Ripley's K-, H-, and L-functions are used increasingly to identify clustering of proteins in membrane microdomains. In this approach, aggregation (or clustering) is identified if the average number ...of proteins within a distance r of another protein is statistically greater than that expected for a random distribution. However, it is not entirely clear how the function may be used to quantitatively determine the size of domains in which clustering occurs. Here, we evaluate the extent to which the domain radius can be determined by different interpretations of Ripley's K-statistic in a theoretical, idealized context. We also evaluate the measures for noisy experimental data and use Monte Carlo simulations to separate the effects of different types of experimental noise. We find that the radius of maximal aggregation approximates the domain radius, while identifying the domain boundary with the minimum of the derivative of H(r) is highly accurate in idealized conditions. The accuracy of both measures is impacted by the noise present in experimental data; for example, here, the presence of a large fraction of particles distributed as monomers and interdomain interactions. These findings help to delineate the limitations and potential of Ripley's K in real-life scenarios.
Prostate cancer develops through a stochastic mechanism whereby precancerous lesions on occasion progress to multifocal adenocarcinoma. Analysis of human benign and cancer prostate tissues revealed ...heterogeneous loss of TGF-β signaling in the cancer-associated stromal fibroblastic cell compartment. To test the hypothesis that prostate cancer progression is dependent on the heterogeneous TGF-β responsive microenvironment, a tissue recombination experiment was designed in which the ratio of TGF-β responsive and nonresponsive stromal cells was varied. Although 100% TGF-β responsive stromal cells supported benign prostate growth and 100% TGF-β nonresponsive stromal cells resulted in precancerous lesions, only the mixture of TGF-β responsive and nonresponsive stromal cells resulted in adenocarcinoma. A computational model was used to resolve a mechanism of tumorigenic progression in which proliferation and invasion occur in two independent steps mediated by distinct stromally derived paracrine signals produced by TGF-β nonresponsive and responsive stromal cells. Complex spatial relationships of stromal and epithelial cells were incorporated into the model on the basis of experimental data. Informed by incorporation of experimentally derived spatial parameters for complex stromal-epithelial relationships, the computational model indicated ranges for the relative production of paracrine factors by each cell type and provided bounds for the diffusive range of the molecules. Because SDF-1 satisfied model predictions for an invasion-promoting paracrine factor, a more focused computational model was subsequently used to investigate whether SDF-1 was the invasion signal. Simulations replicating SDF-1 expression data revealed the requirement for cooperative SDF-1 expression, a prediction supported biologically by heterotypic stromal interleukin-1β signaling between fibroblastic cell populations. The cancer stromal field effect supports a functional role for the unaltered fibroblasts as a cooperative mediator of cancer progression.
In mid-October 2014, the number of cases of the West Africa Ebola virus epidemic in Guinea, Sierra Leone and Liberia exceeded 9,000 cases. The early growth dynamics of the epidemic has been ...qualitatively different for each of the three countries. However, it is important to understand these disparate dynamics as trends of a single epidemic spread over regions with similar geographic and cultural aspects, with likely common parameters for transmission rates and reproduction number R0.
We combine a discrete, stochastic SEIR model with a three-scale community network model to demonstrate that the different regional trends may be explained by different community mixing rates. Heuristically, the effect of different community mixing rates may be understood as the observation that two individuals infected by the same chain of transmission are more likely to share the same contacts in a less-mixed community. Local saturation effects occur as the contacts of an infected individual are more likely to already be exposed by the same chain of transmission.
The effects of community mixing, together with stochastic effects, can explain the qualitative difference in the growth of Ebola virus cases in each country, and why the probability of large outbreaks may have recently increased. An increase in the rate of Ebola cases in Guinea in late August, and a local fitting of the transient dynamics of the Ebola cases in Liberia, suggests that the epidemic in Liberia has been more severe, and the epidemic in Guinea is worsening, due to discrete seeding events as the epidemic spreads into new communities.
A relatively simple network model provides insight on the role of local effects such as saturation that would be difficult to otherwise quantify. Our results predict that exponential growth of an epidemic is driven by the exposure of new communities, underscoring the importance of limiting this spread.
