2.9 Reaction-Diffusion Model on Large Spatial Scales
2.9.2 Spatial Inhibition Effects on Large Spatial Scales
Again, we observe through numerical simulations a traveling wave solution. But this time, the solution moves from the left to the right, that means, the death state again invades the region initially occupied by the life state, see Figure 2.18.
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0 0.05 0.1 0.15 0.2 0.25
r (x 10 um)
caspase concentration (x 100 nM)
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0 0.05 0.1 0.15 0.2 0.25
r (x 10 um)
caspase concentration (x 100 nM)
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0 0.05 0.1 0.15 0.2 0.25
r (x 10 um)
caspase concentration (x 100 nM)
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0 0.05 0.1 0.15 0.2 0.25
r (x 10 um)
caspase concentration (x 100 nM)
Figure 2.18: Traveling wave solution which moves from the left to the right forn= 2.5 in case d= 1 for the radially symmetric reaction-diffusion system (2.75) with ˜kd1 = ˜kd2 = ˜kd3 = ˜kd4 after 2400 (left top), 3600 (right top), 4800 (left bottom) and 6000 (right bottom) time steps.
Red line: caspase 8, blue line: caspase 3, green line: pro-caspase 8, black line: pro-caspase 3.
However, this behavior depends on the special choice of initial condition. For initial conditions of the form
X(r,0) =X(d) for 0≤r < r∗ and X(r,0) =X(l) for r∗ ≤r≤2000,
0< r∗ <2000, the formation of a traveling wave solution is observable if r∗ is larger than a critical value rc. In the next section, we investigate the situation when r∗ < rc and determine the critical value rc, so it exists.
For the first numerical simulations, we chose a spherical shell of a certain thickness filled by the death state as initial condition. At the end of Section 2.9.1, we inverted the initial conditions and considered a ball of a certain diameter occupied by the death state whereas the spherical shell was filled with the life state. In both cases, the death state invades the region initially occupied by the life state. But the formation of a traveling wave solution and the associated invasion of the death state should not be possible for all initial conditions: according to Figure 2.19, the ball filled with the death state can be regarded as a small spot located close to the cell membrane instead of the center of the cell. If there was no threshold for r∗, i.e. for the size of the ball initially filled with the death state, then the tiniest activation would lead to the formation of a traveling wave and the invasion of the death state.
extracellular space
membrane cytosol
L D Ω
R
Figure 2.19: The domain Ω modeling a circle (2d) or a ball (3d) with radius R. The initial condition for the reaction-diffusion system is defined by the death state (D) in a small ball and the life state (L) in a spherical shell, i.e. the rest of the domain Ω. Here, Ω does not reflect a model for the cell but a ball with center on the cell membrane. Thus, the death state occupies a small region located close to the cell membrane. Considering radially symmetric concentrations, we only focus on the dynamics within the cell, i.e. in the cytosol.
In order to determine the critical value rc for r∗, we solve the radially symmetric reaction-diffusion system (2.75) with homogeneous Neumann boundary conditions and initial conditions
X(r,0) =X(d) for 0≤r < r∗ and X(r,0) =X(l) for r∗ ≤r≤2000.
To find the critical valuerc forr∗ that enables the formation of a traveling wave solution, we superpose a bisection algorithm. In the following, the notation rd,· denotes the value for r∗ for which the formation of a traveling wave solution succeeds and rl,· denotes the value for r∗ where the formation of a traveling wave solution fails.
Algorithm 2.2.
Choose rl,0 and rd,0
while | rl,i − rd,i | ≥ 2
compute z = 0.5 · (rl,i + rd,i)
solve initial value problem for r∗ = z if { traveling wave is built }
rd,i+1 = z; rl,i+1 = rl,i; else
rl,i+1 = z; rd,i+1 = rd,i; end
i = i+ 1;
end
It remains to explain the decision whether the formation of a traveling wave is performed.
