Mathematical Biology

The Emergence of Spatial Patterns for Diffusion-Coupled Compartments with Activator-Inhibitor Kinetics in 1-D and 2-D

Speaker: 
Merlin Pelz
Date: 
Wed, Jan 11, 2023
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

Since Alan Turing's pioneering publication on morphogenetic pattern formation obtained with reaction-diffusion (RD) systems, it has been the prevailing belief that two-component reaction diffusion systems have to include a fast diffusing inhibiting component (inhibitor) and a much slower diffusing activating component (activator) in order to break symmetry from a uniform steady-state. This time-scale separation is often unbiological for cell signal transduction pathways.
We modify the traditional RD paradigm by considering nonlinear reaction kinetics only inside compartments with reactive boundary conditions to the extra-compartmental space that provides a two-species diffusive coupling. The construction of a nonlinear algebraic system for all existing steady-states enables us to derive a globally coupled matrix eigenvalue problem for the growth rates of eigenperturbations from the symmetric steady-state, on finite domains in 1-D and 2-D and a periodically extended version in 1-D.

We show that the membrane reaction rate ratio of inhibitor rate to activator rate is a key bifurcation parameter leading to robust symmetry-breaking of the compartments. Illustrated with Gierer-Meinhardt, FitzHugh-Nagumo and Rauch-Millonas intra-compartmental reaction kinetics, our compartmental-reaction diffusion system does not require diffusion of inhibitor and activator on vastly different time scales.
Our results elucidate a possible mechanism of the ubiquitous biological cell specialization observed in nature.

Class: 

Influence of the endothelial surface layer on the motion of red blood cells

Speaker: 
Ying Zhang
Date: 
Wed, Oct 26, 2022
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

The endothelial lining of blood vessels presents a large surface area for exchanging materials between blood and tissues. The endothelial surface layer (ESL) plays a critical role in regulating vascular permeability, hindering leukocyte adhesion as well as inhibiting coagulation during inflammation. Changes in the ESL structure are believed to cause vascular hyperpermeability and induce thrombus formation during sepsis. In addition, ESL topography is relevant for the interactions between red blood cells (RBCs) and the vessel wall, including the wall-induced migration of RBCs and formation of a cell-free layer. To investigate the influence of the ESL on the motion of RBCs, we construct two models to represent the ESL using the immersed boundary method in two dimensions. In particular, we use simulations to study how lift force and drag force change over time when a RBC is placed close to the ESL as thethickness, spatial variation, and permeability of the ESL vary. We find that spatial variation has a significant effect on the wall-induced migration of the RBC when the ESL is highly permeable and that the wall-induced migration can be significantly inhibited by the presence of a thick ESL.

Class: 

Agent-based models: from bacterial aggregation to wealth hot-spots

Speaker: 
Theodore Kolokolnikov
Date: 
Wed, Oct 19, 2022
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

Agent-based models are widely used in numerous applications. They have an advantage of being easy to formulate and to implement on a computer. On the other hand, to get any mathematical insight (motivated by, but going beyond computer simulations) often requires looking at the continuum limit where the number of agents becomes large. In this talk I give several examples of agent- based modelling, including bacterial aggregation, spatio-temporal SIR model, and wealth hotspots in society; starting from their derivation to taking their continuum limit, to analysis of the resulting continuum equations.

Class: 

Resource-mediated competition between two plant species with different rates of water intake

Speaker: 
Chunyi Gai
Date: 
Fri, Oct 14, 2022
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

We propose an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. Rather than competing directly, the plants compete through their intake of water, which is a shared resource between them. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects the co-existence and stability of patches of different plant species. We consider two plant types: a “thirsty” species and a “frugal” species, that only differ by the amount of water they consume per unit growth, while being identical in other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is a sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. The pattern consisting of only frugal spots is the most resilient. The next-most-resilient pattern consists of all-thirsty patches, with the mixed pattern being less resilient than either of the homogeneous patterns. We also examine numerically what happens for very large precipitation rates. We find that for a sufficiently high rate, the frugal plant takes over the entire range, outcompeting the thirsty plant.

Class: 

Actomyosin cables by mechanical self-organization

Speaker: 
Mingfeng Qiu
Date: 
Wed, Oct 5, 2022
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

Supracellular actomyosin cables often drive morphogenesis in development. The origin of these cables is poorly understood. We show theoretically and computationally that under external loading, cell-cell junctions capable of mechanical feedback could undergo spontaneous symmetry breaking and establish a dominant path through which tension propagates, giving rise to a contractile cable. This type of cables transmit force perturbation over a long range, and can be modulated by the tissue properties and the external loading magnitude. Our theory is general and highlights the potential role of mechanical signals in guiding development.

Class: 

Rotary Molecular Motors Driven By Transmembrane Ionic Currents

Speaker: 
Charles S Peskin
Date: 
Wed, Sep 28, 2022
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

There are two rotary motors in biology, ATP synthase and the bacterial flagellar motor. Both are driven by transmembrane ionic currents. We consider an idealized model of such a motor, essentially an electrostatic turbine. The model has a rotor and a stator, which are closely fitting cylinders. Attached to the rotor is a fixed density of negative charge, with helical symmetry. Positive ions move longitudinally by drift and diffusion on the stator. A key assumption is local electroneutrality of the combined charge distribution. With this setup we derive explicit formulae for the transmembrane current and the angular velocity of the rotor in terms of the transmembrane electrochemical potential difference of the positive ions and the mechanical torque on the motor. This relationship between "forces" and "fluxes" turns out to be linear, and given by a symmetric positive definite matrix, as anticipated by non-equilibrium thermodynamics, although we do not make any use of that formalism in deriving the result. The equal off-diagonal terms of this 2x2 matrix describe the electromechanical coupling of the motor. Although macroscopic, the model can be used as a foundation for stochastic simulation via the Einstein relation.

