Multiscale modeling approach typical of systems biology tends to mix continuous, discrete, \
deterministic, and probabilistic submodels. To prevent the loss of blood following a break \
in blood vessels, components in blood and the vessel wall interact rapidly to form a clot t\
o limit hemorrhage. In this talk we will describe a multiscale hybrid model of thrombus for\
mation consisting of components for modeling viscous, incompressible blood plasma; coagulat\
ion pathway; quiescent and activated platelets; blood cells; activating chemicals; fibrinog\
en; the vessel walls and their interactions. At macro scale blood flow is described by the \
incompressible Navier-Stokes equations and is numerically solved using the projection metho\
d. At micro scale, cell movement, cell-cell adhesion, cell-flow and cell-vessel wall intera\
ctions are described through an extended stochastic discrete Cellular Potts Model (CPM). Si\
mulation results show development of an inhomogeneous internal structure of the clot confir\
med by the preliminary experimental data. It is also demonstrated that dependence of the cl\
ot size on the blood flow rate in simulations is close to the one observed experimentally.

In the second half of the talk a continuous limit will be discussed of a two-dimensional st\
ochastic CPM describing cells moving in a medium and reacting to each other through direct \
contact, cell-cell adhesion, and long range chemotaxis. Contrary to classical Keller-Segel \
model, solutions of the obtained equation do not collapse in finite time and can be used ev\
en when relative volume occupied by cells is quite large. A very good agreement was demonst\
rated between CPM Monte Carlo simulations and numerical solutions of the obtained macroscop\
ic nonlinear diffusion equation. Combination of microscopic and macroscopic models was used\
to simulate growth of structures similar to early vascular networks.

Xu, Z., Chen, N., Kamocka, M.M., Rosen, E.D., and M.S. Alber [2008], Multiscale Model of Th\
rombus Development, Journal of the Royal Society Interface 5 705-722.

Alber, M., Chen, N., Lushnikov, P., and S. Newman [2007], Continuous macroscopic limit of a\
discrete stochastic model for interaction of living cells, Physical Review Letters 99 1681\
02.

Lushnikov, P.P., Chen, N., and M.S. Alber, Macroscopic dynamics of biological cells interac\
ting via chemotaxis and direct contact, Physical Review E (to appear).

I will discuss the difficulties which arise when one considers the
long time behavior of a stochastically forced PDE. I will try to
highlight that there are different cases which require very different
ideas. Some cases can be seen as extensions of what is done in finite
dimensions, others require new tools and ideas. I will concentrate on
the case of degenerately forced SPDEs. I will describe an extension of
Hormander's "sum of squares theorem" to hypo-elliptic operators in
infinite dimensions. I will discuss the concert examples of the 2D
Navier Stokes equations on the torus and sphere as well as a class of
reaction diffusion equations. In these contexts the discussion will
center on the transfer of randomness between scales.

he central axis of the famous DNA double helix is
often topologically constrained or even circular.
The topology of this axis can influence which
proteins interact with the underlying DNA. Subsequently, in all cells there are
proteins whose primary function is to change the DNA axis
topology -- for example converting a torus link into an unknot.
Additionally, there are several protein families that change the axis
topology as a by-product of their interaction with DNA.

This talk will describe typical DNA conformations, and the families of
proteins that change these conformations. I'll present a few examples
illustrating how 3-manifold topology has been useful in understanding certain
DNA-protein interactions, and discuss the most common topological techniques
used to attack these problems.

Reflectionless measures are interesting objects because
they arise as limiting measures of spectral measures of arbitrary
Schr"odinger operators with some absolutely continuous spectrum.
In this talk, I'd like to review the definition and some background
material and then discuss more recent work, joint with Alexei Poltoratski,
on reflectionless measures.