3. Mathematical Model of the Receptor Clustering on the Cell
3.3 Model of the Particle Translation and Particle RotationRotation
3.3.5 Particle Model for the Different Particle Configurations
by
gUi(¯ξU,ϕ¯U) = 1 zU
zU
X
k=1,k6=i
81√ 3
2π3 ψUi;Uk(π
3 −ψUi;Uk)(π
3 +ψUi;Uk)×
×H(rcut−RUi;Uk)·rcut−RUi;Uk
rcut .
Again, the set ΛUi contains the three binding sites of the cluster units, RUi;Uk denotes the distance between the cluster units Ui and Uk and ψUi;Uk describes the disorientation of the binding site lying closest to the vector eUiUk, relative to eUiUk, see Figure 3.7. We mention that the structure of the function gUi agrees with the structure of the function gTi, and so, the explanation of the single terms can be transferred.
Ui
Uk1
Uk3
Uk2
ψUi;Uk1
ψUi;Uk2
ψUi;Uk3
Figure 3.7: The cluster units Ukj, j= 1,2,3,cause a rotation of the cluster unit Ui according to the disorientation ψUi;Ukj. Since the distance of the cluster unit Uk3 to Ui is smaller than the distance of Uk1 to Ui, the influence of the cluster unit Uk3 on the rotation of Ui is larger.
In summary, the weighted average of the three disorientations yields the final rotation of the cluster unit Ui.
With the forces ¯Finteract and the functions g for the different particle types at hand, the particle translation and the particle rotation is completely described by the stochastic differential equations (3.6) and (3.19).
differential equations for the different particle configurations introduced in Section 3.3.3.
Since we consider in each case a huge amount of particles, we obtain a large system of stochastic differential equations nonlinearly coupled by the interaction forces ¯Finteract and the functions g describing the mutual influence on the particle rotation. For the description of the motion of a single particle, two stochastic differential equations for the particle translation and one stochastic differential equation for the particle rotation are required.
Before we list the systems of stochastic differential equation for each particle config-uration, we first specify the parameters in equation (3.6) and equation (3.19) since the values of µand ζ depend on the particle type, especially their size. Therefore, we distin-guish in the following the parameters µM, µD, µT, µL, µU, ζM, ζD, ζT, ζL and ζU for the different particle types.
a.i) Death Receptor Monomers and Death Receptor Ligands
We consider a simulation domain with zM death receptor monomers and zL death receptor ligands. Thus, we obtain a system of 3(zM +zL) stochastic differential equations
d¯xMi = 6µ2MF¯Mi(¯ξM,¯ξD,¯ξL,ϕ¯M,ϕ¯D,ϕ¯L)d¯t+√
2µMdfWtrans,Mi,¯t, d¯xLj = 6µ2LF¯Lj(¯ξM,¯ξD,¯ξT,ξ¯L,ϕ¯M,ϕ¯D,ϕ¯T,ϕ¯L)d¯t+√
2µLdWftrans,Lj,¯t, d ¯ϕMi = κµ2MζM2gMi(¯ξM,ξ¯L,ϕ¯M,ϕ¯L)d¯t+√
2µMζMdfWrot,Mi,¯t, d ¯ϕLj = κµ2LζL2gLj(¯ξM,ξ¯D,ξ¯T,¯ξL,ϕ¯M,ϕ¯D,ϕ¯T,ϕ¯L)d¯t+√
2µLζLdfWrot,Lj,¯t, i = 1, . . . , zM;j = 1, . . . , zL;zD = 0;zT = 0. The system of stochastic differential equations is nonlinearly coupled by the functions ¯FMi,F¯Lj, gMi and gLj.
a.ii) Death Receptor Dimers and Death Receptor Ligands
If all death receptors exist in a dimeric pre-form, we again obtain a system of stochastic differential equations with two equations for the particle translation and one equation for the particle rotation. Since we consider zD death receptors dimers and zL death receptor ligands, we get a system of 3(zD+zL) equations
d¯xDk = 6µ2DF¯Dk(¯ξM,ξ¯D,ξ¯L,ϕ¯M,ϕ¯D,ϕ¯L)d¯t+√
2µDdWftrans,Dk,¯t, d¯xLj = 6µ2LF¯Lj(¯ξM,¯ξD,¯ξT,ξ¯L,ϕ¯M,ϕ¯D,ϕ¯T,ϕ¯L)d¯t+√
2µLdfWtrans,Lj,t¯, d ¯ϕDk = κµ2DζD2gDk(¯ξD,¯ξL,ϕ¯D,ϕ¯L)d¯t+√
2µDζDdfWrot,Dk,¯t, d ¯ϕLj = κµ2LζL2gLj(¯ξM,ξ¯D,ξ¯T,¯ξL,ϕ¯M,ϕ¯D,ϕ¯T,ϕ¯L)d¯t+√
2µLζLdfWrot,Lj,¯t, k = 1, . . . , zD;j = 1, . . . , zL;zM = 0;zT = 0. The structure of the system seems the same as for death receptor monomers, but the difference lies in the structure of the functions ¯FDk and gDk.
