Cancer Risk and the Somatic Cell Lineage Tree

All the cells of a multicellular organism are the product of cell divisions that trace out a single binary tree, the so-called cell lineage tree. Because cell divisions are accompanied by replication errors, the shape of the cell lineage tree is one of the key determinants of how somatic evolution, which can potentially lead to cancer, proceeds. Cancer initiation usually requires the accumulation of a certain number of driver mutations. By mapping the accumulation of driver mutations into a graph theoretical problem, we show that in leading order of the mutation rate the probability of collecting a given number of driver mutations depends only on the distribution of the lineage lengths (irrespective of any other details of the cell lineage tree), and we derive a simple analytical formula for this probability. Our results are crucial in understanding how natural selection can shape the cell lineage trees of multicellular organisms in order to reduce their lifetime risk of cancer. In particular, our results highlight the significance of the longest cell lineages. Our analytical formula also provides a tool to quantify cancer susceptibility in theoretical models of tissue development and maintenance, as well as for empirical data on cell linage trees.Significance StatementA series of cell divisions starting from a single cell produce and maintain tissues of multicellular organisms. Somatic evolution, including the development of cancer, takes place along the cell lineage tree traced out by these cell divisions. A fundamental question in cancer research is how the lifetime risk of cancer depends on the properties of an arbitrary cell lineage tree. Here we show that for small mutation rates (which is the case in reality) the distribution of the lineage lengths alone determines cancer risk, and that this risk can be described by a simple analytical formula. Our results have far-reaching implications not only for cancer research, but also for evolutionary biology in general.

Remarks on Cosmological Bulk Viscosity in Different Epochs

The intention of this paper is mainly two-fold. First, we point out a striking numerical agreement between the bulk viscosity in the lepton era calculated by Husdal (2016) and our own calculations of the present-day bulk viscosity when the functional form is ζ ∼ ρ . From a phenomenological point of view, we thus seem to have an ansatz for the viscosity, which bridges the infancy of the Universe (∼1 s) with the present. This can also be looked upon as a kind of symmetry between the early-time cosmology and the present-day cosmology: it is quite remarkable that the kinetic theory-based bulk viscosity in the early universe and the experimentally-based bulk viscosity in the present universe can be covered by the same simple analytical formula. Second, we consider the Kasner universe as a typical anisotropic model of Bianchi-Type I, investigating whether this geometrical model is compatible with constant viscosity coefficients in the fluid. Perhaps surprisingly, the existence of a shear viscosity turns out to be incompatible with the Kasner model. By contrast, a bulk viscosity is non-problematic in the isotropic version of the model. In the special case of a Zel’dovich (stiff) fluid, the three equal exponents in the Kasner metric are even determined by the bulk viscosity alone, independent of the value of the fluid energy density. We also give a brief comparison with some other recent approaches to viscous cosmology.

Upper Bounds for the Conversion Efficiency of Diluted Blackbody Radiation Energy into Work

A new formula has been proposed for the Landsberg–Tonge function \chi (\varepsilon ) entering the entropy density flux of the diluted blackbody radiation of dilution factor ε. Two models have been proposed for the conversion of diluted blackbody radiation energy into work. The Carnot and Petela–Landsberg–Press relationships do not provide accurate upper bounds for the real conversion efficiency and in some cases they wrongly estimate positive output work when the converter of radiation energy into work does not operate. Four upper bounds for the conversion efficiency have been derived. The most accurate upper bound efficiency requires the numerical solution of an algebraic equation for the optimum absorber temperature while the second best upper bound efficiency has the advantage that it is a simple analytical formula.

Hysteresis loop reversing by applying Langevin approximation

Purpose The purpose of this paper is to model one of the unsolved problems of magnetism, the reversal of hysteresis loops, in an analytical way. The mathematical models, describing the multiphase steel used in engineering practice, without any exception, are unsuited to provide a way to reverse the hysteretic process. In this paper, a proposal is put forward to model it by using analytical expressions, applying the reversal of the Langevin function. This model works with a high accuracy, giving useful answers to a long unsolved magnetic problem, the lack of reversibility of the hysteresis loop. The use of the proposal is shown by applying the reversal of Langevin function to a sinusoidal and a triangular waveform, the two most frequently used waveforms in research, test and industrial applications. Schematic representations are given for the wave reconstruction by using the proposed method. Design/methodology/approach The unsolved reversibility of the hysteresis loop is approached by a simple analytical formula, providing close approximation for most applications. Findings The proposed solution, applying the reversal of Langevin function, to the problem provides a good practical solution. Research limitations/implications The simple analytical formula has been applied to a number of loops of widely different shapes and sizes with excellent results. Practical implications The proposed solution provides a missing mathematical tool to an unsolved problem for practical applications. Social implications The solution proposed will reduce the work required and provide replacement for expensive complex test instrumentation. Originality/value To the best of the authors’ knowledge, this approach used in this study is the first successful approach in this field, irrespective of the required waveform, and is completely independent of the model used by the user.

