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X-Machines for Agent-Based Modeling: FLAME Perspectives
7.3.1 Stochastic Modeling
The stochastic Bicoid protein reaction diffusion system was implemented
as a simulation of 100 compartments along the A-P axis, each with length
h = 5µm, which is the average size of one nucleus. The proteins diffuse along
the compartments based on time and concentrations.
Bicoid 1 ⇌ Bicoid i ⇌ Bicoid n
(7.1)
where ⇌ represents the diffusion for i = 1 to n. The Bicoid protein has a life
and degrades in all compartments along the axis, until it translates into bicoid
mRNA in compartment 1 to form the anterior pole of embryo. Details of the
procedure are as follows:
Algorithm: Bicoid reaction-diffusion stochastic simulation
Input: Model parameters; final time.
Output: Bicoid molecular numbers along 100 compartments: m.
Start m = 0; t = 0;
Repeat:
1. Generate two random numbers which are uniformly distributed in
(0, 1): r(1) and r(2).
2. Calculate propensity functions of all the reactions: a = a1 + a2 +
a3 + a4.
3. Calculate the time when next reaction occurs: t + τ , where τ =
1/a ln(1/r(1)).
4. Decide which reaction occurs at Pt + τ : find jǫR such that
Σ
j−1
i=1 a i /a ≤ r(2) <
j
i=1 a i /a,
5. Update numbers of reactants and products in j-th reaction and set
t ← t + τ until time > f inaltime.
The diffusion between neighboring compartments takes place in both directions based on a rate d, related to a diffusion constant of a deterministic model
d = D/h2. The vector m contains a number of molecules along the N = 100
compartments or bins. This is based on the equations in [121].
7.3.2 Converting to an Agent-Based Model
Figure 7.5 shows a structured view of the embryo to understand how a protein diffuses through the length of the embryo. As assumed with the stochastic
model, the embryo cell is divided into 100 compartments, with the source in
the first compartment. The source produces proteins at a certain rate r, and
depending on a rate d, individual proteins diffuse into the next compartment
X-Machines for Agent-Based Modeling: FLAME Perspectives
7.3.1 Stochastic Modeling
The stochastic Bicoid protein reaction diffusion system was implemented
as a simulation of 100 compartments along the A-P axis, each with length
h = 5µm, which is the average size of one nucleus. The proteins diffuse along
the compartments based on time and concentrations.
Bicoid 1 ⇌ Bicoid i ⇌ Bicoid n
(7.1)
where ⇌ represents the diffusion for i = 1 to n. The Bicoid protein has a life
and degrades in all compartments along the axis, until it translates into bicoid
mRNA in compartment 1 to form the anterior pole of embryo. Details of the
procedure are as follows:
Algorithm: Bicoid reaction-diffusion stochastic simulation
Input: Model parameters; final time.
Output: Bicoid molecular numbers along 100 compartments: m.
Start m = 0; t = 0;
Repeat:
1. Generate two random numbers which are uniformly distributed in
(0, 1): r(1) and r(2).
2. Calculate propensity functions of all the reactions: a = a1 + a2 +
a3 + a4.
3. Calculate the time when next reaction occurs: t + τ , where τ =
1/a ln(1/r(1)).
4. Decide which reaction occurs at Pt + τ : find jǫR such that
Σ
j−1
i=1 a i /a ≤ r(2) <
j
i=1 a i /a,
5. Update numbers of reactants and products in j-th reaction and set
t ← t + τ until time > f inaltime.
The diffusion between neighboring compartments takes place in both directions based on a rate d, related to a diffusion constant of a deterministic model
d = D/h2. The vector m contains a number of molecules along the N = 100
compartments or bins. This is based on the equations in [121].
7.3.2 Converting to an Agent-Based Model
Figure 7.5 shows a structured view of the embryo to understand how a protein diffuses through the length of the embryo. As assumed with the stochastic
model, the embryo cell is divided into 100 compartments, with the source in
the first compartment. The source produces proteins at a certain rate r, and
depending on a rate d, individual proteins diffuse into the next compartment