## Monte Carlo Simulation of Single Molecule Kinetics

The following two Mathematica programs can be used to simulate single-molecule data sets for systems that interconvert between two distinct states, i.e. fluorescent on-state and non-fluorescent off-state.

The One-conformation Model program simulates single-molecule data for a simple first-order reaction shown in Fig. 1a. The molecule leaves state A for state B with a forward rate, ka, and it converts back to state A with a backward reaction rate, kb. State A and B are assumed to be distinguishable spectroscopically in single-molecule experiments, i.e. state A is fluorescent (on) and state B is non-fluorescent (off). The program generates both single-molecule fluorescence traces and dwell times that the system resides in the two fluorescent states (on-times and off-times).

There are four input parameters that can be changed in the program, including step, Prob, Turnover and Time.
1) step: it determines the time resolution and is regarded as the unit time in the simulation. step = 0.01 means the program generates data every 0.01 second.
2) Prob: it’s a 2¥1 vector. Prob[1] = 1 - exp(-kb*step) is the probability of the molecule leaving the off-state; Prob[2] =1 - exp(-ka*step) is the probability of leaving the on-state. When the step size is small so that ka*step and kb*step are small, Prob[1] = kb*step and Prob[2] = ka*step. For example, if kb = 2 s-1, ka = 3 s-1 and step = 0.01 s, Prob[1] = 0.02 and Prob[2] = 0.03.
3) Turnover: this parameter determines the total number of turnover to be generated in the simulation. One can set this number depending on the desired number of on-times and off-times, which is also the number of turnovers, to be generated from the program.
4) Time: it defines the total number of data points and the length of the single molecule trace to be generated. For example, if Time = 10000, with a step size of 0.01 second, the length of the single molecule trace is 100 seconds (Time * step). One can roughly estimate this parameter from the number of turnovers to be generated and the forward as well as backward reaction rates, or simply put in a fairly large value.
In the end, the simulated single molecule fluorescence trace is exported to “C:\\simulation\\data\\one-conform.dat” and the on-times as well as off-times are exported to “C:\\simulation\\data\\on-conform-on-off.dat”. The output locations can be changed.

The Two-conformation Model program simulates single-molecule data for systems that involve a conformational fluctuation that results in reaction rates randomly switching between distinct values as shown in Fig. 1b. In conformation 1, the system turns over between the fluorescent on-state, A1, and non-fluorescent off-state, B1, with reaction rate, ka, 1 and kb, 1, respectively. In conformation 2, the system turns over between the fluorescent on-state, A2, and non-fluorescent off-state, B2, with different reaction rates, ka, 2 and kb, 2, respectively. The two on-states, A1 and A2, are fluorescently indistinguishable, so as the two off-states, B1 and B2. The two conformations can interconvert between each other with different rates, ga,i and gb,i (i = 1, 2). The program generates single-molecule data, including single-molecule fluorescence trajectories, dwell times that the system resides in the two fluorescent states (on-times and off-times), conformation trajectories, and dwell times that the system resides in the two conformation states.

There are five input parameters that can be changed in the program, including step, Prob, cProb, Turnover and Time.
1) step: it determines the time resolution and is regarded as the unit time in the simulation. step = 0.01 means the program generates data every 0.01 second.
2) Prob: it’s a 2¥2 matrix, describing the probability of leaving a specific state. In the above two-conformation model, there are 4 possible microscopic states, including fluorescent states A1 & A2, and non-fluorescent states B1 & B2. Prob[i, 1] = 1 - exp(-(kb,i + gb,i)*step) is the probability of leaving the non-fluorescent state, Bi, in conformation i (i = 1, 2); Prob[i, 2] = 1 - exp(-(ka,i + ga,i)*step) is the probability of leaving the fluorescent states, Ai, in conformation i. When the step size is very small, Prob[i, 1] = (kb,i + gb,i)*step and Prob[i, 2] = (ka,i + ga,i)*step. For example, when kb,1 = ka,1 = 1 s-1, kb,2 = ka,2 = 5 s-1, and gb,1 = ga,1 = gb,2 = ga,2 = 0.3 s-1, Prob[1, 1] = 0.013, Prob[1, 2] = 0.013, Prob[2, 1] = 0.053, and Prob[2, 2] = 0.053.
3) cProb: it’s a 2¥2 matrix, describing the probability of the system converting into a different conformation state, not a different fluorescent state, given that it is leaving a specific state. Therefore, cProb[i, 1] = gb,i /(kb,i + gb,i) is the probability of interconversion between the two non-fluorescent states Bi, and cProb[i, 2] = ga,i /(ka,i + ga,i) (i =1, 2) is the probability of interconversion between the two fluorescent states Ai. For example, when kb,1 = ka,1 = 1 s-1, kb,2 = ka,2 = 5 s-1, and gb,1 = ga,1 = gb,2 = ga,2 = 0.3 s-1, cProb[1, 1] = 0.2308; cProb[1, 2] = 0.2308, cProb[2, 1] = 0.0566, and cProb[2, 2] = 0.0566.
4) Turnover: this parameter determines the total number of turnover to be generated in the simulation. One can set this number depending on the desired number of on-times and off-times, which is also the number of turnovers, to be generated from the program.
5) Time: it defines the total number of data points and the length of the single molecule trace to be generated. For example, if Time = 10000, with a step size of 0.01 second, the length of the single molecule trace is 100 seconds (Time * step). One can roughly estimate this parameter from the number of turnovers to be generated and the forward as well as backward reaction rates, or simply put in a fairly large value.
In the end, the simulated single molecule fluorescence trace is exported to “C:\\simulation\\data\\trace.dat”; the on-times as well as off-times are exported to “C:\\simulation\\data\\on-off.dat”; the time evolution of the conformational state of the system is exported to “C:\\simulation\\data\\conf-trace.dat”; and the time durations that the system resides in the two conformational states are exported to C:\\simulation\\data\\c1-c2.dat”. The output locations can be changed.

Mathematica files;
one-conformation.nb
two-conformation

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