Biomedical subjects

M Kimmel

Publications and source records attributed to M Kimmel.

At least 19 recordsLinked to original sources

PubMed1998

Application of a time-dependent coalescence process for inferring the history of population size changes from DNA sequence data.

Distribution of pairwise differences of nucleotides from data on a sample of DNA sequences from a given segment of the genome has been used in the past to draw inferences about the past history of population size changes. However, all earlier methods assume a given model of population size changes (such as sudden expansion), parameters of which (e.g., time and amplitude of expansion) are fitted to the observed distributions of nucleotide differences among pairwise comparisons of all DNA sequences in the sample. Our theory indicates that for any time-dependent population size, N(tau) (in which time tau is counted backward from present), a time-dependent coalescence process yields the distribution, p(tau), of the time of coalescence between two DNA sequences randomly drawn from the population. Prediction of p(tau) and N(tau) requires the use of a reverse Laplace transform known to be unstable. Nevertheless, simulated data obtained from three models of monotone population change (stepwise, exponential, and logistic) indicate that the pattern of a past population size change leaves its signature on the pattern of DNA polymorphism. Application of the theory to the published mtDNA sequences indicates that the current mtDNA sequence variation is not inconsistent with a logistic growth of the human population.

Algorithms↗

PubMed1998

Dynamic balance of segregation distortion and selection maintains normal allele sizes at the myotonic dystrophy locus.

Myotonic dystrophy (DM), an autosomal dominant neurological disorder, is caused by CTG-repeat expansions at the DMPK locus, with affected individuals having > or = 50 repeats of this trinucleotide. Reduced reproductive fitness of affected individuals and decreased viability of congenital DM have been noted. Expanded CTG-repeat alleles are highly unstable, predominantly yielding even higher repeat sizes. Preferential transmission of longer alleles from heterozygous mothers within the normal size range of alleles also is observed. In view of these observations, it is worth examining how DM has been maintained in human populations for hundreds of generations. We present an analysis of the dynamic properties of a model of joint effects of segregation distortion and selection (intensity of which increases with allele sizes of an individual's genotype). Our mathematical formulation and numerical analyses demonstrate that a weak segregation distortion during female meiosis, together with selection of comparable intensity (within the normal allele size range), can maintain an equilibrium distribution of allele frequencies. Genetic drift, acting in conjunction with the occasional contraction of alleles by mutation, can contribute to the balance of segregation distortion and mutation, in the sense that even weaker selection can explain the observed allele frequencies. The model is applied to CTG-repeat size distributions at the DMPK locus, observed in normal individuals from world populations.

Alleles↗

PubMed1998

Signatures of population expansion in microsatellite repeat data.

To examine the signature of population expansion on genetic variability at microsatellite loci, we consider a population that evolves according to the time-continuous Moran model, with growing population size and mutations that follow a general asymmetric stepwise mutation model. We present calculations of expected allele-size variance and homozygosity at a locus in such a model for several variants of growth, including stepwise, exponential, and logistic growth. These calculations in particular prove that population bottleneck followed by growth in size causes an imbalance between allele size variance and heterozygosity, characterized by the variance being transiently higher than expected under equilibrium conditions. This effect is, in a sense, analogous to that demonstrated before for the infinite allele model, where the number of alleles transiently increases after a stepwise growth of population. We analyze a set of data on tetranucleotide repeats that reveals the imbalance expected under the assumption of bottleneck followed by population growth in two out of three major racial groups. The imbalance is strongest in Asians, intermediate in Europeans, and absent in Africans. This finding is consistent with previous findings by others concerning the population expansion of modern humans, with the bottleneck event being most ancient in Africans, most recent in Asians, and intermediate in Europeans. Nevertheless, the imbalance index alone cannot reliably estimate the time of initiation of population expansion.

Genetics, Population↗

PubMed1997

Computer simulation of expansions of DNA triplet repeats in the fragile X syndrome and Huntington's disease.

