L3

Simulating the long time and length scales associated with DNA self-assembly
and DNA nanotechnology is not currently feasible with models at an atomic level
of detail. We, therefore, developed oxDNA a coarse-grained representation of
DNA that aims to capture the fundamental structural, thermodynamic and
mechanical properties of double-stranded and single-stranded DNA, which we have
subsequently applied to study a wide variety of DNA biophysical properties and
DNA nanotechnological systems.

Ductal carcinoma is one of the most common cancers among women, and the main cause of death is the formation of metastases. The development of metastases is caused by cancer cells that migrate from the primary tumour site (the mammary duct) through the blood vessels and extravasating they initiate metastasis. In my talk, I present a multi-compartment mathematical model which mimics the dynamics of tumoural cells in the mammary duct, in the circulatory system and in the bone. Using a branching process approach, the model describes the relation between the survival times and the four markers mainly involved in metastatic breast cancer (EPCAM, CD47, CD44 and MET). In particular, the model takes into account the gene expression profile of circulating tumour cells to predict personalised survival probability. Gene expression data of metastatic breast cancer have been used to validate the model. The administration of drugs as bisphosphonates is also included in order to analyse the dynamic changes induced by the therapy.

Stochastic and deterministic processes are merged together to describe cancer progression and obtain personalised survival analysis based on the gene expression levels of each patient. The main aim of the talk is showing that Mathematics can have a strong impact in speeding cancer research, predicting survival probability and selecting the best cancer treatment. 

The first half of the lecture will begin by reviewing what is known about the
neural representation of faces in the primate visual system. How does the
visual system represent the spatial structure of faces, facial identity and
expression? We then discuss how depression is associated with negative
cognitive biases in the recognition of facial expression, whereby depressed
people interpret facial expressions more negatively. The second half of the
lecture presents computer simulations aimed at understanding how these facial
representations may develop through visual experience. We show how neural
representations of expression are linked to particular spatial relationships
between facial features. Building on this, we show how the synaptic connections
in the model may be rewired by visual training to eliminate the negative
cognitive biases seen in depression.

It is well known that stochasticity can play a fundamental role in 
various biochemical processes, such as cell regulatory networks and 
enzyme cascades. Isothermal, well-mixed systems can be adequately 
modeled by Markov processes and, for such systems, methods such as 
Gillespie's algorithm are typically employed. While such schemes are 
easy to implement and are exact, the computational cost of simulating 
such systems can become prohibitive as the frequency of the reaction 
events increases. This has motivated numerous coarse grained schemes, 
where the ``fast'' reactions are approximated either using Langevin 
dynamics or deterministically.  While such approaches provide a good 
approximation for systems where all reactants are present in large 
concentrations,  the approximation breaks down when the fast chemical 
species exist in small concentrations,  giving rise to significant 
errors in the simulation.  This is particularly problematic when using 
such methods to compute statistics of extinction times for chemical 
species, as well as computing observables of cell cycle models.  In this 
talk, we present a hybrid scheme for simulating well-mixed stochastic 
kinetics, using Gillepsie--type dynamics to simulate the network in 
regions of low reactant concentration, and chemical langevin dynamics 
when the concentrations of all species is large.  These two regimes are 
coupled via an intermediate region in which a ``blended'' jump-diffusion 
model is introduced.  Examples of gene regulatory networks involving 
reactions occurring at multiple scales, as well as a cell-cycle model 
are simulated, using the exact and hybrid scheme, and compared, both in 
terms weak error, as well as computational cost.

This is joint work with A. Duncan (Imperial) and R. Erban (Oxford)

Mammalian lung morphology is well optimized for efficient bulk transport of gases, yet most lung morphogenesis occurs prenatally, when the lung is filled with liquid - and at birth it is immediately ready to function when filled with gas. Lung morphogenesis is regulated by numerous mechanical inputs including fluid secretion, fetal breathing movements, and peristalsis. We generally understand which of these broad mechanisms apply, and whether they increase or decrease overall size and/or branching. However, we do not generally have a clear understanding of the intermediate mechanisms actuating the morphogenetic control. We have studied this aspect of lung morphogenesis from several angles using mathematical/mechanical/transport models tailored to specific questions. How does lumen pressure interact with different locations and tissues in the lung? Is static pressure equivalent to dynamic pressure? Of the many plausible cellular mechanisms of mechanosensing in the prenatal lung, which are compatible with the actual mechanical situation? We will present our models and results which suggest that some hypothesized intermediate mechanisms are not as plausible as they at first seem.

Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum. 

We consider the impact of spatial heterogeneities on the dynamics of 
localized patterns in systems of partial differential equations (in one 
spatial dimension). We will mostly focus on the most simple possible 
heterogeneity: a small jump-like defect that appears in models in which 
some parameters change in value as the spatial variable x crosses 
through a critical value -- which can be due to natural inhomogeneities, 
as is typically the case in ecological models, or can be imposed on the 
model for engineering purposes, as in Josephson junctions. Even such a 
small, simplified heterogeneity may have a crucial impact on the 
dynamics of the PDE. We will especially consider the effect of the 
heterogeneity on the existence of defect solutions, which boils down to 
finding heteroclinic (or homoclinic) orbits in an n-dimensional 
dynamical system in `time' x, for which the vector field for x > 0 
differs slightly from that for x < 0 (under the assumption that there is 
such an orbit in the homogeneous problem). Both the dimension of the 
problem and the nature of the linearized system near the limit points 
have a remarkably rich impact on the defect solutions. We complement the 
general approach by considering two explicit examples: a heterogeneous 
extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous 
generalized FitzHugh–Nagumo system (n = 6).

We study the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters such as strength of protrusion by actin filaments and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion perturbed by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters. In the subcritical regime, the well-posedness of the problem is proved (M. Mizuhara et al., 2015). Our main focus is the supercritical regime where we established surprising features of the motion of the interface such as discontinuities of velocities and hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in the 2D model. Because of properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. This is joint work with V. Rybalko and M. Potomkin.