Invariant Region for the Reaction-Diffusion SystemSystem

Before we solve the reaction-diffusion system (2.14) numerically, we first check the bound-edness of the solution in the L^∞-norm. For this purpose, we show in the following the existence of an invariant region and state it explicitly.

In order to apply the theory of invariant regions established in [57, Chapter 14.§B], we restrict ourselves to the one-dimensional case, i.e. Ω = [−1,1], and the Laplace operator

is replaced by the second derivative _∂x^∂22. Thus, the reaction-diffusion system is written as

∂Xa

∂t = D˜₁^∂_∂x^2X2^a + ˜k_c1X_iY_a−k˜_d1X_a,

∂Ya

∂t = D˜2∂²Ya

∂x² + ˜kc2YiX_aⁿ−˜kd2Ya,

∂Xi

∂t = D˜₃^∂_∂x^2X2ⁱ −˜k_c1X_iY_a−˜k_d3X_i+ ˜k_p1,

∂Yi

∂t = D˜₄^∂_∂x^2Y2ⁱ −˜k_c2Y_iX_aⁿ−˜k_d4Y_i+ ˜k_p2,

(2.45)

with the boundary conditions

∂Xa

∂x _|x|=1

= ∂Ya

∂x _|x|=1

= ∂Xi

∂x _|x|=1

= ∂Yi

∂x _|x|=1

= 0 (2.46)

and the initial condition

(X_a, Y_a, X_i, Y_i)(x,0) = (X_a,0, Y_a,0, X_i,0, Y_i,0), x∈Ω. (2.47) Again, the reaction, degradation and production terms of (2.45) are summarized in the vector-valued functionf, namely

f(X_a, Y_a, X_i, Y_i) =













˜k_c1X_iY_a−˜k_d1X_a

˜k_c2Y_iX_aⁿ−k˜_d2Y_a

−˜k_c1X_iY_a−˜k_d3X_i+ ˜k_p1

−˜kc2YiX_aⁿ−˜kd4Yi+ ˜kp2













. (2.48)

First of all, we state the definition of an invariant region for reaction-diffusion systems.

Definition 2.7. [57, Definition 14.5] A closed subset Σ ⊂ R⁴ is called a (posi-tively) invariant region for the initial value problem (2.45), (2.47), if any solution (X_a, Y_a, X_i, Y_i)(x, t) with initial values in Σ satisfies (X_a, Y_a, X_i, Y_i)(x, t)∈Σ for allx∈Ω and for allt∈[0, T).

According to [57, Corollary 14.8], a rectangular domain of the form

Σ_R ={(X_a, Y_a, X_i, Y_i)∈R⁴,0≤X_a ≤a,0≤Y_a ≤b, (2.49) 0≤X_i ≤c,0≤Y_i ≤d}

would be preferred for an invariant region for the reaction-diffusion system (2.45), since the diffusion matrix ˜Dis diagonal. We show in Appendix A.3 that the domain Σ_Ris unsuitable for the reaction-diffusion system (2.45) although the diffusion matrix is diagonal. Instead

of that, a suitable choice is

Σ :={(X_a, Y_a, X_i, Y_i)∈R⁴, 0≤X_i ≤ ˜k_p1

˜k_d3,0≤X_a ≤ 1 m_x(

k˜_p1

k˜_d3 −X_i), (2.50) 0≤Y_i ≤ k˜_p2

k˜_d4,0≤Y_a≤ 1 m_y(

k˜_p2

k˜_d4 −Y_i)}

with 1 ≤ m_x ≤k˜_d1/˜k_d3 and 1 ≤ m_y ≤ k˜_d2/˜k_d4. Obviously, the concentrations Y_i of pro-caspase 3 and X_i of pro-caspase 8 are bounded from above by the concentration of the life state and from below by zero. The concentrationsYa of caspase 3 andXaof caspase 8 possess the same lower bound but the upper bound depends on the concentration of the corresponding pro-caspase. We have the following

Theorem 2.8.

