based system that is ready to replace manual quantification for mouse scratching behavior in different itch models and for drug screening.

We study equilibrium configurations of double-stranded DNA in a cylindrical viral capsid. The state of the encapsidated DNA consists of a disordered inner core enclosed by an ordered outer region, next to the capsid wall. The DNA configuration is described by a unit helical vector field, tangent to an associated centre curve, passing through properly selected locations. We postulate an expression for the energy of the encapsulated DNA based on that of columnar chromonic liquid crystals. A thorough analysis of the Euler–Lagrange equations yields multiple solutions. We demonstrate that there is a trivial, non-helical solution, together with two solutions with non-zero helicity of opposite sign. Using bifurcation analysis, we derive the conditions for local stability and determine when the preferred coiling state is helical. The bifurcation parameters are the ratio of the twist versus the bend moduli of DNA and the ratio between the sizes of the ordered and the disordered regions.

Molecular motors that translocate DNA are ubiquitous in nature. During morphogenesis of double-stranded DNA bacteriophages, a molecular motor drives the viral genome inside a protein capsid. Several models have been proposed for the three-dimensional geometry of the packaged genome, but very little is known of the signature of the molecular packaging motor. For instance, biophysical experiments show that in some systems, DNA rotates during the packaging reaction, but most current biophysical models fail to incorporate this property. Furthermore, studies including rotation mechanisms have reached contradictory conclusions. In this study, we compare the geometrical signatures imposed by different possible mechanisms for the packaging motors: rotation, revolution, and rotation with revolution. We used a previously proposed kinetic Monte Carlo model of the motor, combined with Brownian dynamics simulations of DNA to simulate deterministic and stochastic motor models. We find that rotation is necessary for the accumulation of DNA writhe and for the chiral organization of the genome. We observe that although in the initial steps of the packaging reaction, the torsional strain of the genome is released by rotation of the molecule, in the later stages, it is released by the accumulation of writhe. We suggest that the molecular motor plays a key role in determining the final structure of the encapsidated genome in bacteriophages.

The three dimensional organization of genomes remains mostly unknown due to their high degree of condensation. Biophysical studies predict that condensation promotes the topological entanglement of chromatin fibers and the inhibition of function. How organisms balance between functionally active genomes and a high degree of condensation remains to be determined. Here we hypothesize that the Rabl configuration, characterized by the attachment of centromeres and telomeres to the nuclear envelope, helps to reduce the topological entanglement of chromosomes. To test this hypothesis we developed a novel method to quantify chromosome entanglement complexity in 3D reconstructions obtained from Chromosome Conformation Capture (CCC) data. Applying this method to published data of the yeast genome, we show that computational models implementing the attachment of telomeres or centromeres alone are not sufficient to obtain the reduced entanglement complexity observed in 3D reconstructions. It is only when the centromeres and telomeres are attached to the nuclear envelope (i.e. the Rabl configuration) that the complexity of entanglement of the genome is comparable to that of the 3D reconstructions. We therefore suggest that the Rabl configuration is an essential player in the simplification of the entanglement of chromatin fibers.

We study equilibrium configurations of hexagonal columnar liquid crystals in the context of characterizing packing structures of bacteriophage viruses in a protein capsid. These are viruses that infect bacteria and are currently the focus of intense research efforts, with the goal of finding new therapies for bacteria-resistant antibiotics. The energy that we propose consists of the Oseen–Frank free energy of nematic liquid crystals that penalizes bending of the columnar directions, in addition to the cross-sectional elastic energy accounting for distortions of the transverse hexagonal structure; we also consider the isotropic contribution of the core and the energy of the unknown interface between the outer ordered region of the capsid and the inner disordered core. The problem becomes of free boundary type, with constraints. We show that the concentric, azimuthal, spool-like configuration is the absolute minimizer. Moreover, we present examples of toroidal structures formed by DNA in free solution and compare them with the analogous ones occurring in experiments with other types of lyotropic liquid crystals, such as food dyes and additives.This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

Unraveling the mechanisms of packing of DNA inside viral capsids is of fundamental importance to understanding the spread of viruses. It could also help develop new applications to targeted drug delivery devices for a large range of therapies. In this article, we present a robust, predictive mathematical model and its numerical implementation to aid the study and design of bacteriophage viruses for application purposes. Exploiting the analogies between the columnar hexagonal chromonic phases of encapsidated viral DNA and chromonic aggregates formed by plank-shaped molecular compounds, we develop a first-principles effective mechanical model of DNA packing in a viral capsid. The proposed expression of the packing energy, which combines relevant aspects of the liquid crystal theory, is developed from the model of hexagonal columnar phases, together with that describing configurations of polymeric liquid crystals. The method also outlines a parameter selection strategy that uses available data for a collection of viruses, aimed at applications to viral design. The outcome of the work is a mathematical model and its numerical algorithm, based on the method of finite elements, and computer simulations to identify and label the ordered and disordered regions of the capsid and calculate the inner pressure. It also presents the tools for the local reconstruction of the DNA "scaffolding" and the center curve of the filament within the capsid.