The expansion of DNA triplet repeats has been shown to be responsible for about a dozen hereditary diseases. In this paper we are concerned with a computer model of such expansion, applied to the fragile X syndrome and Huntington's disease, for which enough quantitative data have been collected. The nucleotide sequence associated with the fragile X consists of CGG repeats and is located inside the FMR1 gene. In normal individuals there is a variable number of triplet repeats less than 60; in asymptomatic carriers the number of repeats is 60-200 (premutation). From the premutation range, the number of triplet repeats can increase within one generation to more than 200 producing affected individuals. In Huntington's disease the CAG repeats are located inside the HD gene. In normal individuals the number of repeats varies from around 11, up to 34. In the intermediate range (34-37 repeats), the mutability is increased, frequently leading to alleles of more than 37 repeats, and the disease phenotype. The rapid increase of the number of triplet repeats in affected individuals has been proposed to be due to the formation of folded DNA structures (hairpins) and their repair or misrepair. In order to determine if this proposed mechanism is adequate to account for the rapid increase of repeats and the large number of repeats in affected individuals we developed a mathematical model that includes the known mechanisms of hairpin formation, and strand synthesis and repair. Simulations based on the model using realistic probabilities of hairpin formation produced results that corresponded with the observed range of repeats and transition probabilities from normal to affected individuals. Similar modelling has been published for the Huntington's disease data. However, in this paper we demonstrate that a uniform approach works for fragile X and Huntington's disease, although the detailed assumptions of the model have to be different. These difference provide insight into the mechanisms of expansion in both cases. Among these insights is that an apparent threshold in the number of repeats for rapid expansion, and the preference for expansion over contraction, may be accounted for by relative probabilities of hairpin formation, replication, slippage and repair.

Computer Simulation↗

PubMed1997

Estimating clonal heterogeneity and interexperiment variability with the bifurcating autoregressive model for cell lineage data.

We utilize an extension of the variance-components models for cell lineage data in Huggins and Staudte (R.M. Huggins and R.G. Staudte, Variance components models for dependent cell populations. J. Am. Stat. Assoc. 89:19-29 (1994) to analyze NIH3T3 cells grown in two different media. This modeling approach has the advantage of a simple built-in correlation structure between familial members and allows for estimating experimental effects, rather than treating them as random effects. In addition, this methodology gives robust estimates of model parameters together with standard errors required for statistical inference. The importance of clonal heterogeneity and interexperiment variability in modeling eukaryotic cell cycles was previously pointed out by Kuczek and Axelrod (T. Kuczek and D.E. Axelrod, The importance of clonal heterogeneity and interexperimental variability in modeling the eukaryotic cell cycle. Math. Biosci. 79:87-96 (1986). This analysis confirms significantly positive sister-sister correlation when cells are grown in rich or poor medium and negative mother-daughter correlation when cells are grown in poor medium. However, for cells grown in rich medium, Kuczek and Axelrod's analysis gives negative mother-daughter correlations, whereas this analysis gives significant positive mother-daughter correlations.

3T3 Cells↗

PubMed1997

Relative mutation rates at di-, tri-, and tetranucleotide microsatellite loci.

Using the generalized stepwise mutation model, we propose a method of estimating the relative mutation rates of microsatellite loci, grouped by the repeat motif. Applying ANOVA to the distributions of the allele sizes at microsatellite loci from a set of populations, grouped by repeat motif types, we estimated the effect of population size differences and mutation rate differences among loci. This provides an estimate of motif-type-specific mutation rates up to a multiplicative constant. Applications to four different sets of di-, tri-, and tetranucleotide loci from a number of human populations reveal that, on average, the non-disease-causing microsatellite loci have mutation rates inversely related to their motif sizes. The dinucleotides appear to have mutation rates 1.5-2 times higher than the tetranucleotides, and the non-disease-causing trinucleotides have mutation rates intermediate between the di- and tetranucleotides. In contrast, the disease-causing trinucleotides have mutation rates 3.9-6.9 times larger than the tetranucleotides. Comparison of these estimates with the direct observations of mutation rates at microsatellites indicates that the earlier suggestion of higher mutation rates of tetranucleotides in comparison with the dinucleotides may stem from a nonrandom sampling of tetranucleotide loci in direct mutation assays.

Analysis of Variance↗

PubMed1997

Minimal residual disease in acute myelogenous leukaemia and myelodysplastic syndromes: a follow-up of patients in clinical remission.

