Multi+ca+predefined

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 * TODO List**
 * REF Force vital
 * REF models of cell reorganization 60's - 90's
 * REF of hydra remixing, and REFs showing they do not

=Multi-level CA: predefined=

In some cases models are formulated with predefined multi-level properties in order to study the impact of such multiple scales or levels. In that case the emergence of higher level entities is assumed and it is not the point of study to determine how they arise but what impact they have once they have arisen. In terms of CAs this means that local rules are defined to be dependent on predefined higher level entities.

//[|Cellular-Potts Model]//
The most famous multi-scale CA model is what is called the Cellular-Potts Model ([|Graner and Glazier 1992]). In this case single biological cells are defined in terms of a local group of lattice sites with the same state (i.e. a certain individual cell). Such an individual cell can further be given properties that make it a particular type of cell.

[[image:binf/CA_example.png width="351" height="176" align="right"]]
In the model, biological cells are defined in terms of a volume (V) and cell type (tau). Such biological cells are assumed to conserve their volume (actual volume) relative to some ideal volume (target volume). Moreover the cell membrane (edge of cell) is assumed to bind to other cells or the medium according to energy bonds (Jij, Jim). The ingenious part of this model is therefore how the local rules depend on the mesoscale porperties of volume and surface energy minimization. This is done by minimizing the free energy with volume conservation, considering the change in energy in the system for every change at the edges of cells in terms of changes of bonds and changes in cell volume. The free energy to be minimized is: H = sum Jij/2 + sum Jim + lambda (v-V)^2 Here, the first term describes the free energy at all edges bordering another cell (of different types j), the second term describes the free energy at all edges bordering the medium, and the last term is a penalty term for deviation of the actual volume v from the target volume V. A change of cell (i.e. copying a neighbour's cell type into one's own) is then determined to happen when: delta H < -B

[[image:http://www.nature.com/nrg/journal/v10/n8/images/nrg2548-i2.jpg width="346" height="129" align="right"]]
(i.e. when H decreases sufficiently by making the change) or with a probability e^-(delta H + B)/M if delta H > -B (i.e. less-favourable changes can still happen to prevent "getting stuck at local optima", but the less-favourable they are (the larger delta H), the less likely they are to happen).It is clear that the decision to copy a lattice cell depends on energy minimization which depends on V.

The CPM provides a good description of cells and interactions with surroundings. Below, several examples of this are given.



//[|The Hydra]//
This is a special organism that after being passed through a gauze (i.e. all cells separated) can reorganize itself into something that resembles its original shape! This led up to the //[|force vital]// in biology to explain how this could occur (REF).

However, is such reorganization much different from oil and water separating after being mixed? Such thoughts led to experiments which model this from the 60's to the 90's (REFS). These attempts failed mainly because they used single pixels in CAs to represent cells and therefore could only get local clumping of cells. As a response they suggested that the original experiments with hydra were not mixed enough (REF)! (Incidentally, those experiments cannot be replicated either!)

//[|Differential] [|cell adhesion]//
With the Cellular-Potts model the 2-scaleness of the model allows for differential cell adhesion. This can achieve cell sorting because in thie model cells can squeeze past each other ([|Graner and Glazier 1992]). In fact using different settings for differential cell adhesion, various forms of cell sorting have been found and these have an interesting parallel to standard ecology competition forms: Here, Jij describes the free energy for a bond between cells of type i and j (e.g. Jii < Jij means that bonding to type i is favoured over bonding to type j, since the free energy should be minimized).
 * Jii = Jjj < Jij: inter cell type bonding less strong than within cell types gives cell sorting (competitive exclusion)
 * Jii = Jjj > Jij: inter cell type bonding stronger than within cell types gives cell mixing (coexistence)
 * Jij < Jim and Jjm < Jim: inter-cell type bonding stronger than with medium gives engulfing.

What is clear is that this 2-scale formalism is a rich formalism which generates many processes at many different scales. However although it is very appropriate to study the patterns and interactions between cells and cell types, it is not formulated in order to be informative above how cells can arise from the molecules from which they are derived.

We will discuss two more examples in which CPM was used to describe cell behaviour.

//"Moving against the flow"//
Käfer et al ([|2006]) described cell movement caused by a gradient in a tissue with two different cell types, that sort through differential cell adhesion (see above). All cells are chemotactically attracted to the source of the gradient. Hence, we would expect all cells to move in the direction of the gradient. However, we observe that some cells move "against the flow". Because the cells are in a "tight, cramped space", this movement is caused by these cells being "pushed away". This can happen if this cell type is larger, has different adhesion settings or is just in the minority to start with.

A strong example of this last point is given in the figure, where the black cells first move to the right (against the gradient), but once they have formed blobs and are locally in the majority, start moving in the direction of the gradient.

//Movement and chemotaxis in a lymph node//
Next we revisit the lymph node, and now consider the movement of T cells. In experimental cell tracking studies, it was shown that T cells move according to "stop-and-go", where cells move in a certain direction for a short amount of time, stop, change direction, move for a short amount of time etc. To study this particular moving mechanism, Beltman et al ([|2007]) used a simple 3D CPM in which they implemented one extra feature: cells had a **persistence of motion**, i.e. a preference to keep moving in the same direction.

In an empty lymph node (i.e. a space that only contains T cells), after some time all cells align and move in the same direction, forming a large "flow of cells". However, in a lymph node there are many obstacles, such as dendritic cells, extracellular matrix scaffolds etc. If this is added to the model, the alignment only happens locally and for short amounts of time, until the cells bump into an obstacle and change their direction. Then, the cells show the "stop-and-go" mechanism! Hence, this model shows that the stop-and-go movement can be explained by simple persistence of motion in a cramped environment with obstacles, and that no internal regulation (e.g. an internal clock) is needed to explain this behaviour.

Finally, we can include chemotaxis into the model, for instance to attract T cells to antigen-presenting dendritic cells (DCs). We might expect chemotaxis to improve the scanning of DCs by T cells, because the T cells can more easily find the DCs. However, when Riggs et al ([|2008]) made a CA-like model of this process, they found that a random search strategy is optimal, and that chemotaxis actually hinders the scanning process. Why does this happen? If T cells are attracted to DCs, after a short period of time clumps of T cells form around the DCs, that hinder the movement of new T cells towards these DCs. However, this is an artifact of the T cells being modeled as fixed, non-deformable blocks. Vroomans et al ([|2012]) later showed that if T cells are modeled as the deformable objects that they are (i.e. by using CPM in stead of a CA-like model), the "old" T cells are pushed away by the new ones, and chemotaxis actually enhances the scanning process. This example warns us to beware of modeling artifacts, and illustrates the importance of using different modeling formalisms.

In conclusion: Cells are deformable, highly viscuous objects that can "wiggle" past each other. Furthermore, cells usually "live" in a cramped environment, with many other cells. The examples show the importance of including these cell characteristics (using CPM) if we want to understand cell movement.

Next: Event-based models