A branching process model of gene amplification following chromosome breakage.

We have devised a mathematical model of gene amplification utilizing recent experimental observations concerning dihydrofolate reductase (DHFR) gene amplification in CHO cells. The mathematical model, based on a biological model which proposes that acentric elements are the initial intermediates in gene amplification, includes the following features: (1) initiation of amplification by chromosomal breakage to produce an acentric structure; (2) replication of acentric DNA, once per cell cycle; (3) dissociation of replicated acentric DNA; (4) unequal segregation of acentric DNA fragments to daughter cells at mitosis; (5) subsequent reintegration of acentric fragments into chromosomes. These processes are assumed to be independent for each element present in a cell at a given time. Thus, processes of unequal segregation and integration may occur in parallel, not necessarily in a unique sequence, and may be reiterated in one or multiple cell cycles. These events are described mathematically as a Galton-Watson branching process with denumerable infinity of object types. This mathematical model qualitatively and quantitatively reproduces the major elements of the dynamical behavior of DHFR genes observed experimentally. The agreement between the mathematical model and the experimental data lends credence to the biological model proposed by Windle et al. (1991), including the importance of chromosome breakage and subsequent gene deletion resulting from resection of the broken chromosome ends as initial events in gene amplification.     

DNA amplification and an unstable arginine gene in Streptomyces lividans 66

Summary Streptomyces lividans 66 produced spontaneous chloramphenicol-sensitive mutants (Cml^s) at a frequency of about 1% of spores. The Cml^s mutant strains were very unstable, giving Arg^- mutants at frequencies of about 25% of spores. All the Arg^- mutants had amplified a particular 5.75 kb DNA fragment into tandem repeats of 250–500 copies per chromosome.     

The dynamics of gene amplification described as a multitype compartmental model and as a branching process.

The present work is aimed at developing the mathematical tools by which the dynamics of gene amplification (GA) can be described in detail. Some discrete compartmental models of GA by disproportionate replication and a general model for other putative GA mechanisms are presented and analyzed. The dynamical distribution of gene copy number in the cell population is calculated with the loss of cells taken either as constant or as copy-number-dependent. Our analysis shows that for a one-copy GA process with constant loss of cells, the relative frequency of single-gene-copy cells (sensitive cells) converges to zero, with the rate of convergence depending on the amplification probability. In contrast, for a one-copy GA process with copy-number-dependent loss of cells, the relative frequency of single-copy cells is bounded, implying a bounded compartment of many-gene-copy cells. Using branching processes theory we calculate the dynamical distribution of the single-gene-copy compartment as well as its extinction probability. Our models are used for estimating treatment prognosis as affected by drug resistance due to GA, showing significant differences in prognosis resulting from small changes in drug dose.     

Stochastic models for horizontal gene transfer: taking a random walk through tree space

Horizontal gene transfer (HGT) plays a critical role in evolution across all domains of life with important biological and medical implications. I propose a simple class of stochastic models to examine HGT using multiple orthologous gene alignments. The models function in a hierarchical phylogenetic framework. The top level of the hierarchy is based on a random walk process in "tree space" that allows for the development of a joint probabilistic distribution over multiple gene trees and an unknown, but estimable species tree. I consider two general forms of random walks. The first form is derived from the subtree prune and regraft (SPR) operator that mirrors the observed effects that HGT has on inferred trees. The second form is based on walks over complete graphs and offers numerically tractable solutions for an increasing number of taxa. The bottom level of the hierarchy utilizes standard phylogenetic models to reconstruct gene trees given multiple gene alignments conditional on the random walk process. I develop a well-mixing Markov chain Monte Carlo algorithm to fit the models in a Bayesian framework. I demonstrate the flexibility of these stochastic models to test competing ideas about HGT by examining the complexity hypothesis. Using 144 orthologous gene alignments from six prokaryotes previously collected and analyzed, Bayesian model selection finds support for (1) the SPR model over the alternative form, (2) the 16S rRNA reconstruction as the most likely species tree, and (3) increased HGT of operational genes compared to informational genes.     

