Example of molecular dynamics representation of cell collision using Protomol. The tests in the results section were designed to illustrate elasticity and adhesion of cells and propose methods of coupling the flow and MD codes.

image:cellcollide.png

Results

The following movies were rendered from the generated Protomol trajectories using VisualMolecularDynamics (University of Illinois at Urbana-Champagn).

Each cell has 20 elements with an inter-element (cytoskeleton) harmonic constant k, element effective diameter of 1.15?, and Lennard-Jones potential 'depth' (adhesion) ?.

Adhesion/Elasticity

Parameters for low adhesion are k = 50 and ? = ? 0.2:

Parameters for high adhesion k = 50, ? = ? 5.0:

Parameters for soft cells/low adhesion k = 10, ? = ? 0.2:

Velocity coupling

As an alternative submerged particles, we can couple the SCE dynamics (SCED) to flow in a similar manner to the current CPM coupling of \Delta E=k_f \Delta d.v_f\,. The CPM coupling has no steady state solution (a better method would be \Delta E=k_f (v_f-v_c).\Delta d\,), so must be modified for SCED. I propose the use of a "Stokes-Langevin" approach where the flow is coupled using the local fluid velocity vf, the fluid damping ? and a stochastic term that satisfies the fluctuation-dissipation theorem. Then

\mathrm{d}\mathbf{v}_c=-\mathbf{M}^{-1}\nabla\mathbf{U}(\mathbf{x})\mathrm{dt}+\gamma(\mathbf{v}_f-\mathbf{v}_c)\mathrm{dt}+\sqrt{2k_{\mathrm{B}}T\gamma}\mathbf{M}^{-\frac{1}{2}}\mathrm{d}\mathbf{W}(\mathrm{t}),

where \mathbf{W}(\mathrm{t}) is a collection of Wiener processes, \mathbf{M} is the diagonal mass matrix and -\nabla\mathbf{U}(\mathbf{x}) are the system forces. For complete notation see the Langevin Leapfrog write-up. In the following simulation the system is solved using the Langevin Leapfrog method. Each SCE is coupled directly to the fluid, ignoring the "shielding" effect of the cell body/membrane, as follows:

A more realistic interaction can be achieved by treating the fluid velocity as if scaled along the vector connecting the SCE to the cell center (sc\,), if the dot product is positive. Here, given u_s=\frac{sc.v_f}{||sc||~||v_f||}, then v_f'=v_f u_s\, if u_s>0\, else v_f'=0\,, with the following results:

The fluid damping could be represented by a tensor ? rather than a scalar ?. The system diffusion tensor D gives rise to ?

\Gamma=\frac{1}{k_bT}D^{-1}M^{-1},

for diagonal mass matrix M.

Input files

Scripts

Blood Cells

Jacob Wenger (CS Undergraduate) has been responsible for writing the python scripts for generating the cell input files for Protomol. Currently he is constructing blood cells, the latest example is shown below.

Breaking news!

We now have colliding blood cells.

Blood Vessel

Some initial results of clot formation in a cylindrical vessel using SCE have been done using Langevin coupling to a flow field.