Copy Number Aberrations, gains and losses of genomic regions, are a hallmark of cancer and can be experimentally detected using microarray comparative genomic hybridization (aCGH). In previous works, we developed a topology based method to analyze aCGH data whose output are regions of the genome where copy number is altered in patients with a predetermined cancer phenotype. We call this method Topological Analysis of array CGH (TAaCGH). Here we combine TAaCGH with machine learning techniques to build classifiers using copy number aberrations. We chose logistic regression on two different binary phenotypes related to breast cancer to illustrate this approach. The first case consists of patients with over-expression of the ERBB2 gene. Over-expression of ERBB2 is commonly regulated by a copy number gain in chromosome arm 17q. TAaCGH found the region 17q11-q22 associated with the phenotype and using logistic regression we reduced this region to 17q12-q21.31 correctly classifying 78% of the ERBB2 positive individuals (sensitivity) in a validation data set. We also analyzed over-expression in Estrogen Receptor (ER), a second phenotype commonly observed in breast cancer patients and found that the region 5p14.3-12 together with six full arms were associated with the phenotype. Our method identified 4p, 6p and 16q as the strongest predictors correctly classifying 76% of ER positives in our validation data set. However, for this set there was a significant increase in the false positive rate (specificity). We suggest that topological and machine learning methods can be combined for prediction of phenotypes using genetic data.

Bacteriophages densely pack their long dsDNA genome inside a protein capsid. The conformation of the viral genome inside the capsid is consistent with a hexagonal liquid crystalline structure. Experiments have confirmed that the details of the hexagonal packing depend on the electrochemistry of the capsid and its environment. In this work, we propose a biophysical model that quantifies the relationship between DNA configurations inside bacteriophage capsids and the types and concentrations of ions present in a biological system. We introduce an expression for the free energy which combines the electrostatic energy with contributions from bending of individual segments of DNA and Lennard-Jones-type interactions between these segments. The equilibrium points of this energy solve a partial differential equation that defines the distributions of DNA and the ions inside the capsid. We develop a computational approach that allows us to simulate much larger systems than what is currently possible using the existing simulations, typically done at a molecular level. In particular, we are able to estimate bending and repulsion between DNA segments as well as the full electrochemistry of the solution, both inside and outside of the capsid. The numerical results show good agreement with existing experiments and molecular dynamics simulations for small capsids.

It is widely recognized that the three-dimensional (3D) architecture of eukaryotic chromatin plays an important role in processes such as gene regulation and cancer-driving gene fusions. Observing or inferring this 3D structure at even modest resolutions had been problematic, since genomes are highly condensed and traditional assays are coarse. However, recently devised high-throughput molecular techniques have changed this situation. Notably, the development of a suite of chromatin conformation capture (CCC) assays has enabled elicitation of contacts - spatially close chromosomal loci - which have provided insights into chromatin architecture. Most analysis of CCC data has focused on the contact level, with less effort directed toward obtaining 3D reconstructions and evaluating the accuracy and reproducibility thereof. While questions of accuracy must be addressed experimentally, questions of reproducibility can be addressed statistically - the purpose of this paper. We use a constrained optimization technique to reconstruct chromatin configurations for a number of closely related yeast datasets and assess reproducibility using four metrics that measure the distance between 3D configurations. The first of these, Procrustes fitting, measures configuration closeness after applying reflection, rotation, translation, and scaling-based alignment of the structures. The others base comparisons on the within-configuration inter-point distance matrix. Inferential results for these metrics rely on suitable permutation approaches. Results indicate that distance matrix-based approaches are preferable to Procrustes analysis, not because of the metrics per se but rather on account of the ability to customize permutation schemes to handle within-chromosome contiguity. It has recently been emphasized that the use of constrained optimization approaches to 3D architecture reconstruction are prone to being trapped in local minima. Our methods of reproducibility assessment provide a means for comparing 3D reconstruction solutions so that we can discern between local and global optima by contrasting solutions under perturbed inputs. © 2014 The Author.

Trypanosomatida parasites, such as Trypanosoma and Leishmania, are the cause of deadly diseases in many third world countries. The three dimensional structure of their mitochondrial DNA, known as kinetoplast DNA (kDNA), is unique since it is organized into several thousands of minicircles that are topologically linked. How and why the minicircles form such a network have remained unanswered questions. In our previous work we have presented a model of network formation that hypothesizes that the network is solely driven by the confinement of minicircles. Our model shows that upon confinement a percolation network forms. This network grows into a space filling network, called saturation network, upon further confinement of minicircles. Our model also shows, in agreement with experimental data, that the mean valence of the network (that is, the average number of minicircles topologically linked to any minicircle in the network) grows linearly with minicircle density. In our previous studies however we disregarded DNA flexibility and used rigid minicircles to model DNA, here we address this limitation by allowing minicircles to be flexible. Our numerical results show that the topological characteristics that describe the growth and topology of the minicircle networks have similar values to those observed in the case of rigid minicircles suggesting that these properties are robust and therefore a potentially adequate description of the networks observed in Trypanosomatid parasites.