In Figure 2.20, we see that for a certain initial condition, the concentrations initially get balanced and the maxima of the active caspase concentrations are decreasing. However,
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0 0.05 0.1 0.15 0.2 0.25
r (x 10 um)
caspase concentration (x 100 nM)
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0 0.05 0.1 0.15 0.2 0.25
r (x 10 um)
caspase concentration (x 100 nM)
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0 0.05 0.1 0.15 0.2 0.25
r (x 10 um)
caspase concentration (x 100 nM)
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0 0.05 0.1 0.15 0.2 0.25
r (x 10 um)
caspase concentration (x 100 nM)
Figure 2.20: The formation of a traveling wave depends on the special choice of initial condi-tion. For values beneath a critical valuerc, the system does not allow the formation of traveling waves. Red line: caspase 8, blue line: caspase 3, green line: caspase 8, black line: pro-caspase 3.
there is a time point t† where the maximum of the active caspase concentrations reaches a minimum and afterward, it is increasing and approaches the death state concentration of active caspase 3 and 8. Finally, a traveling wave solution is built. Thus, we expect the formation of a traveling wave if there exist t1, t2 >0 witht2 > t1 such that
r∈[0,2000]max Xa(r, t2)> max
r∈[0,2000]Xa(r, t1).
For the algorithm, we use the condition
∃k ∈N: max
r∈[0,2000]Xa(r,(k+ 1)·100·∆t)− max
r∈[0,2000]Xa(r, k·100·∆t)>10−3,
thus, the monotony of the maximum of Xa on a discretized time interval. With this condition at hand, we complete the algorithm and a simulation of the algorithm yields series {rl,i}i∈N and {rd,i}i∈N of initial conditions where the formation of traveling waves fails or succeeds. Besides, the series converge towards the critical value rc.
Again, we run the simulation for different scenarios, i.e. for n= 2, n = 2.5, n= 3, and for d= 2,3. We define the function
R(r) =
( 1, r ∈ {rd,i}i∈N,
−1, r ∈ {rl,i}i∈N,
where the value 1 stands for the formation of a traveling wave and, thereby, the death of the cell, and the value -1 stands for the survival of the cell. The numerical simulation yields the critical values r2dc ≈ 9µm and rc3d ≈ 41µm for n = 2, r2dc ≈ 34µm and r3dc ≈ 101µm for n = 2.5, and rc2d ≈67µm and r3dc ≈191µm for n= 3, see Figure 2.21, Figure 2.22 and Figure 2.23. For values larger than rc, a traveling wave is built. We have to mention that the resolution in space is of the order of micrometers, that means, for instance in casen= 2, a value ofr2d= 10µm allows for the formation of a traveling wave.
Values between 9µm and 10µm are not investigated due to the resolution of the spatial discretization with step size ∆r = 0.1 on a scale of L = 10−5m. Again, we obtain a switch-like behavior concerning the initial condition similar to the situation in Section 2.8 on a realistic biological scale. Here, we have the switch-like behavior on a large spatial scale and the critical values for r∗ are of the size of several micrometers.
Remark 2.18. Similar to the study of the switch-like behavior in Section 2.8, we found in this section critical valuesrc for the initial death region, above which the formation of traveling wave solution succeeds. With this critical values at hand, we would be able to compute the volume of the region initially occupied by the death state and with equa-tion (2.33), we would find the corresponding amount of molecules. But, since we consider a large spatial scale, i.e. a cell of size two centimeters, the result for the amount of molecules
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−1.5
−1
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r (x 10 um)
life vs. death
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−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
r (x 10 um)
life vs. death
Figure 2.21: Illustration of the switch-like behavior of the reaction-diffusion system (2.75) on a large spatial scale ford= 2 (left) andd= 3 (right) in casen= 2. The value 1 in they-direction means the death of the cell, the value -1 denotes the survival of the cell.
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−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
r (x 10 um)
life vs. death
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−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
r (x 10 um)
life vs. death
Figure 2.22: Illustration of the switch-like behavior of the reaction-diffusion system (2.75) on a large spatial scale ford= 2 (left) andd= 3 (right) in casen= 2.5.
would be on a very large biological scale. Thus, we restrict ourselves to a qualitative study concerning the switch-like behavior on the large spatial scale.