Let k˜d1 ≥ k˜d3, ˜kd2 ≥ ˜kd4 and D˜ = diag( ˜D1,D˜2,D˜1,D˜2) with D˜3 = ˜D1,D˜4 = ˜D2. Then, the region Σdefined in (2.50)is invariant for the reaction-diffusion system (2.45)-(2.47).

In order to prove Theorem 2.8, we transfer the notation of the invariant region (2.50) to an abstract level to simplify the arguments and apply the general theory of invariant regions. Obviously, Σ defined by (2.50) can be written as

Σ =∩⁸_i=1{(X_a, Y_a, X_i, Y_i)∈R⁴, G_i(X_a, Y_a, X_i, Y_i)≤0}, (2.51) where the functions G_i, i= 1, . . . ,8, are given by

G₁ =−X_i, (2.52)

G2 =−Yi, (2.53)

G₃ =X_i− ˜k_p1

˜k_d3, (2.54)

G4 =Yi− ˜k_p2

k˜_d4, (2.55)

G₅ =−X_a, (2.56)

G6 =Xa− 1 m_x(

k˜_p1

˜k_d3 −Xi), (2.57)

G7 =−Ya, (2.58)

G₈ =Y_a− 1 m_y(k˜_p2

k˜_d4 −Y_i). (2.59)

With the notation of Σ stated in (2.51), we cite the following Theorem 2.9 (Theorem 14.14 in [57]).

Let Σ defined by (2.51) and consider system (2.45) with D˜ positive definite. Suppose

that this system is f-stable. Then Σ is a positively invariant region for (2.45) if and only if the following holds at each boundary point (X_a⁰, Y_a⁰, X_i⁰, Y_i⁰) of Σ, i.e. G_i((X_a⁰, Y_a⁰, X_i⁰, Y_i⁰)) = 0 for some i:

i) ∇G_i is a left eigenvector of D˜ for all x∈Ω.

ii) G_i is quasi-convex at (X_a⁰, Y_a⁰, X_i⁰, Y_i⁰).

iii) ∇G_i·f(X_a, Y_a, X_i, Y_i)≤0 at (X_a⁰, Y_a⁰, X_i⁰, Y_i⁰), where the function f(X_a, Y_a, X_i, Y_i) is given in (2.48).

Before we prove Theorem 2.8 by applying Theorem 2.9, we first recall Gronwall’s inequality in the form given in [16, p. 624].

Lemma 2.10 (Gronwall’s inequality (differential form)).

Let η be a nonnegative, absolutely continuous function on [0, T] which satisfies for a.e. t the differential inequality

η⁰(t)≤γ(t)η(t) +ψ(t),

where γ(t) and ψ(t) are non-negative, summable functions on [0, T]. Then

η(t)≤e

t

R

0

γ(s)ds



η(0) +

t

Z

0

ψ(s)ds





for all 0≤t ≤T.

This inequality is proven in [16].

In order to prove Theorem 2.8, we have to check whether the assumptions of Theorem 2.9 are satisfied. Then, we directly conclude that Σ is an invariant region for the reaction-diffusion system (2.45).

Proof of Theorem 2.8. To prove that Σ is invariant for the reaction-diffusion sys-tem (2.45) we have to check the conditions i)–iii) of Theorem 2.9 for the functions Gi, i = 1, . . . ,8, defined in (2.52)–(2.59). However, condition iii) is weakened due to the f-stability of the reaction-diffusion system, i.e., the strict negativity of the scalar product of ∇G_i and f in [57, Theorem 14.11] is replaced by the condition that the respective scalar product in Theorem 2.9 is less than or equal to zero.

