Q2. What are the future works in "A stochastic analysis of first-order reaction networks" ?
The mathematical formulation that leads to a direct solution of the moment equations for a well-stirred system can be extended to arbitrary networks of well-mixed compartments that are coupled by diffusion. The authors demonstrate that the eigenvalues that govern the evolution in such distributed systems are solutions of a one-parameter family of modified kinetic matrices and thus one can formally display the solution for the first two moments in this case as well. The authors anticipate that the analytical framework presented here will be extended to the stochastic analysis of nonlinear reaction networks, and their analysis of first-order reaction network will lead to insights into the local linear behavior of such networks.
Q3. What is the main objective of many of the analyses treating biological systems?
A major objective of many of the analyses treating biological systems is prediction of the stochastic variations or noise of the concentrations.
Q4. What is the probability distribution of the number of balls in an urn?
Klein treated the number of balls in an urn as a measure of the occupancy of an energy state, and calculated the probability of the number of balls in an urn as a function of the transition probability and the initial distribution.
Q5. What is the proof for the stationary distribution of the number of individuals in each colony?
If one considers each colony to be a distinct species, the open migration process is equivalent to an open conversion reaction system, and the proof for the stationary distribution of the number of individuals in each colony stated by Kelly (1979) may be considered as another proof for the distribution of the number of each species in an open conversion network that the authors derive later.
Q6. What is the evolution of the surface morphology during epitaxial growth?
The evolution of the surface morphology during epitaxial growth involves the nucleation and growth of atomic islands, and these processes may be described by first-order adsorption and desorption reactions coupled with diffusion along the surface.
Q7. What is the way to explain the noise in a well-stirred system?
The mathematical formulation that leads to a direct solution of the moment equations for a well-stirred system can be extended to arbitrary networks of well-mixed compartments that are coupled by diffusion.
Q8. What is the longest characteristic time for the evolution of M and V?
For a closed system, one of the eigenvalues is zero and hence the longest characteristic time for the evolution of M and V will be identical.
Q9. What is the function that identifies a linear combination of species as a?
The function ν̂, which identifies a linear combination of species as a complex is onto, and the relation R has the properties (i) (C(i), C( j)) ∈ R if and only if there exists one and only one reaction of the form C(i)→ C( j), (ii) for every i there is a j = i such that (C(i), C( j)) ∈ R, (iii) (C(i), C(i)) ∈ R.
Q10. What is the mean and variance for the mth species at the steady state?
It follows from (55) that the mean and variance for the mth species at the steady state are given byMm = Nπm = E[Nm ] σ 2(Nm ) = Nπm (1− πm) = E[Nm ] ( 1− E[Nm ]N) . (56)Notice that πm is the steady-state fraction of the mth molecular species in a deterministic description, and since this is fixed by the reaction rates, the variance σ 2(Nm ) does not approach the mean even as N → ∞.