a.iii) Death Receptor Monomers, Death Receptor Dimers and Death Receptor Ligands
Three different particle types are involved in the model where monomeric and dimeric death receptors coexist on the cell membrane. Besides the death recep-tors, the model also includes death receptor ligands, so that we obtain a system of 3(zM+zD+zL) stochastic differential equations
d¯xMi = 6µ2MF¯Mi(¯ξM,¯ξD,¯ξL,ϕ¯M,ϕ¯D,ϕ¯L)d¯t+√
2µMdWftrans,Mi,¯t, d¯xDk = 6µ2DF¯Dk(¯ξM,¯ξD,¯ξL,ϕ¯M,ϕ¯D,ϕ¯L)d¯t+√
2µDdfWtrans,Dk,¯t, d¯xLj = 6µ2LF¯Lj(¯ξM,ξ¯D,ξ¯T,¯ξL,ϕ¯M,ϕ¯D,ϕ¯T,ϕ¯L)d¯t+√
2µLdfWtrans,Lj,¯t, d ¯ϕMi = κµ2MζM2 gMi(¯ξM,¯ξD,¯ξL,ϕ¯M,ϕ¯D,ϕ¯L)d¯t+√
2µMζMdfWrot,Mi,¯t, d ¯ϕDk = κµ2DζD2gDk(¯ξD,ξ¯L,ϕ¯D,ϕ¯L)d¯t+√
2µDζDdfWrot,Dk,¯t, d ¯ϕLj = κµ2LζL2gLj(¯ξM,¯ξD,¯ξT,ξ¯L,ϕ¯M,ϕ¯D,ϕ¯T,ϕ¯L)d¯t+√
2µLζLdfWrot,Lj,¯t, i = 1, . . . , zM;k = 1, . . . , zD;j = 1, . . . , zL;zT = 0. Again, the interaction between the particles is described by the functions ¯FMi, ¯FDk, ¯FLj,gMi, gDk and gLj.
b) Death Receptor Trimers and Death Receptor Ligands
For death receptor trimers we obtain a system of stochastic differential equations with 3(zT+zL) equations
d¯xTι = 6µ2TF¯Tι(¯ξT,¯ξL,ϕ¯T,ϕ¯L)d¯t+√
2µTdWftrans,Tι,t¯, d¯xLj = 6µ2LF¯Lj(¯ξM,¯ξD,¯ξT,ξ¯L,ϕ¯M,ϕ¯D,ϕ¯T,ϕ¯L)d¯t+√
2µLdfWtrans,Lj,t¯, d ¯ϕTι = κµ2TζT2gTι(¯ξT,¯ξL,ϕ¯T,ϕ¯L)d¯t+√
2µTζTdfWrot,Tι,¯t, d ¯ϕLj = κµ2LζL2gLj(¯ξM,ξ¯D,ξ¯T,¯ξL,ϕ¯M,ϕ¯D,ϕ¯T,ϕ¯L)d¯t+√
2µLζLdfWrot,Lj,¯t, ι = 1, . . . , zT;j = 1, . . . , zL;zD = 0;zM = 0, where zT denotes the number of death receptor trimers and zL is the number of death receptor ligands.
c) Cluster Units consisting of Death Receptor Monomers and Death Receptor Ligands
Last but not least, we state the system of stochastic differential equations in case of cluster units. We obtain a system with 3zU equations
d¯xUν = 6µ2UF¯Uν(¯ξU,ϕ¯U)d¯t+√
2µUdWftrans,Uν,¯t, d ¯ϕUν = κµ2UζU2gUν(¯ξU,ϕ¯U)d¯t+√
2µUζUdfWrot,Uν,¯t,
ν = 1, . . . , zU. The equations are nonlinearly coupled by the functions ¯FUν and gUν. This is the most simple of the five introduced systems since we only consider one particle type. The price for the simplicity of the model is the strong assumption that the structure of the cluster units remains fix all the time and the binding between
death receptor ligands and death receptor monomers does not break.
Given a certain initial condition for the coordinates and the orientations for all particles in each of the scenarios above, the random motion of the particles is determined by the Wiener processesWftrans and Wfrot.
Since we are interested in the evolution of the signal competent cluster units, we need appropriate binding conditions which enable a decision whether two particles are bound.
We propose that two particles are bound if the distance between the centers of mass of the particles is smaller than a certain threshold rcut and if the angle between the vector connecting the centers of mass of the particles and the compatible binding site, lying closest to this vector, is smaller than δ. Here, we demand the corresponding binding site to be unoccupied.
So, in this section we finalized the particle model for the different scenarios concerning the various death receptor structures by assembling the stochastic differential equations derived in Section 3.3.1 and Section 3.3.2 to a nonlinearly coupled system of stochastic differential equations. Due to its complex structure, we solve the system of stochastic differential equations numerically. For this purpose, we introduce in the next section the Euler-Maruyama approximation.