1) f-stable

According to [57, Definition 14.10], system (2.45) is called f-stable if whenever f is the limit of functions f_n in the C¹-topology on compacta, then the solution of

the initial value problem (2.45), (2.47) is the limit in the compact-open topology of solutions of the initial value problem (2.45), (2.47) where f is replaced by f_n. We denote with u the solution of (2.45) and with u_n the solution of (2.45) with f_n instead of f, that means

∂u

∂t = ˜D∂²u

∂x² +f(u), (2.60)

∂un

∂t = ˜D∂²un

∂x² +f_n(u_n). (2.61)

Then, subtracting equation (2.61) from equation (2.60) yields

∂(u−u_n)

∂t = ˜D∂²(u−u_n)

∂x² +f(u)−fn(un). (2.62)

We introduce the notation X =L²(Ω,R⁴) and make the ansatz

d

dtku−u_nk²_X = _dt^d R

Ω

|u−u_n|²dx

= 2hu−u_n,_dt^d(u−u_n)i

(2.62)

= 2hu−u_n,D˜^∂2(u−u_∂x2 ⁿ⁾ +f(u)−f_n(u_n)i

= 2hu−u_n,D˜^∂2(u−u_∂x2 ⁿ⁾i+ 2hu−u_n, f(u)−f_n(u_n)i

(2.63)

whereh ·, · i denotes theL²-scalar product. Integration by parts yields with homo-geneous Neumann boundary conditions

hu−u_n,D˜∂²(u−u_n)

∂x² i=−h∂(u−u_n)

∂x ,D˜∂(u−u_n)

∂x i

≤ −d|u˜ −u_n|_H¹_(Ω,R⁴₎ (2.64) where ˜d denotes the smallest entry of the diagonal matrix ˜D. If we put (2.64) into (2.63), we obtain

d

dtku−u_nk²_X ≤ −2 ˜d|u−u_n|_H¹_(Ω,R⁴₎+ 2hu−u_n, f(u)−f_n(u_n)i

≤2|hu−u_n, f(u)−f_n(u_n)i|

≤2|hu−u_n, f(u)−f(u_n)i|+ 2|hu−u_n, f(u_n)−f_n(u_n)i|

≤2ku−u_nk_X · kf(u)−f(u_n)k_X+

+ 2|hu−u_n, f(u_n)−f_n(u_n)i|

≤2ku−u_nk²_X · sup

ρ∈[0,1]

k∇f(u+ρ(u_n−u))k_X+

+ 2|hu−un,sup

vn

kfn(vn)−f(vn)kXi|

≤2ku−u_nk²_X · sup

ρ∈[0,1]

k∇f(u+ρ(u_n−u))k_X+

+ 2 sup

vn

kf_n(v_n)−f(v_n)k_X· |hu−u_n,1i|

≤2ku−u_nk²_X · sup

ρ∈[0,1]

k∇f(u+ρ(u_n−u))k_X+

+ sup

vn

kf_n(v_n)−f(v_n)k_X·(ku−u_nk²_X +k1k²_X)

=ku−unk²_X(2 sup

ρ∈[0,1]

k∇f(u+ρ(un−u))kX+

+ sup

vn

kf_n(v_n)−f(v_n)k_X) +|Ω|²sup

vn

kf_n(v_n)−f(v_n)k_X. Now, we apply Gronwall’s inequality Lemma 2.10 for the functionη(t) =ku−u_nk²_X and, with u|_t=0 = u_n|_t=0, we get

ku(t)−u_n(t)k²_X ≤ e

t

R

0

2 sup_ρ∈[0,1]k∇f(u+ρ(un−u))k_X+sup_vnkfn(vn)−f(vn)k_Xds

×

×|Ω|²

t

R

0

sup_vnkf_n(v_n)−f(v_n)k_Xds).

Taking the supremum with respect to time, we have

sup

t∈[0,T]

ku(t)−u_n(t)k²_X ≤|Ω|²· sup

t∈[0,T] t

Z

0

sup

vn

kf_n(v_n)−f(v_n)k_Xds×

× sup

t∈[0,T]

e

t

R

0

2 sup_ρ∈[0,1]k∇f(u+ρ(u_n−u))k_Xds

×

× sup

t∈[0,T]

e

t

R

0

sup_vnkfn(vn)−f(vn)k_Xds

≤|Ω|²T sup

vn

kf_n(v_n)−f(v_n)k_X×

×e^T·sup_vnkfn(vn)−f(vn)k_X×

×e^{2Tsupρ∈[0,1]k∇f(u+ρ(un−u))kX}

≤|Ω|²T sup

vn

kf_n(v_n)−f(v_n)k_X·e^{2T C}×

×e^{Tsupvnkfn(vn)−f(vn)kX}.

