Bifurcation analysis informs Bayesian inference in the Hes1 feedback loop

<p>Background Ordinary differential equations (ODEs) are an important tool for describing the dynamics of biological systems. However, for ODE models to be useful, their parameters must first be calibrated. Parameter estimation, that is, finding parameter values given experimental data, is an inference problem that can be treated systematically through a Bayesian framework.</p> <p>A Markov chain Monte Carlo approach can then be used to sample from the appropriate posterior probability distributions, provided that suitable prior distributions can be found for the unknown parameter values. Choosing these priors is therefore a vital first step in the inference process. We study here a negative feedback loop in gene regulation where an ODE incorporating a time delay has been proposed as a realistic model and where experimental data is available. Our aim is to show that a priori mathematical analysis can be exploited in the choice of priors.</p> <p>Results By focussing on the onset of oscillatory behaviour through a Hopf Bifurcation, we derive a range of analytical expressions and constraints that link the model parameters to the observed dynamics of the system. Computational tests on both simulated and experimental data emphasise the usefulness of this analysis.</p> <p>Conclusion Mathematical analysis not only gives insights into the possible dynamical behaviour of gene expression models, but can also be used to inform the choice of priors when parameters are inferred from experimental data in a Bayesian setting.</p&gt

Dynamic DNA and human disease: mathematical modelling and statistical inference for myotonic dystrophy type 1 and Huntington disease

Several human genetic diseases, including myotonic dystrophy type 1 (DM1) and Huntington disease (HD), are associated with inheriting an abnormally large unstable DNA simple sequence tandem repeat. These sequences mutate, by changing the number of repeats, many times during the lifetime of those affected, with a bias towards expansion. High repeat numbers are associated with early onset and disease severity. The presence of somatic instability compromises attempts to measure intergenerational repeat dynamics and infer genotype-phenotype relationships. Modelling the progression of repeat length throughout the lifetime of individuals has potential for improving prognostic information as well as providing a deeper understanding of the underlying biological process. Dr Fernando Morales, Dr Anneli Cooper and others from the Monckton lab have characterised more than 25,000 de novo somatic mutations from a large cohort of DM1 patients using single-molecule polymerase chain reaction (SM-PCR). This rich dataset enables us to fully quantify levels of somatic instability across a representative DM1 population for the first time. We establish the relationship between inherited or progenitor allele length, age at sampling and levels of somatic instability using linear regression analysis. We show that the estimated progenitor allele length genotype is significantly better than modal repeat length (the current clinical standard) at predicting age of onset and this novel genotype is the major modifier of the age of onset phenotype. Further we show that somatic variation (adjusted for estimated progenitor allele length and age at sampling) is also a modifier of the age of onset phenotype. Several families form the large cohort, and we find that the level of somatic instability is highly heritable, implying a role for individual-specific trans-acting genetic modifiers. We develop new mathematical models, the main focus of this thesis, by modifying a previously proposed stochastic birth process to incorporate possible contraction. A Bayesian likelihood approach is used as the basis for inference and parameter estimation. We use model comparison analysis to reveal, for the first time, that the expansion bias observed in the distributions of repeat lengths is likely to be the cumulative effect of many expansion and contraction events. We predict that mutation events can occur as frequently as every other day, which matches the timing of regular cell activities such as DNA repair and transcription, but not DNA replication. Mutation rates estimated under the models described above are lower than expected among individuals with inherited repeat lengths less than 100 CTGs, suggesting that these rates may be suppressed at the lower end of the disease causing range. We propose that a length-specific effect may be operating within this range and test this hypothesis by introducing such an effect into the model. To calibrate this extended model, we use blood DNA data from DM1 individuals with small alleles (inherited repeat lengths less than 100 CTGs) and buccal DNA from HD individuals who almost always have inherited repeat lengths less than 100 CAGs. These datasets comprise single DNA molecules sized using SM-PCR. We find statistical support for a general length-specific effect which suppresses mutational rates among the smaller alleles and gives rise to a distinctive pattern in the repeat length distributions. In a novel application of this new model, fitted to a large cohort of DM1 individuals, we also show that this distinctive pattern may help identify individuals whose effective repeat length, with regards to somatic instability, is less than their actual repeat length. A plausible explanation for this distinction is that the expanded repeat tract is compromised by interruptions or other unusual features. For these individuals, we estimate the effective repeat length of their expanded repeat tracts and contribute to the on-going discussion about the effect of interruptions on phenotype. The interpretation of the levels of somatic instability in many of the affected tissues in the triplet repeat diseases is hindered by complex cell compositions. We extend our model to two cell populations whose repeat lengths have different rates of mutation (fast and slow). Swami et al. have recently characterised repeat length distributions in end stage HD brain. Applying our model, we infer for each frontal cortex HD dataset the likely relative weight of these cell populations and their corresponding contribution towards somatic variation. By comparison with data from laser captured single cells we conclude that the neuronal repeat lengths most likely mutate at a higher rate than glial repeat lengths, explaining the characteristic skewed distributions observed in mixed cell tissue from the brain. We confirm that individual-specific mutation rates in neurons are, in addition to the inherited repeat length, a modifier of age of onset. Our results support a model of disease progression where individuals with the same inherited repeat length may reach age of onset, as much as 30 years earlier, because of greater somatic expansions underpinned by higher mutational rates. Therapies aimed at reducing somatic expansions would therefore have considerable benefits with regard to extending the age of onset. Currently clinical diagnosis of DM1 is based on a measure of repeat length from blood cells, but variance in modal length only accounts for between 20 - 40% of the variance in age of onset and, therefore, is not a an accurate predictive tool. We show that in principle progenitor allele length improves the inverse correlation with age of onset over the traditional model length measure. We make use of second blood samples that are now available from 40 DM1 individuals. We show that inherited repeat length and the mutation rates underlying repeat length instability in blood, inferred from samples at two time points rather than one, are better predictors of age of onset than the traditional modal length measure. Our results are a step towards providing better prognostic information for DM1 individuals and their families. They should also lead to better predictions for drug/therapy response, which is emerging as key to successful clinical trials. Microsatellites are another type of tandem repeat found in the genome with high levels of intergenerational and somatic mutation. Differences between individuals make microsatellites very useful biomarkers and they have many applications in forensics and medicine. As well as a general application to other expanded repeat diseases, the mathematical models developed here could be used to better understand instability at other mutational hotspots such as microsatellites

