How to Interpret the Parameters?

    We consider the system as a stochastic process, and the probability of a ribosome to change its state is described by the rate constants ω. Now we interpret these parameters:

    For the simplest chemical reaction with reaction rate k, by law of mass actions, we see that:

    Therefore, in time interval Δt, the probability of A to transform into B should be:

    Hence, if we set ω = k, ω can be interpreted as the probability "per Δt" for the reaction to occur on A. While the actual probability should be p = ωΔt or with more accuracy, .

    The rate constants can be obtained from experiments for E-coli as:

    Moreover, the initiation and termination process are described by two parameters α and β, denoting the reaction rate of a fresh ribosome binds to the first l sites of mRNA and the probability of a ribosome at the rear of the mRNA strand to get detached from the mRNA strand in the time interval Δt.

Theoretical Analysis of the Model

    We first specify some assumptions and notations. P_μ(i) with μ denotes the probability of finding a ribosome in state μ which is 'working' on site i and is assumed to be the site which is covered by the leftmost site of the ribosome. is the probability of finding a ribosome at i. The ribosome can shift forward only if the sites with index from i+1 to i+l are all empty. Assume that the probabilities of finding a ribosome at any sites are independent. Then the probability Q(i) for the required sites are empty is .

    Thus the differential equations are:

    Use Runge-Kutta method to get the solution. Here we assume that, if there were no detachments, the ribosomes are completely recycled.

    However, this will not be used in our programming.

Monte Carlo Methods

    We use Monte Carlo method in our programming for convenience. Here is the specification of the algorithm:

    Note that Each Monte Carlo Step(MCS) represents a time interval Δt = 0.001s in our simulation.

    The mRNA strand of N codons is implemented by an array mRNA of length N, which takes value from 0 to 5 in each entry. 1 to 5 represents the state of ribosome on the certain position, while 0 represents the site is empty, or is not occupied by a "working" ribosome.

    1. Set the initial condition: all the sites on the mRNA strand is empty except the site in the first entry.

    2. Begin a new MCS:

        a) Select an entry with value 1 (ribosome in state 1), generate a random number. If the random number is less than , then change the value to 2, otherwise proceed;

        b) Select an entry with value 2 (ribosome in state 2), generate a random number. If the random number is less than , then change the value to 1, otherwise proceed;

        c) Select an entry with value 2 (ribosome in state 2), generate a random number. If the random number is less than , then change the value to 3, otherwise proceed;

        d) Select an entry with value 3 (ribosome in state 3), generate a random number. If the random number is less than , then change the value to 4, otherwise proceed;

        e) Select an entry with value 4 (ribosome in state 4), generate a random number. If the random number is less than , then change the value to 5, otherwise proceed;

        f) Select an entry with value 5 (ribosome in state 5), generate a random number. If the random number is less than , and that assume the index of the selected entry is i, if entries with index from i+1 to i+l are all of value 0, then change mRNA[i] to 0 and mRNA[i+1] to 1;

        g) Look at the last entry, assume it has index N (when use MATLAB to program) or index N-1 (when use C++ to program), generate a random number. If the value of the last entry is 5 and the random number is less than , then change the last entry to 0.

        h) Look at the first a few entries. If the first l entries are 0, then generate a random number. If the random number is less than , then change the first entry to 1.

    3. Iterate Step 2, until maximal number of Monte Carlo Steps are reached.