Twin concordance: a set theoretic and probability theory approach

The concepts and assumptions used in twin concordance studies are made explicit with a mathematical treatment. The logic connecting those assumptions to the usual interpretations of twin studies is substantiated in a rigorous manner through the use of elementary set and probability theory.     

A control theoretic model of the forearm

In this paper, a control theoretic model of the forearm is developed and analyzed, and a computational method for predicting muscle activations necessary to generate specified motions is described. A detailed geometric model of the forearm kinematics, including the carrying angle and models of how the biceps and the supinator tendons wrap around the bones, is used. Also, including a dynamics model, the final model is a system of differential equations where the muscle activations play the role of control signals. Due to the large number of muscles, the problem of finding muscle activations is redundant, and this problem is solved by an optimization procedure. The computed muscle activations for ballistic movements clearly recaptures the triphasic ABC (Activation-Braking-Clamping) pattern. It is also transparent, from the muscle activation patterns, how the muscles cooperate and counteract in order to accomplish desired motions. A comparison with previously reported experimental data is included and the model predictions can be seen to be partially in agreement with the experimental data.     

Identifiability of a stochastic model for cell debris in flow cytometry

 One of the most important problems in recovering DNA distribution from flow cytometric DNA measurements is the presence of background noise. In this paper, we analyse a probabilistic model recently proposed for background debris distribution and based on a specific probabilistic mechanism for the DNA fragmentation process of the cell nucleus. In particular, we carry out some sufficient conditions to uniquely identify the original DNA distribution from the flow cytometric data. Received: 15 June 1997 / Revised version: 18 November 1997     

The limiting conditional probability distribution in a stochastic model of T cell repertoire maintenance

The limiting conditional probability distribution (LCD) has been much studied in the field of mathematical biology, particularly in the context of epidemiology and the persistence of epidemics. However, it has not yet been applied to the immune system. One of the characteristic features of the T cell repertoire is its diversity. This diversity declines in old age, whence the concepts of extinction and persistence are also relevant to the immune system. In this paper we model T cell repertoire maintenance by means of a continuous-time birth and death process on the positive integers, where the origin is an absorbing state. We show that eventual extinction is guaranteed. The late-time behaviour of the process before extinction takes place is modelled by the LCD, which we prove always exists for the process studied here. In most cases, analytic expressions for the LCD cannot be computed but the probability distribution may be approximated by means of the stationary probability distributions of two related processes. We show how these approximations are related to the LCD of the original process and use them to study the LCD in two special cases. We also make use of the large N expansion to derive a further approximation to the LCD. The accuracy of the various approximations is then analysed.     

The development of cell lineages: A sequential model

Abstract. The concept of cell lineage and the empirical characterization of specific lineages provide valuable insight into the problems of developmental biology. Of central interest is the decision-making process that results in the diversification of cell lines. Studies of the haemopoietic system, in which stem cells can be committed to one of at least six pathways of differentiation, have suggested that the restriction of differentiation potentials is a progressive and stochastic process. We have recently proposed an alternative model which hypothesizes that lineage potentials during haemopoiesis are expressed individually and in a predetermined sequence as progenitor cells mature. The model first arises from experimental studies which show that both normal myeloid progenitor cells and a human promyeloid cell line, which are able to differentiate towards either neutrophils or monocytes, express these potentials sequentially in culture. The close linear relationship between other haemopoietic progenitor cells is inferred from collective data from studies of bipotent progenitor cells and of haemopoietic proliferative disorders. If the development of haemopoietic cell lineages shows a tendency to follow a particular program, such a mechanism is likely to operate throughout development. In this paper we consider the evidence in favour of programmed events within progenitor cells implementing diversification, and the implications of predetermined and restricted pathways of embryonic development.     