Under the assumption thatf_n converges to f in the C¹-topology, the limit n→ ∞ yields

limn→∞sup_t∈[0,T]ku(t)−u_n(t)k²_X ≤ C(T˜ ) limn→∞sup_vnkf_n(v_n)−f(v_n)k_X×

×limn→∞e^T·sup_vnkfn(vn)−f(vn)k_X

= 0,

thus, the solution u_n of the reaction-diffusion system (2.45) with f_n instead of f converges to the solution u of the reaction-diffusion system (2.45) with the reaction kinetics f if the function fn converges to f. Hence, the reaction-diffusion system (2.45) is f-stable.

2) Condition i)

The first conditioni) provides for the gradient of the functions G_i to be a left eigen-vector to the diffusion matrix ˜D. Obviously,∇G_i, i= 1, . . . ,8,is either a multiple of a standard basis vector e₁, . . . , e₄ ofR⁴ or one of the vectors (1,0,1/m_x,0)^>=∇G₆ and (0,1,0,1/my)^> =∇G8. Of course, the standard basis vectors are left eigenvec-tors to the diffusion matrix and the veceigenvec-tors ∇G₆ and ∇G₈ are left eigenvalues due to the special choice ˜D₁ = ˜D₃ and ˜D₂ = ˜D₄.

3) Condition ii)

The second condition demands the quasi-convexity of the functions G_i, i= 1, . . . ,8, i.e., whenever ∇G_i·ξ = 0 then ξ^>∇²G_iξ ≥ 0, cf. [57, Definition 14.6]. It is easy to see that the functions G_i are quasi-convex since they are linear and the second derivative of the functions G_i, i = 1, . . . ,8, vanishes identically. Therefore, the condition ξ^>∇²G_iξ ≥ 0 is satisfied trivially for vectors ξ with ∇G_i·ξ = 0, i = 1, . . . ,8.

4) Condition iii)

Finally, to show that f points into Σ on ∂Σ, we check that ∇G_i· f|_G

i=0 ≤ 0 for i = 1, . . . ,8. Due to the f-stability of the reaction-diffusion system (2.45), we do not have to show that f pointsstrictly into Σ on ∂Σ.

For the function G₁ = −X_i, the gradient is given by ∇G₁ = (0,0,−1,0)^> and we obtain

∇G1·f|_G

1=0 = ˜kc1XiYa+ ˜kd3Xi−k˜p1

Xi=0 =−k˜p1 <0.

Therefore, Xi ≥ 0. Next, we consider the function G2 = −Yi with the gradient

∇G2 = (0,0,0,−1)^>. Similarly, we get

∇G₂·f|_G

2=0 = ˜k_c2Y_iX_aⁿ+ ˜k_d4Y_i−k˜_p2

Yi=0 =−k˜_p2 <0, that implies Y_i ≥0. For G₃ =X_i−^k˜_˜^p1

kd3, the gradient is given by∇G₃ = (0,0,1,0)^>. Then, condition iii) for G₃ reads

∇G₃·f|_G

3=0 = −˜k_c1X_iY_a−˜k_d3X_i+ ˜k_p1

Xi=˜kp1/˜kd3

=−˜k_c1 k˜p1

k˜_d3 ·Y_a≤0,

and we get X_i ≤k˜_p1/k˜_d3. To complete the bounds for Y_i, we consider the function G₄ =Y_i− ^˜k_˜^p2

kd4 with the gradient ∇G₄ = (0,0,0,1)^>. We obtainY_i ≤k˜_p2/˜k_d4 since

∇G₄·f|_G

4=0 =−k˜_c2Y_iX_aⁿ−˜k_d4Y_i+ ˜k_p2|_Y

i=˜kp2/˜kd4 =−˜k_c2 k˜p2

k˜_d4 ·X_aⁿ≤0.