Class: 

Agent-based modelling and topological data analysis of zebrafish patterns

Speaker: 
Bjorn Sandstede
Date: 
Wed, Sep 21, 2022
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
Mathematical Biology Seminar
Abstract: 

Patterns are widespread in nature and often form during early development due to the self-organization of cells or other independent agents. One example are zebrafish (Danio rerio): wild-type zebrafish have regular black and gold stripes, while mutants and other fish feature spotty and patchy patterns. Qualitatively, these patterns display impressive consistency and redundancy, yet variability inevitably exists on both microscopic and macroscopic scales. I will first discuss an agent-based model that suggests that both consistency and richness of patterning on zebrafish stems from the presence of redundancy in iridophore interactions. In the second part of my talk, I will focus on how we can quantify features and variability of patterns to facilitate predictive analyses. I will discuss an approach based on topological data analysis for quantifying both agent-level features and global pattern attributes on a large scale. The proposed methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns and predicts stripe and spot statistics as a function of varying cellular communication. This is joint work with Alexandria Volkening and Melissa McGuirl.

Class: 

Combined modeling and experimental study of the interplay between tissue growth and shape regulation during Drosophila wing disc development

Speaker: 
Mark Alber
Date: 
Wed, Sep 7, 2022
Location: 
UBC, Vancouver, Canada
Online
Conference: 
UBC Math Biology Seminar Series
Abstract: 

The regulation and maintenance of an organ’s shape is a major outstanding question in developmental biology. The Drosophila wing imaginal disc serves as a powerful system for elucidating design principles of the shape formation in epithelial morphogenesis. Yet, even simple epithelial systems such as the wing disc are extremely complex. A tissue’s shape emerges from the integration of many biochemical and biophysical interactions between proteins, subcellular components, and cell-cell and cell-ECM interactions. How cellular mechanical properties affect tissue size and patterning of cell identities on the apical surface of the wing disc pouch has been intensively investigated. However, less effort has focused on studying the mechanisms governing the shape of the wing disc in the cross-section. Both the significance and difficulty of such studies are due in part to the need to consider the composite nature of the material consisting of multiple cell layers and cell-ECM interactions as well as the elongated shape of columnar cells. Results obtained using iterative approach combining multiscale computational modelling and quantitative experimental approach will be used in this talk to discuss direct and indirect roles of subcellular mechanical forces, nuclear positioning, and extracellular matrix in shaping the major axis of the wing pouch during the larval stage in fruit flies, which serves as a prototypical system for investigating epithelial morphogenesis. The research findings demonstrate that subcellular mechanical forces can effectively generate the curved tissue profile, while extracellular matrix is necessary for preserving the bent shape even in the absence of subcellular mechanical forces once the shape is generated. The developed integrated multiscale modelling environment can be readily extended to generate and test hypothesized novel mechanisms of developmental dynamics of other systems, including organoids that consist of several cellular and extracellular matrix layers.

Class: 

Adventures with Partial Identification in Studies of Marked Individuals

Speaker: 
Simon Bonner
Date: 
Thu, Mar 17, 2022
Location: 
PIMS, University of Victoria
Online
Zoom
Conference: 
PIMS-UVic Distinguished Colloquium
Abstract: 

Monitoring marked individuals is a common strategy in studies of wild animals (referred to as mark-recapture or capture-recapture experiments) and hard to track human populations (referred to as multi-list methods or multiple-systems estimation). A standard assumption of these techniques is that individuals can be identified uniquely and without error, but this can be violated in many ways. In some cases, it may not be possible to identify individuals uniquely because of the study design or the choice of marks. Other times, errors may occur so that individuals are incorrectly identified. I will discuss work with my collaborators over the past 10 years developing methods to account for problems that arise when are only individuals are only partially identified. I will present theoretical aspects of this research, including an introduction to the latent multinomial model and algebraic statistics, and also describe applications to studies of species ranging from the golden mantella (an endangered frog endemic to Madagascar measuring only 20 mm) to the whale shark (the largest know species of fish, measuring up to 19m).

Class: 

Directional sensing and signal integration by immune cells

Speaker: 
Sean Collins
Date: 
Wed, Mar 23, 2022
Location: 
Zoom
Online
Conference: 
Mathematical Biology Seminar
Abstract: 

Human neutrophils and other immune cells sense chemical gradients to navigate to sites of injury, infection, and inflammation in the body. Impressively, these cells can detect gradients that differ by as little as about 1% in concentration across the length of the cell. Abstract models suggest that they may do this by integrating opposing local positive and long-range negative signals generated by receptors. However, the molecular basis for signal processing remains unclear. To investigate models of sensing, we developed experimental tools to control receptors with light while measuring downstream signaling responses with spatial resolution in single cells. We are directly measuring responses to both local and cell-wide receptor activation to determine the wiring of signal processing. While we do not see evidence for long-range negative signals, we do see a subcellular context-dependence of signal transmission. We propose that signal transmission from receptors happens locally, but cell-wide polarity biases sensing to maintain persistent migration and achieve temporal averaging to promote directional accuracy.

Class: 

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