318 research outputs found
Mass concentration in a nonlocal model of clonal selection
Self-renewal is a constitutive property of stem cells. Testing the cancer
stem cell hypothesis requires investigation of the impact of self-renewal on
cancer expansion. To understand better this impact, we propose a mathematical
model describing dynamics of a continuum of cell clones structured by the
self-renewal potential. The model is an extension of the finite
multi-compartment models of interactions between normal and cancer cells in
acute leukemias. It takes a form of a system of integro-differential equations
with a nonlinear and nonlocal coupling, which describes regulatory feedback
loops in cell proliferation and differentiation process. We show that such
coupling leads to mass concentration in points corresponding to maximum of the
self-renewal potential and the model solutions tend asymptotically to a linear
combination of Dirac measures. Furthermore, using a Lyapunov function
constructed for a finite dimensional counterpart of the model, we prove that
the total mass of the solution converges to a globally stable equilibrium.
Additionally, we show stability of the model in space of positive Radon
measures equipped with flat metric. The analytical results are illustrated by
numerical simulations
A structured population model of clonal selection in acute leukemias with multiple maturation stages
Funding: TS and AM-C were supported by research funding from the German Research Foundation DFG (SFB 873; subproject B08). TL gratefully acknowledges support from the Heidelberg Graduate School (HGS).Recent progress in genetic techniques has shed light on the complex co-evolution of malignant cell clones in leukemias. However, several aspects of clonal selection still remain unclear. In this paper, we present a multi-compartmental continuously structured population model of selection dynamics in acute leukemias, which consists of a system of coupled integro-differential equations. Our model can be analysed in a more efficient way than classical models formulated in terms of ordinary differential equations. Exploiting the analytical tractability of this model, we investigate how clonal selection is shaped by the self-renewal fraction and the proliferation rate of leukemic cells at different maturation stages. We integrate analytical results with numerical solutions of a calibrated version of the model based on real patient data. In summary, our mathematical results formalise the biological notion that clonal selection is driven by the self-renewal fraction of leukemic stem cells and the clones that possess the highest value of this parameter are ultimately selected. Moreover, we demonstrate that the self-renewal fraction and the proliferation rate of non-stem cells do not have a substantial impact on clonal selection. Taken together, our results indicate that interclonal variability in the self-renewal fraction of leukemic stem cells provides the necessary substrate for clonal selection to act upon.PostprintPeer reviewe
Asymptotic methods for delay equations.
Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006
A dynamical model of genetic networks describes cell differentiation
Cell differentiation is a complex phenomenon whereby a stem cell becomes progressively more specialized and eventually gives rise to a specific cell type. Differentiation can be either stochastic or, when appropriate signals are present, it can be driven to take a specific route. Induced pluripotency has also been recently obtained by overexpressing some genes in a differentiated cell. Here we show that a stochastic dynamical model of genetic networks can satisfactorily describe all these important features of differentiation, and others. The model is based on the emergent properties of generic genetic networks, it does not refer to specific control circuits and it can therefore hold for a wide class of lineages. The model points to a peculiar role of cellular noise in differentiation, which has never been hypothesized so far, and leads to non trivial predictions which could be subject to experimental testing
Effect of Dedifferentiation on Time to Mutation Acquisition in Stem Cell-Driven Cancers
Accumulating evidence suggests that many tumors have a hierarchical
organization, with the bulk of the tumor composed of relatively differentiated
short-lived progenitor cells that are maintained by a small population of
undifferentiated long-lived cancer stem cells. It is unclear, however, whether
cancer stem cells originate from normal stem cells or from dedifferentiated
progenitor cells. To address this, we mathematically modeled the effect of
dedifferentiation on carcinogenesis. We considered a hybrid
stochastic-deterministic model of mutation accumulation in both stem cells and
progenitors, including dedifferentiation of progenitor cells to a stem
cell-like state. We performed exact computer simulations of the emergence of
tumor subpopulations with two mutations, and we derived semi-analytical
estimates for the waiting time distribution to fixation. Our results suggest
that dedifferentiation may play an important role in carcinogenesis, depending
on how stem cell homeostasis is maintained. If the stem cell population size is
held strictly constant (due to all divisions being asymmetric), we found that
dedifferentiation acts like a positive selective force in the stem cell
population and thus speeds carcinogenesis. If the stem cell population size is
allowed to vary stochastically with density-dependent reproduction rates
(allowing both symmetric and asymmetric divisions), we found that
dedifferentiation beyond a critical threshold leads to exponential growth of
the stem cell population. Thus, dedifferentiation may play a crucial role, the
common modeling assumption of constant stem cell population size may not be
adequate, and further progress in understanding carcinogenesis demands a more
detailed mechanistic understanding of stem cell homeostasis
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