The gradient of the function G₅ =−X_a is given by ∇G₅ = (−1,0,0,0)^>. Then, we get

∇G₅·f|_G

5=0 = −k˜_c1X_iY_a+ ˜k_d1X_a Xa=0

=−˜k_c1X_iY_a ≤0.

Thus,X_a≥0. The upper bound for X_a is described by the function G₆ =X_a−_m¹

x(^˜k_˜^p1

kd3 −X_i) with the gradient ∇G₆ = (1,0,_m¹

x,0)^>. Now, the scalar product of the gradient and the function f reads

∇G₆·f|_G

6=0 = ˜k_c1X_iY_a−k˜_d1X_a+ _m¹

x(−k˜_c1X_iY_a−k˜_d3X_i+ ˜k_p1)

= ˜k_c1X_iY_a(1− _m¹

x)−^k˜_m^d1

x(^k˜_˜^p1

kd3 −X_i)− ^˜k_m^d3

xX_i+^k˜_m^p1

x

= ˜k_c1X_iY_a(1− 1 m_x)

| {z }

≤0

+_m¹

x (˜k_d1−˜k_d3)

| {z }

≥0

(X_i−k˜_p1 k˜d3

)

| {z }

≤0

≤ 0.

It follows X_a ≤ _m¹

x(^k˜_˜^p1

kd3 −X_i). The bounds for Y_a are on the one side given by the function G₇ = −Y_a with the gradient ∇G₇ = (0,−1,0,0)^>. Then, condition iii) yields

∇G₇·f|_G

7=0 = −k˜_c2Y_iX_aⁿ+ ˜k_d2Y_a

Ya=0 =−˜k_c2Y_iX_aⁿ≤0,

that implies Ya ≥ 0. On the other side, the function G8 = Ya − _m¹

y(^˜k_˜^p2

kd4 −Yi) determines the upper bound for Ya. With the gradient ∇G8 = (0,1,0,_m¹

y)^>, we obtain

∇G₈·f|_G

8=0 = ˜k_c2Y_iX_aⁿ−˜k_d2Y_a+ _m¹

y(−k˜_c2Y_iX_aⁿ−˜k_d4Y_i+ ˜k_p2)

= ˜kc2YiX_aⁿ(1− _m¹

y)− ^˜k_m^d2

y(^˜k_˜k^p2

d4 −Yi)− ^˜k_m^d4

yYi+ ^˜k_m^p2

y

= ˜k_c2Y_iX_aⁿ(1− 1 my

)

| {z }

≤0

+_m¹

y (˜k_d2−k˜_d4)

| {z }

≥0

(Y_i− ˜k_p2

˜k_d4)

| {z }

≤0

≤ 0.

Finally, the estimate Y_a ≤ _m¹

y(^˜k_˜^p2

kd4 −Y_i) holds. So, we showed that f points into Σ on∂Σ.

In summary, we checked conditions i)–iii) of Theorem 2.9 for the region Σ and further confirmed thef-stability of the reaction-diffusion system. The positive definiteness of the diffusion matrix ˜Dholds trivially and all assumptions of Theorem 2.9 are satisfied. Thus, Σ is invariant for the reaction-diffusion system (2.45) with ˜D = diag( ˜D₁,D˜₂,D˜₁,D˜₂) for

˜k_d1 ≥k˜_d3 and ˜k_d2 ≥˜k_d4. Remark 2.11. In the proof of Theorem 2.8, the lower bound of the parameters mx and my are revealed. Actually, the upper bounds for mx and my do not play a role in the proof. Nevertheless, the values form_xand m_y have to guarantee that the invariant region contains the death stateX^(d) = (Xa^(d), Ya^(d), X_i^(d), Y_i^(d)). Thus, we ask for the conditions

(i) X_a^(d) ≤ 1 m_x(k˜_p1

k˜_d3 −X_i^(d)), (ii) Y_a^(d) ≤ 1

m_y(

˜k_p2

˜k_d4 −Y_i^(d)).