Cell cycle variability: The oscillator model of the cell cycle yields transition probability alpha and beta type curves

The limit cycle concept of cell replication attributes cell cycle variability to continuous random modulation of the rates of reactions forming the intracellular control system believed to be responsible for replication processes. It is shown that this model can yield frequency histograms and both alpha and beta type accumulative distribution curves (with respect to generation times and also to cycle phases) which are of the various forms seen experimentally. The results thus provide additional support for this concept.     

A probability model for the fusion of nucleoli in metabolic cells

The nucleoli are assumed to be originally of negligible size and located at random in the nucleus, considered as a sphere. They increase to a maximum size during interphase. Fusion occurs if the nucleoli touch during the size increase process. The probability of this event is calculated for various values of the maximum size of the nucleolus as compared with the size of the nucleus. A set of experimental data is analysed, using the results obtained. It turns out that the frequency of fusion is significantly higher than the assumption of initial random location would indicate. For further details on experimental material and methods, see Darvey &; Driscoll (1972).     

A control theoretic paradigm for cell signaling networks: a simple complexity for a sensitive robustness

The fields of molecular biology and cell biology are being flooded with complex genomic and proteomic datasets of large dimensions. We now recognize that each molecule in the cell and tissue can no longer be viewed as an isolated entity. Instead, each molecule must be considered as one member of an interacting network. Consequently, there is an urgent need for mathematical models to understand the behavior of cell signaling networks in health and in disease.     

Stochastic model for tumor control probability: effects of cell cycle and (a)symmetric proliferation

Background

Estimating the required dose in radiotherapy is of crucial importance since the administrated dose should be sufficient to eradicate the tumor and at the same time should inflict minimal damage on normal cells. The probability that a given dose and schedule of ionizing radiation eradicates all the tumor cells in a given tissue is called the tumor control probability (TCP), and is often used to compare various treatment strategies used in radiation therapy.

Method

In this paper, we aim to investigate the effects of including cell-cycle phase on the TCP by analyzing a stochastic model of a tumor comprised of actively dividing cells and quiescent cells with different radiation sensitivities. Moreover, we use a novel numerical approach based on the method of characteristics for partial differential equations, validated by the Gillespie algorithm, to compute the TCP as a function of time.

Results

We derive an exact phase-diagram for the steady-state TCP of the model and show that at high, clinically-relevant doses of radiation, the distinction between active and quiescent tumor cells (i.e. accounting for cell-cycle effects) becomes of negligible importance in terms of its effect on the TCP curve. However, for very low doses of radiation, these proportions become significant determinants of the TCP. We also present the results of TCP as a function of time for different values of asymmetric division factor.

Conclusion

We observe that our results differ from the results in the literature using similar existing models, even though similar parameters values are used, and the reasons for this are discussed.

     

A probability distribution model of small -scale species richness in plant communities

We devised a probability distribution model that best expressed species richness per quadrat in grassland communities, and clarified the mechanism by which the mean richness per quadrat was always larger than the variance among quadrats. Our model will aid in the understanding of community structures, and allow comparisons among different communities. The model was constructed based on relatively simple theoretical assumptions about the mechanisms in play in target communities. We assumed in the model that the number of species occurring in an actual quadrat, j, is the sum of “the fundamental number of species”, k (constant), and “a fluctuating number of species”, i (a Poisson variate with the mean of μ); that is, j = k + i, where i, j and k are non-negative integers. The probability that j species occur in a quadrat is given by a Poisson-like distribution (extended Poisson), with two parameters k and μ. The mean species richness in the probability distribution is expressed by λ (= k + μ), and the variance is λ − k. The proposed model afforded a good fit for the observed frequency distribution of species richness per quadrat. If even one species is common among many quadrats, the mean number of species per quadrat is greater than the variance. The greater the number of common species among quadrats is, the larger is the value of k, and then the more pronounced is the difference between the mean and the variance (although the variance does not change). We fitted the model to 55 datasets collected by ourselves from grasslands in various locations (Tibet, Inner Mongolia, Slovakia, or Japan), with varying quadrat size (0.25, 0.0625, or 0.01 m²), and under differing management status (various stocking densities).