Pre-Impact Configuration Designing of a Robot Manipulator for Impact Minimization

This paper studies the collision problem of a robot manipulator and presents a method to minimize the impact force by pre-impact configuration designing. First, a general dynamic model of a robot manipulator capturing a target is established by spatial operator algebra (SOA) and a simple analytical formula of the impact force is obtained. Compared with former models proposed in literatures, this model has simpler form, wider range of applications, O(n) computation complexity, and the system Jacobian matrix can be provided as a production of the configuration matrix and the joint matrix. Second, this work utilizes the impulse ellipsoid to analyze the influence of the pre-impact configuration and the impact direction on the impact force. To illustrate the inertia message of each body in the joint space, a new concept of inertia quasi-ellipsoid (IQE) is introduced. We find that the impulse ellipsoid is constituted of the inertia ellipsoids of the robot manipulator and the target, while each inertia ellipsoid is composed of a series of inertia quasi-ellipsoids. When all inertia quasi-ellipsoids exhibit maximum (minimum) coupling, the impulse ellipsoid should be the flattest (roundest). Finally, this paper provides the analytical expression of the impulse ellipsoid, and the eigenvalues and eigenvectors are used as measurements to illustrate the size and direction of the impulse ellipsoid. With this measurement, the desired pre-impact configuration and the impact direction with minimum impact force can be easily solved. The validity and efficiency of this method are verified by a PUMA robot and a spatial robot.

Quantifying the contribution of asymptomatic infection to the cumulative incidence

SUMMARYMany infectious diseases in humans may manifest with no or mild symptoms. While numerous studies have estimated the proportion of infectious individuals in whom symptoms are absent during the entire course of infection, the contribution of asymptomatic cases to the overall cumulative incidence is difficult to untangle. Here, with a mathematical model, we provide a simple analytical formula to quantify this contribution and highlight the potential for large errors that can arise when naively estimating it.

A simple analytical formula to compute the residual Mutual Information between pairs of data vectors

ABSTRACTSummaryThe Mutual Information of pairs of data vectors, for example sequence alignment positions or gene expression profiles, is a quantitative measure of the interdependence between the data. However, data vectors based on a finite number of samples retain non-zero Mutual Information values even for completely random data, which is referred to as background or residual Mutual Information. Estimates of the residual Mutual Information have so far been obtained through heuristic or numerical approximations. Here we introduce a simple analytical formula for the computation of the residual Mutual Information that yields precise values and does not require the joint probabilities between the vector elements as input.Availability and ImplementationA C program arMI is available at http://mathbio.crick.ac.uk/wiki/Software#arMI. Using an input alignment in FASTA format or alternatively an internally created random alignment of specified length and depth, the program computes three types of Mutual information: (i) Shannon’s Mutual Information between all pairs of alignment columns; (ii) the numerical residual Mutual Information by using the same formula on the randomised (shuffled) data; (iii) the analytical residual Mutual Information introduced here. The package depends on the GNU Scientific Library, which is used for vector and matrix operations, factorial expressions and random number generation (Galassi et al., 2009). Reference alignments and result data are included in the program package in the folder ‘tests’. The R environment was used for statistics and plotting (R Core Team, 2014)[email protected] MaterialA detailed derivation of the analytical formula is given in the Supplementary Material.

Comparison between different proximity potentials and the double-folding model for spherical–deformed interacting nuclei

The Coulomb barrier parameters have been calculated for a spherical–deformed interacting pair of nuclei using 14 different versions of the proximity approaches and a simple analytical formula for the Coulomb part of the heavy ion potential. The results of these proximity versions have been compared with more accurate results obtained from the double-folding model (DFM). We have considered the interacting pair 48Ca + 238Pu as an example and assumed the presence of the quadrupole, octupole, and hexadecapole deformation parameters for 238Pu. The orientation angle dependence of the Coulomb barrier parameters has been computed for different sets of deformation parameters. We found that the proximity types named Prox77, BW Prox91, AW Prox95, Bass Prox77, and Bass Prox80 are the best ones of the available 14 versions of the proximity approaches for calculating the nuclear part of the interaction potential for a spherical–deformed pair of nuclei.

Asymptotic Analysis of the Maxwell Garnett Formula Using the Two-Phase Composite Model

In this paper, a two-phase computational model of a composite (TwPM) with cylindrical inclusion of square cross-sections and small sizes is proposed. The applied asymptotic techniques devoted to the analysis of the constructed TwPM are validated through a comparison of the results reported by others. A simple analytical formula of the reduced heat transfer parameter for small size of inclusions is derived, and an asymptotic character of Maxwell Garnett (MG) formula is illustrated.