A minimally parametrized branching process explaining plateau phase of qPCR amplification

Quantitative polymerase chain reaction (qPCR) is a commonly used molecular biology technique for measuring the concentration of a target nucleic acid sequence in a sample. The whole qPCR amplification process usually consists of an exponential, a linear and a plateau phase. In qPCR experiments, amplification curves of samples with different template concentrations often, even though not always, have the same plateau height. The biological theory for this phenomenon is that the plateau height is determined by reaction kinetics. Does it mean that the target concentration has no effect on the final plateau height? We proposed a branching process based on Michaelis–Menten kinetics. Our model can describe all phases of qPCR amplification despite its simplicity (it depends on only one parameter). We theoretically showed, through almost sure convergence, that amplification curves will eventually plateau at finite values in any experiment, under any condition. We conclude that the plateau height is largely determined by reaction kinetics but could also be affected by the template concentration. This is in accordance with the current biological theory.

     

Random walk models of worker sorting in ant colonies

Sorting can be an important mechanism for the transfer of information from one level of biological organization to another. Here we study the algorithm underlying worker sorting in Leptothorax ant colonies. Worker sorting is related to task allocation and therefore to the adaptive advantages associated with an efficient system for the division of labour in ant colonies. We considered four spatially explicit individual-based models founded on two-dimensional correlated random walk. Our aim was to establish whether sorting at the level of the worker population could occur with minimal assumptions about the behavioural algorithm of individual workers. The behaviour of an individual worker in the models could be summarized by the rule "move if you can, turn always". We assume that the turning angle of a worker is individually specific and negatively dependent on the magnitude of an internal parameter micro which could be regarded as a measure of individual experience or task specialization. All four models attained a level of worker sortedness that was compatible with results from experiments onLeptothorax ant colonies. We found that the presence of a sorting pivot, such as the nest wall or an attraction force towards the centre of the worker population, was crucial for sorting. We make a distinction between such pivots and templates and discuss the biological implications of their difference.     

A generalization of the optimal models of arterial branching

Optimal models of arterial branching angles are usually based on the assumption that the equation relating flow and radius is given byf=kr ³, as proposed by Murray in 1926. An exception is the model of Uylings (1977), in which he allowed the exponent ofr to vary from 2.33 to 3.0. Theoretical considerations coupled with empirical evidence suggest that the cubic flow equation may not be appropriate to describe the branching pattern of the arterial tree. The optimal models are modified to accommodate a more general flow equationf=kr ^x . Models that minimize a geometric feature such as surface or volume are sensitive to variations inx in a different way from those which minimize flow-related parameters, such as power loss due to friction and shear stress.     

Multi-type branching models to describe cell differentiation programs

Cell proliferation and differentiation is described by a multi-type branching process, a probability model that defines the inheritance of cell type. Cell type is defined by (i) a repression index related to the time required for S-phase entry and (ii) phenotype as determined by cell markers and division history. The inheritance of cell type is expressed as the expected number and type of progeny cells produced by a mother cell given her type. Expressions for the expected number and type of cells produced by a multi-cellular (bulk culture) system are derived from the general model by making the simplifying assumption that cell generation times are independent. The multi-type Smith-Martin model (MSM) makes the further assumption that cell generation times are lag-exponentially distributed with phenotype transitions occurring just before entry into S-phase. The inheritance-modified MSM (IMSM) model includes the influence of generation time memory so that mother and daughter generation times are correlated. The expansion of human cord blood CD34⁺ cells by haematopoietic growth factors was division tracked in bulk culture using carboxyfluorescein diacetate, succinimidyl ester (CFDA-SE). The MSM model was fitted to division tracking data to indentify cell cycle length, and the rates of CD34 antigen down-regulation and apoptosis. The IMSM model was estimated for mouse granulocyte-macrophage progenitors using live cell imaging data. Multi-type branching models describe cell differentiation dynamics at both single- and multi-cell scales, providing a new paradigm for systematic analysis of stem and progenitor cell development.     

Synthesis of mathematical models of branching axons and dendrites

A synthesis was made of models of branching neuronal cable structures from a full set of standard basic models. The study aimed to produce an instrument of mathematical modelling making it possible to reflect true life morphological and electrophysiological characteristics of axons and dendrites, discarding some of the restrictions and simplifications characterizing existing models of the structures mentioned. Equivalent electrical circuits of branching axons and dendrites were set up with in-series and node connections of standard four-terminal networks corresponding to basic segments with active or passive membrane. Equations were obtained for electrical processes in branching neuronal neurites, generalized in the case of multiple binary branching with arbitrary symmetry and branching structure. A difference scheme common to the whole class of models contemplated was produced and the algorithm of a numerical solution to the difference equations thus obtained was elaborated. The instrument described makes it possible to synthesize diverse models of branching axons and dendrites, offering considerably greater opportunities for modelling the main electrophysiological processes developing in these structures of electrotonus, propagation of excitation, and interaction between these two factors.State University Commemorating Tricentenary of Russo-Ukrainian Union. Dnepropetrovsk. Translated from Neirofiziologiya, Vol. 20, No. 4, pp. 471–479, July–August, 1988.     

