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Delay chemical master equation: direct and closed-form solutions.

Leier A, Marquez-Lago TT - Proc. Math. Phys. Eng. Sci. (2015)

Bottom Line: In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME).However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME.In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation.

View Article: PubMed Central - PubMed

Affiliation: Okinawa Institute of Science and Technology , Onna-son, Okinawa, Japan.

ABSTRACT

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

No MeSH data available.


Related in: MedlinePlus

Time evolution of the DCME solution with initial state (M,P)=(0,0). Different slices correspond to different time points. Each rectangle represents a state (M,P). Its colour refers to the probability of observing the system in this state at that time.
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RSPA20150049F4: Time evolution of the DCME solution with initial state (M,P)=(0,0). Different slices correspond to different time points. Each rectangle represents a state (M,P). Its colour refers to the probability of observing the system in this state at that time.

Mentions: Figure 4 presents the time evolution of the DCME solution of our system for r=7, μ=μm=μp=0.2, and k=km=kp=1, when starting at state (M,P)=(0,0) and assuming a single DNA. Figure 5 shows a comparison of the DCMC direct numerical solution against statistics obtained from independent SSA simulations. The number of molecules M and P were limited to values between [0, 9] and [0, 14], respectively. For this state space, the error of the FSP is around 0.1%, and results show a remarkably good fit.Figure 4.


Delay chemical master equation: direct and closed-form solutions.

Leier A, Marquez-Lago TT - Proc. Math. Phys. Eng. Sci. (2015)

Time evolution of the DCME solution with initial state (M,P)=(0,0). Different slices correspond to different time points. Each rectangle represents a state (M,P). Its colour refers to the probability of observing the system in this state at that time.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4528653&req=5

RSPA20150049F4: Time evolution of the DCME solution with initial state (M,P)=(0,0). Different slices correspond to different time points. Each rectangle represents a state (M,P). Its colour refers to the probability of observing the system in this state at that time.
Mentions: Figure 4 presents the time evolution of the DCME solution of our system for r=7, μ=μm=μp=0.2, and k=km=kp=1, when starting at state (M,P)=(0,0) and assuming a single DNA. Figure 5 shows a comparison of the DCMC direct numerical solution against statistics obtained from independent SSA simulations. The number of molecules M and P were limited to values between [0, 9] and [0, 14], respectively. For this state space, the error of the FSP is around 0.1%, and results show a remarkably good fit.Figure 4.

Bottom Line: In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME).However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME.In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation.

View Article: PubMed Central - PubMed

Affiliation: Okinawa Institute of Science and Technology , Onna-son, Okinawa, Japan.

ABSTRACT

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

No MeSH data available.


Related in: MedlinePlus