The mitochondrial DNA of trypanosomes, called kinetoplast DNA (kDNA) contains thousands of minicircles that are topologically linked into a single structure that resembles a medieval chainmail. This biological chainmail is characterized by two parameters: the link type between minicircles, and the number of minicircles linked to each minicircle (i.e. the minicircle valence). In previous works, a protocol was proposed to determine the mean value of the minicircle valence. In these experiments, minicircles were excised from the network and the products compared with those obtained from fragmenting idealized structures. These idealized structures assumed a negligible variance in the distribution of valences of the initial network. It is therefore unclear to what extent this theoretical analysis captures the true topology of the kDNA network when kDNA samples are extracted from unsynchronized cells or from cells with silenced kDNA replication genes. Subsequent studies proposed that there is a critical percolation density during network formation. We asked whether this density can be estimated using fragmentation reactions. The goal of this work is to develop a mathematical method that can be used to estimate the mean valence of networks when the variance of the valence is non-negligible. We first show microscopy data on Crithidia fasciculata that, in agreement with the original experimental results, show a distribution of valences with nonzero variance. Second, we use computer simulations of network fragmentation to show that the predicted and actual mean valence are different when the valence distribution has nonzero variance. We propose a more general mathematical formulation and computer simulations of kDNA fragmentation to estimate this value. Last, we show that fragmentation experiments may lead to errors in the estimation of the critical percolation density since the collapsing density depends on the initial density of the network, and on the fragmentation reaction.

Genomic technologies measure thousands of molecular signals with the goal of understanding complex biological processes. In cancer these molecular signals have been used to characterize disease subtypes, signaling pathways and to identify subsets of patients with specific prognosis. However molecular signals for any disease type are so vast and complex that novel mathematical approaches are required for further analyses. Persistent and computational homology provide a new method for these analyses. In our previous work we presented a new homology-based supervised classification method to identify copy number aberrations from comparative genomic hybridization arrays. In this work we first propose a theoretical framework for our classification method and second we extend our analysis to gene expression data. We analyze a published breast cancer data set and find that that our method can distinguish most, but not all, different breast cancer subtypes. This result suggests that specific relationships between genes, captured by our algorithm, help distinguish between breast cancer subtypes. We propose that topological methods can be used for the classification and clustering of gene expression profiles.

Networks of topologically linked minicircle polymers are found in diverse natural systems and are a subject of intense research in nanotechonology. In a recent report the authors introduced a new theoretical model to study the effects of polymer density on the formation and on the topological properties of minicircle networks. Three key topological characteristics were identified in the formation and characterization of a network: the critical percolation density, the average saturation density and the mean valence of the network. In this work we report how these characteristics change when an orientation bias is imposed on the minicircles forming the network. We observe that such restrictions have significant effects on the key topological characteristics of the network. In particular while the effects of restriction of the tilting angle can be predicted we find that those of the azimuthal angle can have somewhat unexpected results.

The experimental method comparative genomic hybridization (CGH) array provides a full picture of chromosome amplifications and deletions of an individual's genome. The multi-dimensional nature of these data, however, has not been fully explored. Current methods for CGH analysis focus on specific markers, but deeper understanding can be gained by examining the properties of the overall genomic profile. In this paper we propose a novel method for characterizing CGH profiles mathematically using algebraic homology. We apply our method to breast cancer CGH profiles from patients that have been subjected to different treatments. We show that our results obtained using homology are in agreement with previous studies and are able to distinguish between frequency of cancer recurrence in chemotherapy and non-treated patient populations.

A number of idealized models have been proposed to explain the long range organization of the DNA in bacteriophages. However, none of these models can account for the distributions of complex knots found when examining DNA extracted from bacteriophage P4 capsids. Furthermore, these models do not consider possible chirality biases in the arrangement of the DNA molecule inside the capsid. In this paper, we address these two issues by proposing a randomized version of one of the most popular models: the coaxially spooled model. We present analytical and numerical results for the properties of the random polygons (knots) generated using this model. We show that such model can easily generate complex knotted conformations and although it accounts for some chirality of the organization of the DNA molecules inside bacteriophage capsids does not fully explain the experimental data.

DNA knots formed under extreme conditions of condensation, as in bacteriophage P4, are difficult to analyze experimentally and theoretically. In this paper, we propose to use the uniform random polygon model as a supplementary method to the existing methods for generating random knots in confinement. The uniform random polygon model allows us to sample knots with large crossing numbers and also to generate large diagrammatically prime knot diagrams. We show numerically that uniform random polygons sample knots with large minimum crossing numbers and certain complicated knot invariants (as those observed experimentally). We do this in terms of the knot determinants or colorings. Our numerical results suggest that the average determinant of a uniform random polygon of n vertices grows faster than . We also investigate the complexity of prime knot diagrams. We show rigorously that the probability that a randomly selected 2D uniform random polygon of n vertices is almost diagrammatically prime goes to 1 as n goes to infinity. Furthermore, the average number of crossings in such a diagram is at the order of O(n2). Therefore, the two-dimensional uniform random polygons offer an effective way in sampling large (prime) knots, which can be useful in various applications.

Cancer progression is due to the accumulation of recurrent genomic alterations that induce growth advantage and clonal expansion. Most of these genomic changes can be detected using the array comparative genomic hybridization (CGH) technique. The accurate classification of these genomic alterations is expected to have an important impact on translational and basic research. Here we review recent advances in CGH technology used in the characterization of different features of breast cancer. First, we present bioinformatics methods that have been developed for the analysis of CGH arrays; next, we discuss the use of array CGH technology to classify tumor stages and to identify and stratify subgroups of patients with different prognoses and clinical behaviors. We finish our review with a discussion of how CGH arrays are being used to identify oncogenes, tumor suppressor genes, and breast cancer susceptibility genes.

