S10
Supporting Information
for
Stochastic mapping of first order reaction networks: a systematic comparison of the stochastic and deterministic approaches
Gábor Lente
Department of Inorganic and Analytical Chemistry, University of Debrecen, Debrecen, Hungary. Fax: 36-52-518-660; Tel: 36-52-512-900 Ext.: 22373; E-mail:
Derivation of equations appearing in the manuscript:
Equation 11:
Equation (11) is the number of different solutions of Diophantine Equation (10). Equation (2.25) in reference 34 gives this formula directly. In combinatorics, this is identical to the number of multisets of (n - 1) length on (N0 + 1) symbols.
Equation 12:
The case n = 1 does not make sense as no reaction is possible with only a single substance. For n = 2, the formula is simplified as follows:
As 0 £ a2 £ N0, this formula returns a different integer between 1 and N0 + 1 to each state, therefore, it is suitable enumerating function.
The rest of the proof will be based on mathematical induction. Suppose that the suitability of the formula is already proved for n = q. It will be shown that the formula is then appropriate for n = q + 1 as well.
The structure of the formula is such that states with aq+1 = 0 are enumerated first, then states with aq+1 = 1 and so on. Equation (11) gives the number of states with aq+1 = k as follows:
Therefore, the sum of the number of states occurring before aq+1 = k is given as:
The enumerating function for n = q + 1 can be constructed as:
The last part of this series of equations is exactly the enumerating function for n = q + 1.
Equation 18:
Through a suitable linear combination of differential equations appearing in Equation (13) and using the definition of expectations in Equation (14), an equation fully analogous Equation (4) can be obtained. The suitable linear combination of the parts of Equation (3) is the following when the differential equation for the hth variable is derived:
From this equation, it is not easy to see the transformations leading to the form analogous to Equation (4). However, a somewhat more heuristic line of thought can also be used. The expectation for variable ai can only change by one in processes where Ai is consumed or produced. Therefore, the following differential equation can be stated using the infinitesimal transition probabilities:
Transforming this equation gives the following formula:
This differential equation is fully analogous to Equation (4). As the sum of all Pi values is 1, whereas the sum of all expectations must be N0, the relationship given in Equation (18) must hold.
Equation 20:
Using a sequence of thought very similar to that presented for the derivation of Equation (18):
Equation (20) follows directly from the last line using Equation (18).
Equation 21:
Using a sequence of thought very similar to that presented for the derivation of Equation (18):
The form in the least line can be easily re-arranged to Equation (21).
Equation 22:
When all the molecules are in the same state initially, they are also indistinguishable. Therefore, the probability given in this equation is simply obtained as the product of the probabilities that individual molecules are in a given state (P functions) multiplied by the appropriate combinatorial factor.
Equation 23:
The probability that an individual is exactly in the Ai state is Pi. Therefore, the probability that exactly N of the N0 independent molecules is in the Ai state is obtained from a straightforward binomial expression shown in Equation (23).
Equation 24:
The known standard deviation of the binomial distribution shown in Equation (23) yields Equation (24).
Equation 25:
The equation follows from a simple binomial distribution.
Equation 26:
The Equation follows from a combination of Equations (23) and (25).
Equation 29:
For the product molecule in the defined case:
Therefore, the expectation and standard deviation of the number of A2 molecules:
The condition for defining the stochastic region is that the standard deviation should be higher than 1 % of the expectation:
Equation 30:
If N (< N0) molecules remains out of the initial N0 in the process, the probability function for k1,2t can be written as follows:
The corresponding distribution function is:
The expectation for k1,2t will be calculated by using the following known integral:
The expectation is calculated as follows:
A straightforward rearrangement of this last equation gives Equation (28).
Equation 31:
This equation can be proved using the distribution function introduced during the derivation of Equation (28).
The standard deviation is then calculated as
A straightforward rearrangement of this last equation gives Equation (29).
Equation 32:
For the product molecule in the defined case:
Therefore, the expectation and standard deviation of the number of A2 molecules:
The condition for defining the stochastic region is that the standard deviation should be higher than 1 % of the expectation:
Equation 33:
For the reactant and intermediate molecules in the defined case:
Therefore, the expectation and standard deviation of the number of A2 molecules:
The condition for defining the stochastic region is that the standard deviation should be higher than 1 % of the expectation:
Equations 34-36:
This equations can be derived by the standard method of solving linear, first order, ordinary differential equations.
Fig. S1 Three-dimensional stochastic map for the irreversible first order reaction using the number of product molecules as the target variable.
Fig. S2 Three-dimensional stochastic map for the reversible first order reaction using the number of product molecules as the target variable.
Fig. S3 Three-dimensional stochastic map for the two consecutive first order reactions using the number of intermediate molecules as the target variable.
Fig. S4 Expectation of the number of intermediate molecules in two consecutive first order reactions as a function of dimensionless time. Red solid line: expectation values. Blue dotted lines: standard errors (±s) of the expectation. Green markers: a simulation using the Gillespie algorithm.