The majority of patients with acute myelogenous leukaemia (AML) and myelodysplastic syndromes (MDS) relapse, especially those with unfavourable cytogenetics. This study was designed to investigate the presence and frequency of minimal residual disease (MRD) in patients with AML or MDS (n=35) and numerical abnormalities of chromosomes 6, 7, 8, 9, 10, 17 and 18 in clinical remission by using a combination of fluorescence activated cell sorting (FACS), fluorescence in-situ hybridization (FISH) and labelling with bromodeoxyuridine (BUdR). The technique enables the detection of as few as three leukaemic cells in 10(5) normal cells. MRD was detected in 33/35 patients in complete remission (CR). 16 patients relapsed (8/11 with monosomy 7, 4/17 with trisomy 8, and 4/7 with other cytogenetic abnormalities) after a median of 4.8 months (range 3-13). Levels of MRD (P=0.007) and proliferation index (P=0.011) were significantly higher in patients with monosomy 7 than in patients with trisomy 8 or other cytogenetic abnormalities. The percentage of cells in S-phase, the number of abnormal cells and cytogenetic class were related to time to relapse (P=0.001) with S-phase being the single most important prognostic factor (P=0.0001). We conclude that the combination of FACS/FISH/BUdR, which determines the number, phenotype and proliferation rate of very rare leukaemic cells in patients with AML or MDS in clinical remission, provides information that is useful in the identification of patients with high and low likelihood of relapse.

Acute Disease↗

PubMed1996

A discrete-time, multi-type generational inheritance branching process model of cell proliferation.

Mammalian cell populations, such as tumors, may contain subpopulations differing in parameters such as cell lifetimes, even if the populations are derived from single cells. The mode of inheritance of cell lifetimes has previously been the subject of experimental and mathematical investigation. To obtain data on cell lifetimes over more cell generations then previously available, Axelrod et al. [Cell Prolif. 26:235-249(1988)] measured the number of cells in primary colonies and secondary colonies derived form the primary colonies. The experimental results indicated large variance of cells per colony and highly significant correlations between the numbers of cells in primary and secondary colonies. To mathematically model these results we derive, for previously uninvestigated multi-type Galton-Watson branching process models, the covariance of the cell counts in the primary and secondary colonies. As a result, we are able to successfully model the data with two subpopulations having differing proliferation rates, in which the proliferation rate of a daughter cell is primarily determined by the proliferation rate of its mother. Interestingly, simulations display a trade-off between high values of variances and correlation coefficients. The values obtained from experiment are located on the boundary of the region attainable by simulation.

Animals↗

PubMed1996

Measures of variation at DNA repeat loci under a general stepwise mutation model.

Polymorphisms at tandem repeat loci are caused by mutations with allele sizes occasionally altered by more than one repeat unit in both forward and backward directions. Such mutational changes may occur with asymmetric probabilities. Therefore, a one-step symmetric stepwise mutation model may not be appropriate for studying the population dynamics at all repeat loci. In this work, we evaluated the expectation and variance of the within-population variance of the allele size distribution in a finite population, and the expected homozygosity at a locus by the coalescence approach under a general stepwise mutation model, where mutational transitions of allele sizes can be arbitrary, including being asymmetric. Under the special cases of symmetric one-step, two-step, and multi-step geometric distributions of mutations, our general results reduce to the corresponding results obtained by earlier investigators. The general results indicate that in a finite population, which has reached a steady state under the (general stepwise) mutation and drift balance, the within-population variance of allele sizes has a simple expectation (i.e., proportional to Nnu, the product of the mutation rate, nu, and effective population size, N). However, its stochastic variance is a quadratic function of this composite parameter, Nnu. Furthermore, this second-order variance does not decay with the number of alleles sampled from a population. Application of this theory to data on allele size distributions in unrelated Caucasians from the CEPH pedigree (obtained from the Genome Data Base) shows that the relationship of the variance and mean of within-population variance of allele sizes at tandem repeat loci, grouped by their chromosomal assignment, has a trend compatible with the theory. However, there is an indication that the second-order variance is generally underestimated. One reason for this departure might be that the CEPH sample may not represent a single homogeneous population that reached equilibrium at all tandem repeat loci.

DNA↗

PubMed1996

Dynamics of repeat polymorphisms under a forward-backward mutation model: within- and between-population variability at microsatellite loci.