In this paper, we study the topological entanglement of uniform random polygons in a confined space. We derive the formula for the mean squared linking number of such polygons. For a fixed simple closed curve in the confined space, we rigorously show that the linking probability between this curve and a uniform random polygon of n vertices is at least . Our numerical study also indicates that the linking probability between two uniform random polygons (in a confined space), of m and n vertices respectively, is bounded below by . In particular, the linking probability between two uniform random polygons, both of n vertices, is bounded below by .

DNA copy number aberrations (CNAs) are of biological and medical interest because they help identify regulatory mechanisms underlying tumor initiation and evolution. Identification of tumor-driving CNAs (driver CNAs) however remains a challenging task, because they are frequently hidden by CNAs that are the product of random events that take place during tumor evolution. Experimental detection of CNAs is commonly accomplished through array comparative genomic hybridization (aCGH) assays followed by supervised and/or unsupervised statistical methods that combine the segmented profiles of all patients to identify driver CNAs. Here, we extend a previously-presented supervised algorithm for the identification of CNAs that is based on a topological representation of the data. Our method associates a two-dimensional (2D) point cloud with each aCGH profile and generates a sequence of simplicial complexes, mathematical objects that generalize the concept of a graph. This representation of the data permits segmenting the data at different resolutions and identifying CNAs by interrogating the topological properties of these simplicial complexes. We tested our approach on a published dataset with the goal of identifying specific breast cancer CNAs associated with specific molecular subtypes. Identification of CNAs associated with each subtype was performed by analyzing each subtype separately from the others and by taking the rest of the subtypes as the control. Our results found a new amplification in 11q at the location of the progesterone receptor in the Luminal A subtype. Aberrations in the Luminal B subtype were found only upon removal of the basal-like subtype from the control set. Under those conditions, all regions found in the original publication, except for 17q, were confirmed, all aberrations, except those in chromosome arms 8q and 12q were confirmed in the basal-like subtype. These two chromosome arms, however, were detected only upon removal of three patients with exceedingly large copy number values. More importantly, we detected 10 and 21 additional regions in the Luminal B and basal-like subtypes, respectively. Most of the additional regions were either validated on an independent dataset and/or using GISTIC. Furthermore, we found three new CNAs in the basal-like subtype: a combination of gains and losses in 1p, a gain in 2p and a loss in 14q. Based on these results, we suggest that topological approaches that incorporate multiresolution analyses and that interrogate topological properties of the data can help in the identification of copy number changes in cancer.

We here address two problems concerning the writhe of random polygons. First, we study the behavior of the mean writhe as a function length. Second, we study the variance of the writhe. Suppose that we are dealing with a set of random polygons with the same length and knot type, which could be the model of some circular DNA with the same topological property. In general, a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons; if the mean writhe is non-zero then the knot is chiral. How accurate is this method? For example, if for a specific knot type K the mean writhe decreased to zero as the length of the polygons increased, then this method would be limited in the case of long polygons. Furthermore, we conjecture that the sign of the mean writhe is a topological invariant of chiral knots. This sign appears to be the same as that of an 'ideal' conformation of the knot. We provide numerical evidence to support these claims, and we propose a new nomenclature of knots based on the sign of their expected writhes. This nomenclature can be of particular interest to applied scientists. The second part of our study focuses on the variance of the writhe, a problem that has not received much attention in the past. In this case, we focused on the equilateral random polygons. We give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons (of length n) behaves as a linear function of the length of the equilateral random polygon.

The circular chromosome of Escherichia coli is organized into independently supercoiled loops, or topological domains. We investigated the organization and size of these domains in vivo and in vitro. Using the expression of >300 supercoiling-sensitive genes to gauge local chromosomal supercoiling, we quantitatively measured the spread of relaxation from double-strand breaks generated in vivo and thereby calculated the distance to the nearest domain boundary. In a complementary approach, we gently isolated chromosomes and examined the lengths of individual supercoiled loops by electron microscopy. The results from these two very different methods agree remarkably well. By comparing our results to Monte Carlo simulations of domain organization models, we conclude that domain barriers are not placed stably at fixed sites on the chromosome but instead are effectively randomly distributed. We find that domains are much smaller than previously reported, ∼10 kb on average. We discuss the implications of these findings and present models for how domain barriers may be generated and displaced during the cell cycle in a stochastic fashion.

Analysing chromosome aberrations induced by low linear energy transfer (LET) radiation in order to characterize systematic spatial clustering among the 22 human autosomes in human lymphocytes and to compare their relative participation in interchanges.A multicolour fluorescence in situ hybridization (mFISH) data set, specifying colour junctions in metaphases of human peripheral blood lymphocytes 72 h after in vitro exposure to low LET radiation, was analysed separately and in combination with previously published results. Monte Carlo computer simulations and mathematical modelling guided data analysis.Statistical tests on aberration data confirmed two clusters of chromosomes, [1, 16, 17, 19, 22] and [13, 14, 15, 21, 22], as having their members being on average closer to each other than randomness would predict. The first set has been reported previously to be near the centre of the interphase nucleus and to be formed mainly by gene-rich chromosomes, while the second set comprises the nucleolus chromosomes. The results suggest a possible interplay between chromosome positioning and transcription. A number of other clusters suggested in the literature were not confirmed and considerable randomness of chromosome-chromosome juxtapositions was present. In addition, and consistent with previous results, it was found that chromosome participation in interchanges is approximately proportional to the two-thirds power of the DNA content.