Deep learnability: using neural networks to quantify language similarity and learnability

Learning a second language (L2) usually progresses faster if a learner's L2 is similar to their first language (L1). Yet global similarity between languages is difficult to quantify, obscuring its precise effect on learnability. Further, the combinatorial explosion of possible L1 and L2 language pairs, combined with the difficulty of controlling for idiosyncratic differences across language pairs and language learners, limits the generalisability of the experimental approach. In this study, we present a different approach, employing artificial languages and artificial learners. We built a set of five artificial languages whose underlying grammars and vocabulary were manipulated to ensure a known degree of similarity between each pair of languages. We next built a series of neural network models for each language, and sequentially trained them on pairs of languages. These models thus represented L1 speakers learning L2s. By observing the change in activity of the cells between the L1-speaker model and the L2-learner model, we estimated how much change was needed for the model to learn the new language. We then compared the change for each L1/L2 bilingual model to the underlying similarity across each language pair. The results showed that this approach can not only recover the facilitative effect of similarity on L2 acquisition, but can also offer new insights into the differential effects across different domains of similarity. These findings serve as a proof of concept for a generalisable approach that can be applied to natural languages

Deep learning for real-time single-pixel video

Single-pixel cameras capture images without the requirement for a multi-pixel sensor, enabling the use of state-of-the-art detector technologies and providing a potentially low-cost solution for sensing beyond the visible spectrum. One limitation of single-pixel cameras is the inherent trade-off between image resolution and frame rate, with current compressive (compressed) sensing techniques being unable to support real-time video. In this work we demonstrate the application of deep learning with convolutional auto-encoder networks to recover real-time 128 × 128 pixel video at 30 frames-per-second from a single-pixel camera sampling at a compression ratio of 2%. In addition, by training the network on a large database of images we are able to optimise the first layer of the convolutional network, equivalent to optimising the basis used for scanning the image intensities. This work develops and implements a novel approach to solving the inverse problem for single-pixel cameras efficiently and represents a significant step towards real-time operation of computational imagers. By learning from examples in a particular context, our approach opens up the possibility of high resolution for task-specific adaptation, with importance for applications in gas sensing, 3D imaging and metrology

