Cell sorting with variable cluster size: a Smoluchowski equation approach

Cell segregation is an widespread phenomena in nature and has
interested physicists since the last 50 years. It opens the
possibility of studying a system composed of many interacting active
identical elements, both theoretically and experimentally. A typical
cell sorting experiment measures the evolution of clusters sizes, or
also the size of the interface between the two tissues at stake. The
dynamics underlying cell migration drives the cell segregation, which
is directly related to cluster formation, where the endoderm cells
attach each other forming groups. This development evolves through
cluster diffusion and depends on cluster cross section and cell
affinity. In the context of active media cluster growth may present
unexpected exponents when compared to non-active matter. When clusters
are formed by inert particles it is expected that the diffusion scales
inversely with the cluster mass, in the case of active matter that
does not hold and this is central to define the segregation time
scales. Also, finite size effects are important since they impose
deviation from power law solutions. To approach this problem from a
theoretical point of view we use the Smoluchowski
fragmentation-coagulation equation with an adapted coagulation kernel
and a fragmentation kernel. It is found that the underlying growth
power laws may be hidden depending on initial cluster sizes, sample
size and fragmentation constant. The average cluster size solutions
found with the Smoluchowski equation are used to fit the data
resulting from the simulations and the power law behavior can be
clearly separated out of the minimum and maximum cluster size limits.