Derivation of Equations for the Particle Translation

The particle translation is described by the particle coordinates x and the velocity v, whereas the relationship v = ˙x holds. The momentum p is given by p = m·v and a variation in the momentum is caused by a force F, thus _dt^d(mv) = F. The mass m of the particles is constant, hence m·v˙ = F. We assume that the force which affects the particles is composed of a random forceσ_transX˜_t, a friction force F_fric and a force F_interact for the interaction between the particles. The friction force is proportional to the particle velocity and has the form Ffric =−βv with the friction coefficient β. Thus, the particle translation is modeled by the Langevin equation

mv˙ =Finteract−βv+σtransX˜t. (3.1)

We assume that the components of the random force ˜X_t describe a white noise. Then, the relation between the fluctuation of the random force σtrans and β is given by the Einstein relation σtrans = √

2kBT β, cf. [54, Chapter 8], where kB is the Boltzmann constant andT is the temperature. Besides, we assume that the friction force is of Stokes type, so that the coefficientβ has the form β = 6πηR, where ηis the viscosity of the cell membrane and R is the radius of the particle. The masses of the death receptors and death receptor ligands are m_receptor ≈ 55 kDa and m_ligand ≈ 51 kDa, respectively. With 1 kDa ≈ 1.6605·10⁻²⁴kg, these masses correspond to m_receptor ≈ 9.133·10⁻²³kg and m_ligand ≈ 8.469·10⁻²³kg. By the way, we do not distinguish the masses of monomeric, dimeric and trimeric pre-forms of death receptors since the masses of the particles do not play any role in the model as we will see later.

In order to transform equation (3.1) into a dimensionless form, we introduce the mag-nitudes ¯x = x/L,¯t = t/τ and ¯F_interact = L/ε·F_interact, where ε is the binding energy between the particles,L is the length scale of the cell and τ is the time scale. Thus, the dimensionless velocity is ¯v =τ /L·v and the dimensionless acceleration is ¯v⁰ =τ²/L·v,˙

where the prime denotes the derivative with respect to ¯t. The binding energy between two death receptors is approximately ε_RR ≈6k_BT, and for a death receptor and a death receptor ligand we have ε_RL ≈ 60k_BT [24]. Thus, we set ε = 6k_BT and multiply the external force by a factor ten, if the interaction occurs between a death receptor and a death receptor ligand.

With these magnitudes, we obtain from (3.1) L

τ²m·v¯⁰ = ε

L·F¯_interact−βL

τ ·v¯+p

2k_BT β·X˜_t. (3.2)

The length scale of the cell is assumed to be L= 10µm and for the time scale we choose τ = 1 s. The coefficient on the left-hand side of equation (3.2) is of the order 10⁻²⁷ for the chosen length scale and the given masses, and the coefficients on the right-hand side of (3.2) are of the order 10⁻¹⁵to 10⁻¹². Thus, we simplify (3.2) by omitting the left-hand side. Hence, equation (3.2) becomes

¯ v= ετ

L²β ·F¯_interact+ s

2k_BT τ²

L²β ·X˜_t= 6k_BT τ

L²β ·F¯_interact+ s

2k_BT τ²

L²β ·X˜_t. (3.3) Introducing the parameter µ defined by µ² := ^k_L^B2^{T τ}β , we obtain from (3.3)

¯ v= d¯x

d¯t = 6µ²·F¯_interact+√

2µ·X˜¯t with X˜¯t:=√

τX˜_t. (3.4)

Integration of (3.4) with respect to time yields

¯

x¯t= ¯x₀+

¯t

Z

0

6µ²·F¯_interactd¯s+

¯t

Z

0

√2µ·X˜_s¯d¯s

= ¯x₀+

¯t

Z

0

6µ²·F¯_interactd¯s+

¯t

Z

0

√

2µdWf_trans,¯s (3.5)

with dfW_trans,¯s = ˜X_s¯d¯s. The second integral on the right-hand side of (3.5) is an Itˆo stochastic integral with respect to the Wiener process Wf_trans = {Wf_trans,¯t,¯t ≥ 0}. We abbreviate the integral equation (3.5) as an Itˆo stochastic differential equation

d¯x= 6µ²·F¯_interactd¯t+√

2µ·dWf_trans,¯t, (3.6)

where dfW_trans,t¯ are increments of the Wiener process Wf_trans = {fW_trans,t¯,¯t ≥ 0} and ¯x are the coordinates of the particles, cf. [20].