A comprehensive description of chromosome aberrations is introduced that is suitable for all cytogenetic protocols (e.g. solid staining, banding, FISH, mFISH, SKY, bar coding) and for mathematical analyses. "Aberration multigraphs" systematically characterize and interrelate three basic aberration elements: (1) the initial configuration of chromosome breaks; (2) the exchange process, whose cycle structure helps to describe aberration complexity; and (3) the final configuration of rearranged chromosomes, which determines the observed pattern but may contain cryptic misrejoinings in addition. New aberration classification methods and a far-reaching generalization of mPAINT descriptors, applicable to any protocol, emerge. The difficult problem of trying to infer actual exchange processes from cytogenetically observed final patterns is analyzed using computer algorithms, adaptations of known theorems on cubic graphs, and some new graph-theoretical constructs. Results include the following: (1) For a painting protocol, unambiguously inferring the occurrence of a high-order cycle requires a corresponding number of different colors; (2) cycle structure can be computed by a simple trick directly from mPAINT descriptors if the initial configuration has no more than one break per homologue pair; and (3) higher-order cycles are more frequent than the obligate cycle structure specifies. Aberration multigraphs are a powerful new way to describe, classify and quantitatively analyze radiation-induced chromosome aberrations. They pinpoint (but do not eliminate) the problem that, with present cytogenetic techniques, one observed pattern corresponds to many possible initial configurations and exchange processes.

To test quantitatively whether there are systematic chromosome–chromosome associations within human interphase nuclei, interchanges between all possible heterologous pairs of chromosomes were measured with 24-color whole-chromosome painting (multiplex FISH), after damage to interphase lymphocytes by sparsely ionizing radiation in vitro. An excess of interchanges for a specific chromosome pair would indicate spatial proximity between the chromosomes comprising that pair. The experimental design was such that quite small deviations from randomness (extra pairwise interchanges within a group of chromosomes) would be detectable. The only statistically significant chromosome cluster was a group of five chromosomes previously observed to be preferentially located near the center of the nucleus. However, quantitatively, the overall deviation from randomness within the whole genome was small. Thus, whereas some chromosome–chromosome associations are clearly present, at the whole-chromosomal level, the predominant overall pattern appears to be spatially random.

Exchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments rejoin with fragments from other chromosomes or from other regions within the same chromosome. ETCAs are commonly observed in cancer cells and in cells exposed to radiation. The frequency of these chromosome rearrangements is correlated with their spatial proximity, therefore it can be used to infer the three dimensional organization of the genome. Extracting statistical significance of spatial proximity from cancer and radiation data has remained somewhat elusive because of the sparsity of the data. We here propose a new approach to study the three dimensional organization of the genome using algebraic statistics. We test our method on a published data set of irradiated human blood lymphocyte cells. We provide a rigorous method for testing the overall organization of the genome, and in agreement with previous results we find a random relative positioning of chromosomes with the exception of the chromosome pairs {1,22} and {13,14} that have a significantly larger number of ETCAs than the rest of the chromosome pairs suggesting their spatial proximity. We conclude that algebraic methods can successfully be used to analyze genetic data and have potential applications to larger and more complex data sets.

Trypanosomatida parasites, such as trypanosoma and lishmania, are the cause of deadly diseases in many third world countries. A distinctive feature of these organisms is the three dimensional organization of their mitochondrial DNA into maxi and minicircles. In some of these organisms minicircles are confined into a small disk volume and are topologically linked, forming a gigantic linked network. The origins of such a network as well as of its topological properties are mostly unknown. In this paper we quantify the effects of the confinement on the topology of such a minicircle network. We introduce a simple mathematical model in which a collection of randomly oriented minicircles are spread over a rectangular grid. We present analytical and computational results showing that a finite positive critical percolation density exists, that the probability of formation of a highly linked network increases exponentially fast when minicircles are confined, and that the mean minicircle valence (the number of minicircles that a particular minicircle is linked to) increases linearly with density. When these results are interpreted in the context of the mitochondrial DNA of the trypanosome they suggest that confinement plays a key role on the formation of the linked network. This hypothesis is supported by the agreement of our simulations with experimental results that show that the valence grows linearly with density. Our model predicts the existence of a percolation density and that the distribution of minicircle valences is more heterogeneous than initially thought.