Remark on models for amplification of asymmetry

There exists a number of evolutionary models in which a tiny asymmetry in the size of several subpopulations is amplified. In the present paper a counterexample is given based on equally justifiable starting conditions. The existence of similar models leading to contradictory results clearly demonstrates the highly hypothetic nature and questionable value of conclusions drawn from 'by chance' selection models.     

Remark on models for amplification of asymmetry

There exists a number of evolutionary models in which a tiny asymmetry in the size of several subpopulations is amplified. In the present paper a counterexample is given based on equally justifiable starting conditions. The existence of similar models leading to contradictory results clearly demonstrates the highly hypothetic nature and questionalble value of conclusions drawn from by chance selection models.     

A new approach to reconstruction models of dendritic branching patterns

Quantitative models for characterising the detailed branching patterns of dendritic trees aim to explain these patterns either in terms of growth models based on principles of dendritic development or reconstruction models that describe an existing structure by means of a canonical set of elementary properties of dendritic morphology, which when incorporated into an algorithmic procedure will generate samples of dendrites that are statistically indistinguishable in both canonical and emergent features from those of the original sample of real neurons. This article introduces a conceptually new approach to reconstruction modelling based on the single assumption that dendritic segments are built from sequences of units of constant diameter, and that the distribution of the lengths of units of similar diameter is independent of location within a dendritic tree. This assumption in combination with non-parametric methods for estimating univariate and multivariate probability densities leads to an algorithm that significantly reduces the number of basic parameters required to simulate dendritic morphology. It is not necessary to distinguish between stem and terminal segments or to specify daughter branch ratios or dendritic taper. The procedure of sampling probability densities conditioned on local morphological features eliminates the need, for example, to specify daughter branch ratios and dendritic taper since these emerge naturally as a consequence of the conditioning process. Thus several basic parameters of previous reconstruction algorithms become emergent parameters of the new reconstruction process. The new procedure was applied successfully to a sample of 51 interneurons from lamina II/III of the spinal dorsal horn.     

Biased random walk models for chemotaxis and related diffusion approximations

Summary Stochastic models of biased random walk are discussed, which describe the behavior of chemosensitive cells like bacteria or leukocytes in the gradient of a chemotactic factor. In particular the turning frequency and turn angle distributions are derived from certain biological hypotheses on the background of related experimental observations. Under suitable assumptions it is shown that solutions of the underlying differential-integral equation approximately satisfy the well-known Patlak-Keller-Segel diffusion equation, whose coefficients can be expressed in terms of the microscopic parameters. By an appropriate energy functional a precise error estimation of the diffusion approximation is given within the framework of singular perturbation theory.     

The branching angles in computer-generated optimized models of arterial trees

The structure of a complex arterial tree model is generated on the computer using the newly developed method of "constrained constructive optimization." The model tree is grown step by step, at each stage of development fulfilling invariant boundary conditions for pressures and flows. The development of structure is governed by adopting minimum volume inside the vessels as target function. The resulting model tree is analyzed regarding the relations between branching angles and segment radii. Results show good agreement with morphometric measurements on corrosion casts of human coronary arteries reported in the literature.     

Genome position and gene amplification

Background  

Amplifications, regions of focal high-level copy number change, lead to overexpression of oncogenes or drug resistance genes in tumors. Their presence is often associated with poor prognosis; however, the use of amplification as a mechanism for overexpression of a particular gene in tumors varies. To investigate the influence of genome position on propensity to amplify, we integrated a mutant form of the gene encoding dihydrofolate reductase into different positions in the human genome, challenged cells with methotrexate and then studied the genomic alterations arising in drug resistant cells.     

An unstable nuclear gene in phycomyces

A gentic instability in Phycomyces is described that appears to be associated with a single nuclear gene, dar. The wild type is able to take up riboflavin and its toxic analogue, deaza-riboflavin, from nanomolar concentrations in the medium. The mutants are unable to take up riboflavin and are resistant to deaza-riboflavin. Forward and reverse mutation rates are estimated to be 4 x 10^-5 and 2 x 10^-3 per nuclear division. Independently arisen dar mutants do not complement in heterokaryons. The mutant alleles are almost completely recessive. The phenotype of spores is not determined cell-autonomously, but is strongly influenced by the allele ratio among the nuclei in the sporangium of origin.