##### Document Text Contents

Page 71

2.2. An analytical approach to hierarchic stochastic modelling

stochastic process emerging by clustering of IP3 receptors.

An approach to model such complex stochastic systems was coined hierarchic

stochastic modeling (HSM), its theoretical developments started already some

years ago [61, 79]. The successful application to Ca2+ dynamics was achieved

recently [103] and further analytical insights were gained even more recently

[72] and shall be presented in the following.

2.2. An analytical approach to hierarchic stochastic

modelling

In this section the general theoretical framework involved in hierarchic stochastic

modeling (HSM) shall be developed. At first, the general idea to refrain from

a pure Markovian description shall be motivated. The formal consequences

including semi-Markovian processes, the correspondent non-Markovian Master

equations and the concept of probability fluxes will be discussed subsequently.

Finally, a specific Ca2+ model developed in ref. [103] shall be analytically solved

in the context of a first passage time problem.

2.2.1. What is HSM ?

The main goal of HSM is a state space reduction without fully neglecting the

microscopic dynamics of the system. As to be seen in the following, this effectively

implies a semi-Markovian description. However, the greater theoretical challenge

is paid off well by a substantial reduction in the number of free parameters in

the model. Additionally, it makes the theory readily applicable to experiments,

which especially in molecular biology rarely observe microscopic state changes

directly. This is due to the enormous complexity of actual molecular interactions

often found for elementary cellular processes such as transcription, translation or

Ca2+ signaling. The many cooperative interactions translate via combinatorics

to high dimensional state spaces which in turn make the application of standard

methods like the chemical master equation often intractable. The integration

of many microscopic states into one mesoscopic observable state is therefore a

naturally choice for the description of intracellular processes and is the core idea

of HSM.

When describing a receptor channel molecule, often the main question of

interest is if the channel is open or closed. The receptor molecule might have

many internal states where only one corresponds to the channel being open

[22, 93]. However, in a standard Markovian description, all internal state

transitions have to be described, also the ones which are not leading to an

opening event. The classical master equation reads

∂

∂t

P

j

i (t) =

N∑

l=1

[

qliP

j

l (t) − qilP

j

i (t)

]

. (2.1)

Here, N is the number of system states and P ji (t) = P (i, t|j, 0) is the conditional

probability that the system is in state i at time t conditioned by being in state

61

2.2. An analytical approach to hierarchic stochastic modelling

stochastic process emerging by clustering of IP3 receptors.

An approach to model such complex stochastic systems was coined hierarchic

stochastic modeling (HSM), its theoretical developments started already some

years ago [61, 79]. The successful application to Ca2+ dynamics was achieved

recently [103] and further analytical insights were gained even more recently

[72] and shall be presented in the following.

2.2. An analytical approach to hierarchic stochastic

modelling

In this section the general theoretical framework involved in hierarchic stochastic

modeling (HSM) shall be developed. At first, the general idea to refrain from

a pure Markovian description shall be motivated. The formal consequences

including semi-Markovian processes, the correspondent non-Markovian Master

equations and the concept of probability fluxes will be discussed subsequently.

Finally, a specific Ca2+ model developed in ref. [103] shall be analytically solved

in the context of a first passage time problem.

2.2.1. What is HSM ?

The main goal of HSM is a state space reduction without fully neglecting the

microscopic dynamics of the system. As to be seen in the following, this effectively

implies a semi-Markovian description. However, the greater theoretical challenge

is paid off well by a substantial reduction in the number of free parameters in

the model. Additionally, it makes the theory readily applicable to experiments,

which especially in molecular biology rarely observe microscopic state changes

directly. This is due to the enormous complexity of actual molecular interactions

often found for elementary cellular processes such as transcription, translation or

Ca2+ signaling. The many cooperative interactions translate via combinatorics

to high dimensional state spaces which in turn make the application of standard

methods like the chemical master equation often intractable. The integration

of many microscopic states into one mesoscopic observable state is therefore a

naturally choice for the description of intracellular processes and is the core idea

of HSM.

When describing a receptor channel molecule, often the main question of

interest is if the channel is open or closed. The receptor molecule might have

many internal states where only one corresponds to the channel being open

[22, 93]. However, in a standard Markovian description, all internal state

transitions have to be described, also the ones which are not leading to an

opening event. The classical master equation reads

∂

∂t

P

j

i (t) =

N∑

l=1

[

qliP

j

l (t) − qilP

j

i (t)

]

. (2.1)

Here, N is the number of system states and P ji (t) = P (i, t|j, 0) is the conditional

probability that the system is in state i at time t conditioned by being in state

61