During the early phase of the cell cycle the eukaryotic genome is organized into chromosome territories. The geometry of the interface between any two chromosomes remains a matter of debate and may have important functional consequences. The Interchromosomal Network model (introduced by Branco and Pombo) proposes that territories intermingle along their periphery. In order to partially quantify this concept we here investigate the probability that two chromosomes form an unsplittable link. We use the uniform random polygon as a crude model for chromosome territories and we model the interchromosomal network as the common spatial region of two overlapping uniform random polygons. This simple model allows us to derive some rigorous mathematical results as well as to perform computer simulations easily. We find that the probability that one uniform random polygon of length n that partially overlaps a fixed polygon is bounded below by $${1-O(\frac{1}{\sqrt n})}$$ . We use numerical simulations to estimate the dependence of the linking probability of two uniform random polygons (of lengths n and m, respectively) on the amount of overlapping. The degree of overlapping is parametrized by a parameter $${\epsilon\in [0,1]}$$ such that $${\epsilon=0}$$ indicates no overlapping and $${\epsilon=1}$$ indicates total overlapping. We propose that this dependence relation may be modeled as $${f(\varepsilon, m, n) =1-{\frac{a(\epsilon)}{b(\epsilon)\sqrt{mn}+c(\epsilon)}}}$$ . Numerical evidence shows that this model works well when $${\epsilon}$$ is relatively large $${(\varepsilon \ge 0.5)}$$ . We then use these results to model the data published by Branco and Pombo and observe that for the amount of overlapping observed experimentally the URPs have a non-zero probability of forming an unsplittable link.

Type II topoisomerases are enzymes that change the topology of DNA by performing strand-passage. In particular, they unknot knotted DNA very efficiently. Motivated by this experimental observation, we investigate transition probabilities between knots. We use the BFACF algorithm to generate ensembles of polygons in Z3 of fixed knot type. We introduce a novel strand-passage algorithm which generates a Markov chain in knot space. The entries of the corresponding transition probability matrix determine state-transitions in knot space and can track the evolution of different knots after repeated strand-passage events. We outline future applications of this work to DNA unknotting.

In recent years, knot theory and low-dimensional topology have been effectively used to study the topology and geometry of DNA under different spatial constraints, and to solve the topological mechanisms of enzymes such as site-specific recombinases and topoisomerases. Through continuous collaboration and close interaction with experimental biologists, many problems approached and the solutions proposed remain relevant to the biological community, while being mathematically and computationally interesting. In this paper, we illustrate the use of mathematical and computational methods in a variety of DNA topology problems. This is by no means an exhaustive description of techniques and applications, but is rather intended to introduce the reader to the exciting applications of topology to the study of DNA. Many more examples will be found throughout this book.

Kinetoplast DNA (kDNA) is the mitochondrial of DNA of disease causing organisms such as Trypanosoma Brucei (T. Brucei) and Trypanosoma Cruzi (T. Cruzi). In most organisms, KDNA is made of thousands of small circular DNA molecules that are highly condensed and topologically linked forming a gigantic planar network. In our previous work we have developed mathematical and computational models to test the confinement hypothesis, that is that the formation of kDNA minicircle networks is a product of the high DNA condensation achieved in the mitochondrion of these organisms. In these studies we studied three parameters that characterize the growth of the network topology upon confinement: the critical percolation density, the mean saturation density and the mean valence (i.e. the number of mini circles topologically linked to any chosen minicircle). Experimental results on insect-infecting organisms showed that the mean valence is equal to three, forming a structure similar to those found in medieval chain-mails. These same studies hypothesized that this value of the mean valence was driven by the DNA excluded volume. Here we extend our previous work on kDNA by characterizing the effects of DNA excluded volume on the three descriptive parameters. Using computer simulations of polymer swelling we found that (1) in agreement with previous studies the linking probability of two minicircles does not decrease linearly with the distance between the two minicircles, (2) the mean valence grows linearly with the density of minicircles and decreases with the thickness of the excluded volume, (3) the critical percolation and mean saturation densities grow linearly with the thickness of the excluded volume. Our results therefore suggest that the swelling of the DNA molecule, due to electrostatic interactions, has relatively mild implications on the overall topology of the network. Our results also validate our topological descriptors since they appear to reflect the changes in the physical properties of the polymeric chains and at the same time remain faithful to their description of kDNA.

© 2015 Diao et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Kinetoplast DNA (kDNA), a unique mitochondrial structure common to trypanosomatid parasites, contains thousands of DNA minicircles that are densely packed and can be topologically linked into a chain mail-like network. Experimental data indicate that every minicircle in the network is, on average, singly linked to three other minicircles (i.e., has mean valence 3) before replication and to six minicircles in the late stages of replication. The biophysical factors that determine the topology of the network and its changes during the cell cycle remain unknown. Using a mathematical modeling approach, we previously showed that volume confinement alone can drive the formation of the network and that it induces a linear relationship between mean valence and minicircle density. Our modeling also predicted a minicircle valence two orders of magnitude greater than that observed in kDNA. To determine the factors that contribute to this discrepancy we systematically analyzed the relationship between the topological properties of the network (i.e., minicircle density and mean valence) and its biophysical properties such as DNA bending, electrostatic repulsion, and minicircle relative position and orientation. Significantly, our results showed that most of the discrepancy between the theoretical and experimental observations can be accounted for by the orientation of the minicircles with volume exclusion due to electrostatic interactions and DNA bending playing smaller roles. Our results are in agreement with the three dimensional kDNA organization model, initially proposed by Delain and Riou, in which minicircles are oriented almost perpendicular to the horizontal plane of the kDNA disk. We suggest that while minicircle confinement drives the formation of kDNA networks, it is minicircle orientation that regulates the topological complexity of the network.