Förster resonance energy transfer (FRET) has become an important tool to study the submicrometer distribution of proteins and lipids in membranes. Although resolving the two-dimensional distribution ...of fluorophores from FRET is generally underdetermined, a forward approach can be used to determine characteristic FRET “signatures” for interesting classes of microdomain organizations. As a first step toward this goal, we use a stochastic Monte Carlo approach to characterize FRET in the case of molecules randomly distributed within disk-shaped domains. We find that when donors and acceptors are confined within domains, FRET depends very generally on the density of acceptors within domains. An implication of this result is that two domain populations with the same acceptor density cannot be distinguished by this FRET approach even if the domains have different diameters or different numbers of molecules. In contrast, both the domain diameter and molecule number can be resolved by combining this approach with a segregation approach that measures FRET between donors confined in domains and acceptors localized outside domains. These findings delimit where the inverse problem is tractable for this class of distributions and reframe ways FRET can be used to characterize the structure of microdomains such as lipid rafts.
We present a stochastic cellular automaton model for the behavior of limb bud precartilage mesenchymal cells undergoing chondrogenic patterning. This “agent-oriented” model represents cells by points ...on a lattice that obey rules motivated by experimental findings. The “cells” follow these rules as autonomous agents, interacting with other cells and with the microenvironments cell activities produce. The rules include random cell motion, production and lateral deposition of a substrate adhesion molecule (SAM, corresponding to fibronectin), production and release of a diffusible growth factor (“activator,” corresponding to TGF-β) that stimulates production of the SAM, and another diffusible factor (“inhibitor”) that suppresses the activity of the activator. We implemented the cellular automaton on a two-dimensional (2D) square lattice to emulate the quasi-2D micromass culture extensively used to study patterning in avian limb bud precartilage cells. We identified parameters that produce nodular patterns that resemble, in size and distribution, cell condensations in leg-cell cultures, thus establishing a correspondence between in vitro and in silico results. We then studied the in vitro and in silico micromass cultures experimentally. We altered the standard in vitro micromass culture by diluting the initial cell density, transiently exposing it to exogenous activator, suppressing the inhibitor, and constitutively activating fibronectin production. We altered the standard in silico micromass culture in each case by changing the corresponding parameter. In vitro and in silico experiments agreed well. We also used the model to test hypotheses for differences in the in vitro patterns of cells derived from chick embryo forelimb and hindlimb. We discuss the applicability of this model to limb development in vivo and to other organ development.
A lattice gas cellular automation (LGCA) model is used to simulate rippling and aggregation in myxobacteria. An efficient way of representing cells of different cell size, shape and orientation is ...presented that may be easily extended to model later stages of fruiting body formation. This LGCA model is designed to investigate whether a refractory period, a minimum response time, a maximum oscillation period and non-linear dependence of reversals of cells on C-factor are necessary assumptions for rippling. It is shown that a refractory period of 2–3
min, a minimum response time of up to 1
min and no maximum oscillation period best reproduce rippling in the experiments of
Myxococcus xanthus. Non-linear dependence of reversals on C-factor is critical at high cell density. Quantitative simulations demonstrate that the increase in wavelength of ripples when a culture is diluted with non-signaling cells can be explained entirely by the decreased density of C-signaling cells. This result further supports the hypothesis that levels of C-signaling quantitatively depend on and modulate cell density. Analysis of the interpenetrating high density waves shows the presence of a phase shift analogous to the phase shift of interpenetrating solitons. Finally, a model for swarming, aggregation and early fruiting body formation is presented.
Despite a very effective vaccine, active conflict and community distrust during the ongoing DRC Ebola epidemic are undermining control efforts, including a ring vaccination strategy that requires the ...prompt immunization of close contacts of infected individuals. However, in April 2019, it was reported 20% or more of close contacts cannot be reached or refuse vaccination, and it is predicted that the ring vaccination strategy would not be effective with such a high level of inaccessibility. The vaccination strategy is now incorporating a "third ring" community-level vaccination that targets members of communities even if they are not known contacts of Ebola cases. To assess the impact of vaccination strategies for controlling Ebola epidemics in the context of variable levels of community accessibility, we employed an individual-level stochastic transmission model that incorporates four sources of heterogeneity: a proportion of the population is inaccessible for contact tracing and vaccination due to lack of confidence in interventions or geographic inaccessibility, two levels of population mixing resembling household and community transmission, two types of vaccine doses with different time periods until immunity, and transmission rates that depend on spatial distance. Our results indicate that a ring vaccination strategy alone would not be effective for containing the epidemic in the context of significant delays to vaccinating contacts even for low levels of household inaccessibility and affirm the positive impact of a supplemental community vaccination strategy. Our key results are that as levels of inaccessibility increase, there is a qualitative change in the effectiveness of the vaccination strategy. For higher levels of vaccine access, the probability that the epidemic will end steadily increases over time, even if probabilities are lower than they would be otherwise with full community participation. For levels of vaccine access that are too low, however, the vaccination strategies are not expected to be successful in ending the epidemic even though they help lower incidence levels, which saves lives, and makes the epidemic easier to contain and reduces spread to other communities. This qualitative change occurs for both types of vaccination strategies: ring vaccination is effective for containing an outbreak until the levels of inaccessibility exceeds approximately 10% in the context of significant delays to vaccinating contacts, a combined ring and community vaccination strategy is effective until the levels of inaccessibility exceeds approximately 50%. More broadly, our results underscore the need to enhance community engagement to public health interventions in order to enhance the effectiveness of control interventions to ensure outbreak containment.
