13 research outputs found
Hybrid approaches for multiple-species stochastic reaction-diffusion models
Reaction-diffusion models are used to describe systems in fields as diverse
as physics, chemistry, ecology and biology. The fundamental quantities in such
models are individual entities such as atoms and molecules, bacteria, cells or
animals, which move and/or react in a stochastic manner. If the number of
entities is large, accounting for each individual is inefficient, and often
partial differential equation (PDE) models are used in which the stochastic
behaviour of individuals is replaced by a description of the averaged, or mean
behaviour of the system. In some situations the number of individuals is large
in certain regions and small in others. In such cases, a stochastic model may
be inefficient in one region, and a PDE model inaccurate in another. To
overcome this problem, we develop a scheme which couples a stochastic
reaction-diffusion system in one part of the domain with its mean field
analogue, i.e. a discretised PDE model, in the other part of the domain. The
interface in between the two domains occupies exactly one lattice site and is
chosen such that the mean field description is still accurate there. This way
errors due to the flux between the domains are small. Our scheme can account
for multiple dynamic interfaces separating multiple stochastic and
deterministic domains, and the coupling between the domains conserves the total
number of particles. The method preserves stochastic features such as
extinction not observable in the mean field description, and is significantly
faster to simulate on a computer than the pure stochastic model.Comment: 38 pages, 8 figure
Single-Cell Migration in Complex Microenvironments: Mechanics and Signaling Dynamics
Cells are highly dynamic and mechanical automata powered by molecular motors that respond to external cues. Intracellular signaling pathways, either chemical or mechanical, can be activated and spatially coordinated to induce polarized cell states and directional migration. Physiologically, cells navigate through complex microenvironments, typically in three-dimensional (3D) fibrillar networks. In diseases, such as metastatic cancer, they invade across physiological barriers and remodel their local environments through force, matrix degradation, synthesis, and reorganization. Important external factors such as dimensionality, confinement, topographical cues, stiffness, and flow impact the behavior of migrating cells and can each regulate motility. Here, we review recent progress in our understanding of single-cell migration in complex microenvironments.National Cancer Institute (U.S.) (Grant No. 5U01CA177799)National Institutes of Health (U.S.) (Ruth L. Kirschstein National Research Service Award
Stochastic multi-scale models of competition within heterogeneous cellular populations: simulation methods and mean-field analysis
We propose a modelling framework to analyse the stochastic behaviour of
heterogeneous, multi-scale cellular populations. We illustrate our methodology
with a particular example in which we study a population with an
oxygen-regulated proliferation rate. Our formulation is based on an
age-dependent stochastic process. Cells within the population are characterised
by their age. The age-dependent (oxygen-regulated) birth rate is given by a
stochastic model of oxygen-dependent cell cycle progression. We then formulate
an age-dependent birth-and-death process, which dictates the time evolution of
the cell population. The population is under a feedback loop which controls its
steady state size: cells consume oxygen which in turns fuels cell
proliferation. We show that our stochastic model of cell cycle progression
allows for heterogeneity within the cell population induced by stochastic
effects. Such heterogeneous behaviour is reflected in variations in the
proliferation rate. Within this set-up, we have established three main results.
First, we have shown that the age to the G1/S transition, which essentially
determines the birth rate, exhibits a remarkably simple scaling behaviour. This
allows for a huge simplification of our numerical methodology. A further result
is the observation that heterogeneous populations undergo an internal process
of quasi-neutral competition. Finally, we investigated the effects of
cell-cycle-phase dependent therapies (such as radiation therapy) on
heterogeneous populations. In particular, we have studied the case in which the
population contains a quiescent sub-population. Our mean-field analysis and
numerical simulations confirm that, if the survival fraction of the therapy is
too high, rescue of the quiescent population occurs. This gives rise to
emergence of resistance to therapy since the rescued population is less
sensitive to therapy
Coarse-graining and hybrid methods for efficient simulation of stochastic multi-scale models of tumour growth
The development of hybrid methodologies is of current interest in both multi-scale modelling and stochastic reaction-diffusion systems regarding their applications to biology. We formulate a hybrid method for stochastic multi-scale models of cells populations that extends the remit of existing hybrid methods for reaction-diffusion systems. Such method is developed for a stochastic multi-scale model of tumour growth, i.e. population-dynamical models which account for the effects of intrinsic noise affecting both the number of cells and the intracellular dynamics. In order to formulate this method, we develop a coarse-grained approximation for both the full stochastic model and its mean-field limit. Such approximation involves averaging out the age-structure (which accounts for the multi-scale nature of the model) by assuming that the age distribution of the population settles onto equilibrium very fast. We then couple the coarse-grained mean-field model to the full stochastic multi-scale model. By doing so, within the mean-field region, we are neglecting noise in both cell numbers (population) and their birth rates (structure). This implies that, in addition to the issues that arise in stochastic-reaction diffusion systems, we need to account for the age-structure of the population when attempting to couple both descriptions. We exploit our coarse-graining model so that, within the mean-field region, the age-distribution is in equilibrium and we know its explicit form. This allows us to couple both domains consistently, as upon transference of cells from the mean-field to the stochastic region, we sample the equilibrium age distribution. Furthermore, our method allows us to investigate the effects of intracellular noise, i.e. fluctuations of the birth rate, on collective properties such as travelling wave velocity. We show that the combination of population and birth-rate noise gives rise to large fluctuations of the birth rate in the region at the leading edge of front, which cannot be accounted for by the coarse-grained model. Such fluctuations have non-trivial effects on the wave velocity. Beyond the development of a new hybrid method, we thus conclude that birth-rate fluctuations are central to a quantitatively accurate description of invasive phenomena such as tumour growth