Suggested molecular mechanisms for the generation of new tandem repeats of simple sequences indicate that the microsatellite loci evolve via some of forward-backward mutation. We provide a mathematical basis for suggesting a measure of genetic distance between populations based on microsatellite variation. Our results indicate that such a genetic distance measure can remain proportional to the divergence time of populations even when the forward-backward mutations produce variable and/or directionally biased alleles size changes. If the population size and the rate of mutation remain constant, then the measure will be proportional to the time of divergence of populations. This genetic distance is expressed in terms of a ratio of components of variance of allele sizes, based on expressions developed for studying population dynamics of quantitative traits. Application of this measure to data on 18 microsatellite loci in the nine human populations leads to evolutionary trees consistent with the known ethnohistory of the populations.

Alleles↗

PubMed1996

Error detection for genetic data, using likelihood methods.

As genetic maps become denser, the effect of laboratory typing errors becomes more serious. We review a general method for detecting errors in pedigree genotyping data that is a variant of the likelihood-ratio test statistic. It pinpoints individuals and loci with relatively unlikely genotypes. Power and significance studies using Monte Carlo methods are shown by using simulated data with pedigree structures similar to the CEPH pedigrees and a larger experimental pedigree used in the study of idiopathic dilated cardiomyopathy (DCM). The studies show the index detects errors for small values of theta with high power and an acceptable false positive rate. The method was also used to check for errors in DCM laboratory pedigree data and to estimate the error rate in CEPH-chromosome 6 data. The errors flagged by our method in the DCM pedigree were confirmed by the laboratory. The results are consistent with estimated false-positive and false-negative rates obtained using simulation.

Cardiomyopathy, Dilated↗

PubMed1996

Optimal control problems arising in cell-cycle-specific cancer chemotherapy.

We explore mathematical properties of models of cancer chemotherapy including cell-cycle dependence. Using the mathematical methods of control theory, we demonstrate two assertions of interest for the biomedical community: 1 Periodic chemotherapy protocols are close to the optimum for a wide class of models and have additional favourable properties. 2 Two possible approaches, (a) to minimize the final count of malignant cells and the cumulative effect of the drug on normal cells, or (b) to maximize the final count of normal cells and the cumulative effect of the drug on malignant cells, lead to similar principles of optimization. From the mathematical viewpoint, the paper provides a catalogue of simplest mathematical models of cell-cycle dependent chemotherapy. They can be classified based on the number of compartments and types of drug action modelled. In all these models the optimal controls are complicated by the singular and periodic trajectories and multiple solutions. However, efficient numerical methods have been developed. In simpler cases, it is also possible to provide an exhaustive classification of solutions. We also discuss developments in estimation of cell cycle parameters and cell-cycle dependent drug action.

Cell Compartmentation↗

PubMed1995

Mathematical modeling of the loss of telomere sequences.

hortening of telomeres is one of the supposed mechanisms of cellular aging and death. An important question related to this so-called "end-replication" hypothesis is whether it can explain in quantitative detail the dynamic of cell sensescence in vitro and in vivo. A natural way to answer this question is to use mathematical modeling. In this paper, the models were successfully fitted to data on cultured fibroblasts from two different sources assuming that after reaching the Hayflick checkpoint on a single chromosome cells cease to proliferate. The main conclusion is that the end-replication hypothesis provides an explanation for the cell aging process which is quantitatively consistent with the data. As a secondary outcome, estimates were obtained of the rate of shortening of telomeres and several interesting mathematical results for branching processes with infinite type spaces arise.

Animals↗

PubMed1994

Gene amplification by unequal sister chromatid exchange: probabilistic modeling and analysis of drug resistance data.