The mechanisms behind the sub-exponential growth dynamics of the West Africa Ebola virus disease epidemic could be related to improved control of the epidemic and the result of reduced disease ...transmission in spatially constrained contact structures. An individual-based, stochastic network model is used to model immediate and delayed epidemic control in the context of social contact networks and investigate the extent to which the relative role of these factors may be determined during an outbreak. We find that in general, epidemics quickly establish a dynamic equilibrium of infections in the form of a wave of fixed size and speed traveling through the contact network. Both greater epidemic control and limited community mixing decrease the size of an infectious wave. However, for a fixed wave size, epidemic control (in contrast with limited community mixing) results in lower community saturation and a wave that moves more quickly through the contact network. We also found that the level of epidemic control has a disproportionately greater reductive effect on larger waves, so that a small wave requires nearly as much epidemic control as a larger wave to end an epidemic.
Genotypic variation, environmental variation, and their interaction may produce variation in the developmental process and cause phenotypic differences among individuals. Developmental noise, which ...arises during development from stochasticity in cellular and molecular processes when genotype and environment are fixed, also contributes to phenotypic variation. While evolutionary biology has long focused on teasing apart the relative contribution of genes and environment to phenotypic variation, our understanding of the role of developmental noise has lagged due to technical difficulties in directly measuring the contribution of developmental noise. The influence of developmental noise is likely underestimated in studies of phenotypic variation due to intrinsic mechanisms within organisms that stabilize phenotypes and decrease variation. Since we are just beginning to appreciate the extent to which phenotypic variation due to stochasticity is potentially adaptive, the contribution of developmental noise to phenotypic variation must be separated and measured to fully understand its role in evolution. Here, we show that variation in the component of the developmental process corresponding to environmental and genetic factors (here treated together as a unit called the LALI-type) versus the contribution of developmental noise, can be distinguished for leopard gecko (Eublepharis macularius) head color patterns using mathematical simulations that model the role of random variation (corresponding to developmental noise) in patterning. Specifically, we modified the parameters of simulations corresponding to variation in the LALI-type to generate the full range of phenotypic variation in color pattern seen on the heads of eight leopard geckos. We observed that over the range of these parameters, variation in color pattern due to LALI-type variation exceeds that due to developmental noise in the studied gecko cohort. However, the effect of developmental noise on patterning is also substantial. Our approach addresses one of the major goals of evolutionary biology: to quantify the role of stochasticity in shaping phenotypic variation.
Animal color patterns are widely studied in ecology, evolution, and through mathematical modeling. Patterns may vary among distinct body parts such as the head, trunk or tail. As large amounts of ...photographic data is becoming more easily available, there is a growing need for general quantitative methods for capturing and analyzing the full complexity and details of pattern variation. Detailed information on variation in color pattern elements is necessary to understand how patterns are produced and established during development, and which evolutionary forces may constrain such a variation. Here, we develop an approach to capture and analyze variation in melanistic color pattern elements in leopard geckos. We use this data to study the variation among different body parts of leopard geckos and to draw inferences about their development. We compare patterns using 14 different indices such as the ratio of melanistic versus total area, the ellipticity of spots, and the size of spots and use these to define a composite distance between two patterns. Pattern presence/absence among the different body parts indicates a clear pathway of pattern establishment from the head to the back legs. Together with weak within-individual correlation between leg patterns and main body patterns, this suggests that pattern establishment in the head and tail may be independent from the rest of the body. We found that patterns vary greatest in size and density of the spots among body parts and individuals, but little in their average shapes. We also found a correlation between the melanistic patterns of the two front legs, as well as the two back legs, and also between the head, tail and trunk, especially for the density and size of the spots, but not their shape or inter-spot distance. Our data collection and analysis approach can be applied to other organisms to study variation in color patterns between body parts and to address questions on pattern formation and establishment in animals.
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