To model the interaction between two particles, a potential which describes the

inter-action between uncharged and chemically unbounded atoms is necessary. This is typically done with a Lennard-Jones potential, which has an attractive impact given by the Van-der-Waals force and a repulsive impact for very small distances between the particles affected by the Pauli repulsion. For simplicity, we choose for our model the Lennard-Jones-(2n, n) potential

V_LJ^P(r) = εγ_n

σ^P_LJ r

²ⁿ

−α_n σ_LJ^P

r ⁿ!

, (3.7)

wherer is the distance between two particles,σ^P_LJ describes the size of the particles, and ε is the depth of the potential and corresponds to the binding energy. The parameters γ_n and α_n are determined below under suitable conditions for the potential V_LJ^P. The dimensionless form of the Lennard-Jones potential is

V¯_LJ^P(¯r) = γ_n

σ¯_LJ^P

¯ r

²ⁿ

−α_n σ¯^P_LJ

¯ r

ⁿ!

, (3.8)

with ¯r = r/L and ¯σ_LJ^P = σ^P_LJ/L. The interaction force ¯F_interact is given by the negative gradient of the potential ¯V_LJ^P(¯r). First, we ask for the potential that its minimum is −ε, the binding energy of the particles under consideration. The minimum of the potential V_LJ^P is determined by the condition

∂V_LJ^P

∂r =εγn −2n·1 r ·

σ^P_LJ r

2n

+αn·n·1 r ·

σ^P_LJ r

n!

= 0.!

With this, we obtain for the minimizing distance

−2 σ_LJ^P

r ²ⁿ

+α_n σ_LJ^P

r ⁿ

= 0, that implies α_n = 2 σ_LJ^P

r ⁿ

.

Now, we demand on the potential to reach its minimum for r^∗ = 2σ^P_LJ. Thus, we have αn= 2¹⁻ⁿ. The condition V_LJ^P(r^∗) =−ε finally yields the value for γn,

V_LJ^P(r^∗) = γ_nε 1

2 2n

−2¹⁻ⁿ 1

2 n!

=εγ_n −2⁻²ⁿ !

=−ε ⇒ γ_n= 2²ⁿ.

With the values forα_n and γ_n, the Lennard-Jones potential (3.8) reads V¯_LJ^P(¯r) = 2²ⁿ

σ¯_LJ^P

¯ r

2n

−2¹⁻ⁿ σ¯^P_LJ

¯ r

n!

. (3.9)

For the interaction between particles of the same type, except death receptor

monomers, we introduce a further potential with a sole repulsive impact. For this purpose, we consider the repulsive part of the Lennard-Jones potential in (3.9) and define

W¯_LJ^P(¯r) = 2²ⁿ σ¯_LJ^P

¯ r

2n

. (3.10)

Since the interaction between the particles is short-ranged, i.e. on a length scale of several nanometers, we introduce a cut-off radius r_cut for the potentials ¯V_LJ^P(¯r) and ¯W_LJ^P(¯r). For this purpose, we set

V¯_LJ,cut^P (¯r) :=

( V¯_LJ^P(¯r), for ¯r ≤r_cut,

0, for ¯r > r_cut (3.11)

and

W¯_LJ,cut^P (¯r) :=

( W¯_LJ^P(¯r), for ¯r≤r_cut,

0, for ¯r > r_cut. (3.12)

With these potentials at hand and with the fact that the force acting on a particle is given by the negative gradient of the potential, all the ingredients of the stochastic differential equation (3.6) are determined. The structure of the interaction forces for the different particle types will be given later in Section 3.3.4.

In order to describe the motion of the particles in more detail, we introduce a further equation for the particle rotation in the following section.

Im Dokument Mathematical modeling and numerical simulations of the extrinsic pro-apoptotic signaling pathway (Seite 78-81)