Unequal sister chromatid exchange has been proposed as one of several possible mechanisms for gene amplification resulting in tandemly repeated sequences on chromosomes. Two requirements for testing this hypothesis are analytical observations and a mathematical model. Recently observations were reported for the number of tandemly repeated sequences on chromosomes of cells growing in the presence of a toxic drug and the mechanism was proposed to be unequal sister chromatid exchange. We now develop a mathematical model of this process based on the following hypotheses, (i) the extent of slippage between paired sister chromatids is a random variable with geometric distribution, (ii) the number of crossover sites is a random variable with a Poisson distribution, and (iii) cells with less than a threshold number of copies of an essential gene are eliminated when grown in selective conditions. Iterating the model at successive cell divisions results in a Markov chain with a denumerable infinity of states. The resulting distributions of gene copy number per cell at a particular population size are compared to published data on the CAD gene in BHK cells growing in the presence of the drug PALA (Smith et al., 1990, Cell, 63, 1219). The mathematical model can reproduce the observed means and standard deviations of gene copy number per cell and allows construction of confidence region estimates of parameters describing the extent of slippage, density of crossover sites, and strength of selection. An important prediction of the model is that in non-selective conditions the cells with amplified sequences gradually disappear from the population even if they are not at a growth disadvantage, though rare cells with a very large number of amplified sequences might continue to exist. The success of modeling suggests that the proposed mechanism of gene amplification by unequal sister chromatid exchange is consistent with the number of tandemly repeated sequences on chromosomes observed in some circumstances.

Animals↗

PubMed1994

Fluctuation test for two-stage mutations: application to gene amplification.

The determination of mutation rates is an important experimental procedure for characterizing mutation processes. The accepted method of determining mutation rates, the fluctuation test, was introduced by Luria and Delbrück in 1943. Since then it has been applied to various microorganisms and cells. The Luria-Delbrück test is based on a restrictive hypothesis of mutations being due to single irreversible events. However, some inherited changes in phenotype, like gene amplification, may be due to two or more genetic changes, some of which may be reversible. The Luria-Delbrück model of mutation was compared to other models which included reversibility and more than one mutation stage. The Luria-Delbrück model has been confirmed to be consistent with the original bacteriophage resistance data. However, for gene amplification this model gives incompatible estimates of mutation rates by the P0 and r methods. Relaxing the hypotheses of the single-stage models did not improve the fit. In contrast, a two-stage reversible model provided a fit. Analysis of gene amplification data by the two-stage reversible model provides new information, including estimates of rates for each of the two forward stages and of the reverse step.

Animals↗

PubMed1994

Time-continuous branching walk models of unstable gene amplification.

We consider a stochastic mechanism of the loss of resistance of cancer cells to cytotoxic agents, in terms of unstable gene amplification. Two models being different versions of a time-continuous branching random walk are presented. Both models assume strong dependence in replication and segregation of the extrachromosomal elements. The mathematical part of the paper includes the expression for the expected number of cells with a given number of gene copies in terms of modified Bessel functions. This adds to the collection of rare explicit solutions to branching process models. Original asymptotic expansions are also demonstrated. Fitting the model to experimental data yields estimates of the probabilities of gene amplification and deamplification. The thesis of the paper is that purely stochastic mechanisms may explain the dynamics of reversible drug resistance of cancer cells. Various stochastic approaches and their limitations are discussed.

Animals↗

PubMed1994

Two simple models of almost the same population with very different dynamics.

We present our point of view of the controversy regarding complex dynamics of population models. We analyze two nonlinear models of population dynamics which may be used to model the same population, but which display very different dynamics. We prove that Model 1 stays globally asymptotically stable, while Model 2 is known to display instability, oscillations, and apparent period doubting leading to chaos. We also indicate situations in which both models are equivalent. We employ these results to argue that the "complex" behavior of some models may be due to apparently small differences in assumptions.

Animals↗

PubMed1993

Screening for lung cancer. The Mayo Lung Project revisited.

BACKGROUND: The Mayo Lung Project (MLP) reported lung cancer incidence and mortality in a population offered chest radiographs and sputum cytologic screening examinations every 4 months and a population offered only the Mayo Clinic advice to undergo annual examination. No mortality benefit attributable to screening was observed after 6 years of observation and at least 1 year of follow-up. METHODS: The authors describe a simulation study designed to estimate from Mayo data the parameters in a mathematical model of the natural history of lung cancer and to estimate the potential benefit associated with periodic screening of high-risk people starting at 45 years of age. RESULTS: It was found that the mean duration of Stage I non-small cell lung cancer is at least 4 years and that rates of Stage I detectability and curability are less than 25% and 35%, respectively. CONCLUSIONS: A trial of the magnitude, duration, and contamination of the MLP would have a less than 20% probability of showing significant benefit from screening; however, long-term annual screening might result in a modest decrease in lung cancer mortality, ranging from 0% to 13%. A greater benefit would accrue from improved detection and treatment.

Aged↗