For (i), we obtain with (2.24) mxX_a^(d) ≤ ˜k_p1

˜k_d3 − ˜k_p1k˜_d2(˜k_c2(Xa^(d))ⁿ+ ˜k_d4)

(˜k_c1k˜_c2k˜_p2+ ˜k_c2k˜_d2˜k_d3)(Xa^(d))ⁿ+ ˜k_d2k˜_d3˜k_d4. Multiplying by the denominator yields

m_xh

(˜k_c1˜k_c2k˜_p2+ ˜k_c2k˜_d2k˜_d3)(X_a^(d))ⁿ⁺¹+ ˜k_d2˜k_d3k˜_d4X_a^(d)i

≤ k˜_p1 k˜_d3

h

(˜k_c1˜k_c2k˜_p2+ ˜k_c2k˜_d2k˜_d3)(X_a^(d))ⁿ+ ˜k_d2k˜_d3k˜_d4i

−k˜_p1˜k_d2˜k_c2(X_a^(d))ⁿ−˜k_p1k˜_d2k˜_d4)

=

˜k_p1

˜k_d3

k˜c1k˜c2˜kp2(X_a^(d))ⁿ.

SinceXa^(d) is a solution of equation (2.26), we obtain m_x ˜k_p1

˜k_d1

k˜_c1˜k_c2˜k_p2(X_a^(d))ⁿ

!

≤ ˜k_p1 k˜_d3

k˜_c1˜k_c2˜k_p2(X_a^(d))ⁿ,

which is equivalent tom_x ≤k˜_d1/k˜_d3. For (ii), we have with (2.22) and (2.23)

my· k˜_p2˜k_c2(Xa^(d))ⁿ

˜k_d2(˜k_c2(Xa^(d))ⁿ+ ˜k_d4) ≤ k˜_p2

k˜_d4 − ˜k_p2

˜k_c2(Xa^(d))ⁿ+ ˜k_d4. (2.65) Again, multiplying (2.65) by the denominator on the right-hand side of (2.65) and simul-taneously dividing (2.65) by ˜k_p2 yields

m_y

k˜_c2(Xa^(d))ⁿ

k˜_d2 ≤ ˜k_c2(Xa^(d))ⁿ+ ˜k_d4

˜k_d4 −1, which is equivalent tom_y ≤k˜_d2/˜k_d4.

Thus, we see that the parameters m_x and m_y have to be bounded from above dependent on the degradation rates ˜k_di, i= 1, . . . ,4.

Remark 2.12. In case ˜D₁ 6= ˜D₃, ˜D₂ 6= ˜D₄, condition i) in Theorem 2.9 is not satisfied and Theorem 2.9 can not be applied to prove that Σ is an invariant region for the reaction-diffusion system (2.45). Thus, the proof of the existence of an invariant region is more difficult in this case, and one has to pursue a totally different strategy.

Hence, we proved that if the solution (X_a, Y_a, X_i, Y_i)(x, t) once resides in the region Σ, it remains in Σ for all time. Besides the boundedness of the solution of the reaction-diffusion system (2.45), we further obtain an important result for the numerical analysis, since the existence of an invariant region belongs to the conditions of a theorem in Sec-tion 2.8.

Remark 2.13. Concluding, we note that the conditions for the concentrationsX_a, Y_a, X_i and Y_i, namely X_a ≥ 0, Y_a ≥ 0, X_i ≥ 0 and Y_i ≥ 0, are natural for the caspase concentrations due to the positivity of concentrations.

After analyzing the reaction-diffusion system (2.45) with respect to the existence and uniqueness of a solution and after determining an invariant region, we solve the reaction-diffusion system in case of radially symmetric concentration distributions numerically in the subsequent section.

Im Dokument Mathematical modeling and numerical simulations of the extrinsic pro-apoptotic signaling pathway (Seite 41-51)