5 pages, 4 figures.-- PMID: 15958528 [PubMed].-- PMCID: PMC1166588., Icosahedral bacteriophages pack their double-stranded DNA genomes to near-crystalline density and achieve one of the highest levels of DNA condensation found in nature. Despite numerous studies, some essential properties of the packaging geometry of the DNA inside the phage capsid are still unknown. We present a different approach to the problems of randomness and chirality of the packed DNA. We recently showed that most DNA molecules extracted from bacteriophage P4 are highly knotted because of the cyclization of the linear DNA molecule confined in the phage capsid. Here, we show that these knots provide information about the global arrangement of the DNA inside the capsid. First, we analyze the distribution of the viral DNA knots by high-resolution gel electrophoresis. Next, we perform Monte Carlo computer simulations of random knotting for freely jointed polygons confined to spherical volumes. Comparison of the knot distributions obtained by both techniques produces a topological proof of nonrandom packaging of the viral DNA. Moreover, our simulations show that the scarcity of the achiral knot 41 and the predominance of the torus knot 51 over the twist knot 52 observed in the viral distribution of DNA knots cannot be obtained by confinement alone but must include writhe bias in the conformation sampling. These results indicate that the packaging geometry of the DNA inside the viral capsid is writhe-directed., This work was supported by grants from the Ministerio Español de Ciencia y Tecnología (BMC2002-03275), the National Institutes of Health (GM68423-01), and the National Science Foundation (DMS9971169) and by a Burroughs Wellcome Fund Interfaces grant to the Program in Mathematics and Molecular Biology.

Chromosome aberrations are large-scale illegitimate rearrangements of the genome. They are indicative of DNA damage and informative about damage processing pathways. Despite extensive investigations over many years, the mechanisms underlying aberration formation remain controversial. New experimental assays such as multiplex fluorescent in situ hybridyzation (mFISH) allow combinatorial "painting" of chromosomes and are promising for elucidating aberration formation mechanisms. Recently observed mFISH aberration patterns are so complex that computer and graph-theoretical methods are needed for their full analysis. An important part of the analysis is decomposing a chromosome rearrangement process into "cycles." A cycle of order n, characterized formally by the cyclic graph with 2n vertices, indicates that n chromatin breaks take part in a single irreducible reaction. We here describe algorithms for computing cycle structures from experimentally observed or computer-simulated mFISH aberration patterns. We show that analyzing cycles quantitatively can distinguish between different aberration formation mechanisms. In particular, we show that homology-based mechanisms do not generate the large number of complex aberrations, involving higher-order cycles, observed in irradiated human lymphocytes.

A simple molecular mechanics model has been used to investigate optimal spool-like packing conformations of double-stranded DNA molecules in viral capsids with icosahedral symmetry. The model represents an elastic segmented chain by using one pseudoatom for each ten basepairs (roughly one turn of the DNA double helix). Force constants for the various terms in the energy function were chosen to approximate known physical properties, and a radial restraint was used to confine the DNA into a sphere with a volume corresponding to that of a typical bacteriophage capsid. When the DNA fills 90% of the spherical volume, optimal packaging is obtained for coaxially spooled models, but this result does not hold when the void volume is larger. When only 60% of the spherical volume is filled with DNA, the lowest energy structure has two layers, with a coiled core packed at an angle to an outer coaxially spooled shell. This relieves bending strain associated with tight curvature near the poles in a model with 100% coaxial spooling. Interestingly, the supercoiling density of these models is very similar to typical values observed in plasmids in bacterial cells. Potential applications of the methodology are also discussed.

When linear double-stranded DNA is packed inside bacteriophage capsids, it becomes highly compacted. However, the phage is believed to be fully effective only if the DNA is not entangled. Nevertheless, when DNA is extracted from a tailless mutant of the P4 phage, DNA is found to be cyclic and knotted (probability of 0.95). The knot spectrum is very complex, and most of the knots have a large number of crossings. We quantified the frequency and crossing numbers of these knots and concluded that, for the P4 tailless mutant, at least half the knotted molecules are formed while the DNA is still inside the viral capsid rather than during extraction. To analyze the origin of the knots formed inside the capsid, we compared our experimental results to Monte Carlo simulations of random knotting of equilateral polygons in confined volumes. These simulations showed that confinement of closed chains to tightly restricted volumes results in high knotting probabilities and the formation of knots with large crossing numbers. We conclude that the formation of the knots inside the viral capsid is driven mainly by the effects of confinement., J.A. and M.V. were supported by the Program in Mathematics and Molecular Biology through a Burroughs Wellcome Fund Interfaces Grant and by National Science Foundation Grant DMS-9971169. M.V. was supported by a Dirección General de Asuntos del Personal Académico-Universidad Nacional Autónoma de México graduate fellowship. This research was supported also by Ministry of Science of Spain Grant PB98-0487

