Cell-cell dissipation is the requirement that we set out that was least adequately accounted for in the other models that we examined. In this section, we will describe two aspects of cell-cell dissipation. The first describes the relationship between the relative motion of cells at an interface and the force that is induced by this movement. We call this relationship the viscous force model.
The second aspect relates to how we update the positions and shapes of cells from a given distribution of viscous forces within a tissue, which we call the slippage kinematics. This aspect is the area that was not properly accounted for in the model of Palsson and Othmer (Palsson and Othmer [2000]), which calculated the viscous forces based on the movements of the cell centres rather than the slippages at the interfaces (which must involve both the cell centres and shapes).
In this thesis, we will only consider viscous forces that relate to the sliding between two cells, rather than between a cell and a substrate. The reason for developing it in this way was that other models have the substrate interactions so hard-wired that they are difficult to remove. By developing the model with- out cell-substrate interactions, we could ensure that there really is no implicit substrate assumption. Once this framework has been developed, we will see that including cell-substrate viscous forces within the framework is relatively simple.
2.4.1
Viscous Force Model
Viscous forces within the model occur whenever two cells with a shared interface move relative to one another. From the distribution of viscous forces at the interfaces between cells, we can obtain a field of relative cell movements through the viscous force model. This field is used to update the positions and shapes of
cells, which we will describe in the next section. In this section, we will describe the viscous model i.e. how the relative motion of cells at an interface relates to the viscous forces between them.
A cell in contact with certain substrates or other cells can forms adhesive junctions that bind the cell to the other object. These adhesive junctions serve a variety of functions such as anchoring junctions to transmit stress and channel forming junctions to link the cytoplasms of neighbouring cells. These adhesion junctions undergo binding and unbinding as a thermodynamic process. To model this process, we assume that these bonds have two major features:
• Stiffness, describing how difficult it is to stretch the bonds between the adhesion molecules. This has an associated spring constant k.
• Turnover Rate, a timescale τT associated with the average time between
binding and unbinding of adhesion molecules due to thermodynamic fluc- tuations.
To examine the effects of a shear force on the interface between 2 cells consider the situation illustrated in Figure 2.3
a)
b)
Figure 2.3: Description of the viscous model. a) The cell boundaries with ad- hesion molecules before a shear is applied. b) The cell boundaries and adhesion molecules after a shear due to a relative velocity v, the net displacement of the endpoints of an adhesion molecule is given by δx.
A shear, due to a relative movement δx, using Hooke’s Law and a first order approximation requires an energy Es = k(δx)2 per adhesion. The turnover rate
molecule when it unbinds if the two interfaces are moving at relative velocity v is given by δx ≈ vτT. When a molecule unbinds the energy due to extension is
lost as heat so the energy loss per adhesion is: E = k(vτT)2
For an interface of length l with n adhesions per unit length, the resistive force Fv for the entire interface is given by the total energy expended divided by the distance it has moved through so is given by:
Fv = Enl vτT
= kvτTnl
Using vector notation and combining the bonding affinity k, the timescale τT
and the number of adhesions per unit length n into a single viscosity parameter η, we arrive at the viscous model:
Fv = ηlv (2.6)
Using this relation, we can calculate the slippage or relative velocity between cells due to a given viscous force. Therefore, for a given distribution of viscous forces across the tissue we can calculate a velocity field of slippages between cells. This model differs from a finite-range spring model, in that the springs break/ unbind after a characteristic time τT rather than breaking after the spring reaches
a certain length. The consequence of this difference is that there is no elastic ”re- coil”, i.e. if the membranes are moved less than the range of the finite springs, once the shear force is removed the membranes will recoil to their original posi- tions. In the viscous model of this section, even for shear forces lasting less than the timescale τT, there will be a loss of energy and a net displacement after the
shear force has stopped.
In the next section, we will discuss how to use the velocity field calculated from the viscous forces to update the positions and shapes of the cells.
2.4.2
Slippage Kinematics
Having determined the viscous forces, the viscous model defines a slippage rate vF
ij between cells in contact i and j. The positions xi and shapes of the cells devi
must be updated to be consistent with this slippage vector field. In this section, we will discuss the kinematics of how the slippage between two ellipses relate to their change in position and shape (taken from Kabla et al. [2010]).
Figure 2.4: Cells with contact point P at position cij. Slippage is calculated by
considering the movement of P from cell i and cell j due to changes in position dx and shape di.
Consider 2 cells with positions xi and xj with contact point P at cij = cji. If
the positions of the cells change by dxi and dxj, and shape changes by di and
dj. We determine how these quantities relate to the slippage amount vFijδt, by
considering the movement of the contact point P from each cell.
Movement of P as viewed from reference frame of cell i or j (respectively) is:
dP = di(cij − xi)
dP = dj(cji− xj)
dxj − dxi. The movement of P as viewed in this frame is:
dP = dxj− dxi+ dj(cji− xj)
The slippage amount is the difference between these terms so:
vFijdt = dxi− dxj + di.(cij − xi) − dj.(cji− xj) (2.7)
This equation provides a relationship between the slippage amount vF
ijdt and
the shape and position change, δi and δx
i for each cell.