Deep learning: an introduction for applied mathematicians

Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus, approximation theory, optimization and linear algebra. This article provides a very brief introduction to the basic ideas that underlie deep learning from an applied mathematics perspective. Our target audience includes postgraduate and final year undergraduate students in mathematics who are keen to learn about the area. The article may also be useful for instructors in mathematics who wish to enliven their classes with references to the application of deep learning techniques. We focus on three fundamental questions: what is a deep neural network? how is a network trained? what is the stochastic gradient method? We illustrate the ideas with a short MATLAB code that sets up and trains a network. We also show the use of state-of-the art software on a large scale image classification problem. We finish with references to the current literature

Deep learning optimized single-pixel LiDAR

Interest in autonomous transport has led to a demand for 3D imaging technologies capable of resolving fine details at long range. Light detection and ranging (LiDAR) systems have become a key technology in this area, with depth information typically gained through time-of-flight photon-counting measurements of a scanned laser spot. Single-pixel imaging methods offer an alternative approach to spot-scanning, which allows a choice of sampling basis. In this work, we present a prototype LiDAR system, which compressively samples the scene using a deep learning optimized sampling basis and reconstruction algorithms. We demonstrate that this approach improves scene reconstruction quality compared to an orthogonal sampling method, with reflectivity and depth accuracy improvements of 57% and 16%, respectively, for one frame per second acquisition rates. This method may pave the way for improved scan-free LiDAR systems for driverless cars and for fully optimized sampling to decision-making pipelines

Complex history of dog (Canis familiaris) origins and translocations in the Pacific revealed by ancient mitogenomes

Archaeological evidence suggests that dogs were introduced to the islands of Oceania via Island Southeast Asia around 3,300 years ago, and reached the eastern islands of Polynesia by the fourteenth century AD. This dispersal is intimately tied to human expansion, but the involvement of dogs in Pacific migrations is not well understood. Our analyses of seven new complete ancient mitogenomes and five partial mtDNA sequences from archaeological dog specimens from Mainland and Island Southeast Asia and the Pacific suggests at least three dog dispersal events into the region, in addition to the introduction of dingoes to Australia. We see an early introduction of dogs to Island Southeast Asia, which does not appear to extend into the islands of Oceania. A shared haplogroup identified between Iron Age Taiwanese dogs, terminal- Lapita and post-Lapita dogs suggests that at least one dog lineage was introduced to Near Oceania by or as the result of interactions with Austronesian language speakers associated with the Lapita Cultural Complex. We did not find any evidence that these dogs were successfully transported beyond New Guinea. Finally, we identify a widespread dog clade found across the Pacific, including the islands of Polynesia, which likely suggests a post-Lapita dog introduction from southern Island Southeast Asia

Genetic mechanisms of critical illness in COVID-19.

Host-mediated lung inflammation is present1, and drives mortality2, in the critical illness caused by coronavirus disease 2019 (COVID-19). Host genetic variants associated with critical illness may identify mechanistic targets for therapeutic development3. Here we report the results of the GenOMICC (Genetics Of Mortality In Critical Care) genome-wide association study in 2,244 critically ill patients with COVID-19 from 208 UK intensive care units. We have identified and replicated the following new genome-wide significant associations: on chromosome 12q24.13 (rs10735079, P = 1.65 × 10-8) in a gene cluster that encodes antiviral restriction enzyme activators (OAS1, OAS2 and OAS3); on chromosome 19p13.2 (rs74956615, P = 2.3 × 10-8) near the gene that encodes tyrosine kinase 2 (TYK2); on chromosome 19p13.3 (rs2109069, P = 3.98 ×  10-12) within the gene that encodes dipeptidyl peptidase 9 (DPP9); and on chromosome 21q22.1 (rs2236757, P = 4.99 × 10-8) in the interferon receptor gene IFNAR2. We identified potential targets for repurposing of licensed medications: using Mendelian randomization, we found evidence that low expression of IFNAR2, or high expression of TYK2, are associated with life-threatening disease; and transcriptome-wide association in lung tissue revealed that high expression of the monocyte-macrophage chemotactic receptor CCR2 is associated with severe COVID-19. Our results identify robust genetic signals relating to key host antiviral defence mechanisms and mediators of inflammatory organ damage in COVID-19. Both mechanisms may be amenable to targeted treatment with existing drugs. However, large-scale randomized clinical trials will be essential before any change to clinical practice