Aim The success of kidney transplantation relies heavily upon the compatibility of the HLA genes between donor and recipient, but the high degree of polymorphism among these genes in the population makes finding an ideal match difficult. However, it is unclear whether specific amino acids are particularly critical in determining histocompatibility. To investigate this possibility, we introduce a novel method of comparing donor and recipient genotypes and explore its efficacy in predicting transplant rejection. Methods Using typing data of the HLA-A, -B, -C, and -DRB1 genes for 580 paired donors and recipients of a kidney transplant, we developed a method of allele estimation that used population frequency information to estimate amino acid sequence information. Using this sequence data we tested a number of metrics for measuring differences between donor and recipient genotypes, and looked for associations between mismatched residues and chronic rejection. Results Preliminary analysis of a cohort of 348 transplants with Caucasian recipients indicate a high association between residue mismatches in HLA-B and chronic rejection, primarily in residues 69 and 71 with p α p α Conclusions Association between HLA residue mismatches and chronic rejection was observed in cases where the residues’ properties differed significantly, and when the value of the residue is linked to broader motif changes. Future investigations will consider the impact of functional changes in residues in addition to statistical characterization of these predictive sites, and how these factors relate to transplant compatibility assessment.

4 pages, 3 figures.-- PMID: 11433043 [PubMed].-- PMCID: PMC55791., We describe a two-dimensional agarose gel electrophoresis procedure that improves the resolution of knotted DNA molecules. The first gel dimension is run at low voltage, and DNA knots migrate according to their compactness. The second gel dimension is run at high voltage, and DNA knots migrate according to other physical parameters such as shape and flexibility. In comparison with one-dimensional gel electrophoresis, this procedure segregates the knotted DNA molecules from other unknotted forms of DNA, and partially resolves populations of knots that have the same number of crossings. The two-dimensional display may allow quantitative and qualitative characterization of different types of DNA knots simply by gel velocity., This work was supported by grant PB98-0487 of the Ministry of Science of Spain (to J.R.). S.T. was the recipient of a predoctoral fellowship of the Ministry of Science of Spain. J.A., M.E.V. and D.W.S. were supported by the Program in Mathematics and Molecular Biology (PMMB) through a Burroughs Wellcome Fund Interfaces Grant

Motivation: The position of chromosomes in the interphase nucleus is believed to be associated with a number of biological processes. Here, we present a web-based application that helps analyze the relative position of chromosomes during interphase in human cells, based on observed radiogenic chromosome aberrations. The inputs of the program are a table of yields of pairwise chromosome interchanges and a proposed chromosome geometric cluster. Each can either be uploaded or selected from provided datasets. The main outputs are P-values for the proposed chromosome clusters. SCHIP is designed to be used by a number of scientific communities interested in nuclear architecture, including cancer and cell biologists, radiation biologists and mathematical/computational biologists. Availability: http://cramer.stat.ub.es/schip Contact: jarsuaga@cc.ucsf.edu Supplementary information: http://cramer.stat.ub.es/schip/help.htm

The mitochondrial DNA of trypanosomes is organized into a network of topologically linked minicircles. In order to investigate how key topological properties of the network change with minicircle density, the authors introduced, in an earlier study, a mathematical model in which randomly oriented minicircles were placed on the vertices of the simple square lattice. Using this model, the authors rigorously showed that when the density of minicircles increases, percolation clusters form. For higher densities, these percolation clusters are the backbones for networks of minicircles that saturate the entire lattice. An important relevant question is whether these findings are generally true. That is, whether these results are independent of the choice of the lattices on which the model is based. In this paper, we study two additional lattices (namely the honeycomb and the triangular lattices). These regular lattices are selected because they have been proposed for trypanosomes before and after replication. We compare our findings with our earlier results on the square lattice and show that the mathematical statements derived for the square lattice can be extended to these other lattices qualitatively. This finding suggests the universality of these properties. Furthermore, we performed a numerical study which provided data that are consistent with our theoretical analysis, and showed that the effect of the choice of lattices on the key network topological characteristics is rather small.

Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numerically, the minimum length (also called minimum step number) needed to form a particular knot in the simple cubic lattice. Our analytical work is based on improvement of a method introduced by Diao to enumerate conformations of a given knot type for a fixed length. This method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 (which is 24) to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34, respectively. Using an independent method based on the BFACF algorithm, we provide a complete list of numerical estimates (upper bounds) of the minimum step numbers for prime knots up to ten crossings, which are improvements over current published numerical results. We enumerate all minimum lattice knots of a given type and partition them into classes defined by BFACF type 0 moves.

The physical and biological properties of collapsed long polymer chains as well as of highly condensed biopolymers (such as DNA in all organisms) are known to be determined, at least in part, by their topological and geometrical properties. With this purpose of characterizing the topological properties of such condensed systems equilateral random polygons restricted to confined volumes are often used. However, very few analytical results are known. In this paper, we investigate the effect of volume confinement on the mean average crossing number (ACN) of equilateral random polygons. The mean ACN of knots and links under confinement provides a simple alternative measurement for the topological complexity of knots and links in the statistical sense. For an equilateral random polygon of n segments without any volume confinement constrain, it is known that its mean ACN ACN is of the order . Here we model the confining volume as a simple sphere of radius R. We provide an analytical argument which shows that ACN of an equilateral random polygon of n segments under extreme confinement (meaning R n) grows as O(n2). We propose to model the growth of ACN as a(R)n2 + b(R)nln(n) under a less-extreme confinement condition, where a(R) and b(R) are functions of R with R being the radius of the confining sphere. Computer simulations performed show a fairly good fit using this model.