Stochastic dynamic study of optical transition properties of single GFP-like molecules

Abstract

Due to high fluctuations and quantum uncertainty, the processes of single-molecules should be treated by stochastic methods. To study fluorescence time series and their statistical properties, we have applied two stochastic methods, one of which is an analytic method to study the off-time distributions of certain fluorescence transitions and the other is Gillespie’s method of stochastic simulations. These methods have been applied to study the optical transition properties of two single-molecule systems, GFPmut2 and a Dronpa-like molecule, to yield results in approximate agreement with experimental observations on these systems. Rigorous oscillatory time series of GFPmut2 before it unfolds in the presence of denaturants have not been obtained based on the stochastic method used, but, on the other hand, the stochastic treatment puts constraints on the conditions under which such oscillatory behavior is possible. Furthermore, a sensitivity analysis is carried out on GFPmut2 to assess the effects of transition rates on the observables, such as fluorescence intensities.

Appendix: The off-time distribution for the 4-state and 6-state models

For the 6-state model, the probability that it takes less time than t to reach the next green emission from state N is:

$$\begin{array}{@{}rcl@{}} P(T_{N}<t) & =& P(T_{k_{2}}<T_{k_{7}})P(T_{k_{9}}<T_{k_{3}})P(T_{k_{1}}+T_{k_{2}}+T_{k_{9}}<t)\\ && +P(T_{k_{2}}<T_{k_{7}})P(T_{3}<T_{k_{9}})P(T_{k_{1}}+T_{k_{2}}+T_{k_{3}}+T_{k_{4}}<t)\\ && +P(T_{k_{7}}<T_{k_{2}})P(T_{k_{1}}+T_{k_{7}}+T_{N}<t). \end{array} $$

(A.1)

The probability distribution of T _N can be shown to be

$$\begin{array}{@{}rcl@{}} f_{T_{N}}(t) \!& =&\! \frac{dP(T_{N}<t)}{dt}\\ \!& =&\! {{\int}_{0}^{t}}{\int}_{0}^{t_{1}}dt_{1}dt_{2}e^{-k_{7}(t_{1}-t_{2})}e^{-k_{3}(t-t_{1})}k_{1}e^{-k_{1}t_{2}}k_{2}e^{-k_{2}(t_{1}-t_{2})}k_{9}e^{-k_{9}(t-t_{1})}\\ && +{{\int}_{0}^{t}}{\int}_{0}^{t_{1}}{\int}_{0}^{t_{2}}dt_{1}dt_{2}dt_{3}e^{-k_{7}(t_{2}-t_{3})}e^{-k_{9}(t_{1}-t_{2})}k_{1}e^{-k_{1}t_{3}}k_{2}e^{-k_{2}(t_{2}-t_{3})}k_{3}e^{-k_{3}(t_{1}-t_{2})}\\&&\times k_{4}e^{-k_{4}(t-t_{1})}\\ && +{{\int}_{0}^{t}}{\int}_{0}^{t_{1}}dt_{1}dt_{2}e^{-k_{2}(t_{1}-t_{2})}k_{1}e^{-k_{1}t_{2}}k_{7}e^{-k_{7}(t_{1}-t_{2})}f_{T_{N}}(t-t_{1})\\ \!& =&\! \frac{k_{1}k_{2}k_{9}}{-k_{1}+k_{2}+k_{7}}\left( \frac{e^{-k_{1}t}-e^{-k_{3}t-k_{9}t}}{-k_{1}+k_{3}+k_{9}}-\frac{e^{-k_{2}t_{1}-k_{7}t_{1}}-e^{-k_{3}t-k_{9}t}}{-k_{2}-k_{7}+k_{3}+k_{9}}\right)+\frac{k_{1}k_{2}k_{3}k_{4}}{k_{7}-k_{1}+k_{2}}\\ &&\times\left( \frac{\frac{e^{-k_{1}t}-e^{-k_{4}t}}{k_{4}-k_{1}}-\frac{e^{-k_{9}t-k_{3}t}-e^{-k_{4}t}}{-k_{9}-k_{3}+k_{4}}}{k_{9}-k_{1}+k_{3}}-\frac{\frac{e^{-k_{7}t-k_{2}t}-e^{-k_{4}t}}{k_{4}-k_{7}-k_{2}}-\frac{e^{-k_{9}t_{1}-k_{3}t_{1}}-e^{-k_{4}t}}{-k_{9}-k_{3}+k_{4}}}{k_{9}-k_{7}-k_{2}+k_{3}}\right)\\ && +\frac{k_{1}k_{7}}{k_{2}-k_{1}+k_{7}}{{\int}_{0}^{t}}dt_{1}(e^{-k_{1}t_{1}}-e^{-k_{2}t_{1}-k_{7}t_{1}})f_{T_{N}}(t-t_{1}). \end{array} $$

(A.2)

After applying the rule for the Laplace transform of a convolution, (A.2) yields

$$\begin{array}{@{}rcl@{}} \tilde{f}_{T_{N}}(s) & =& \frac{k_{1}k_{2}k_{9}}{-k_{1}+k_{2}+k_{7}}(\frac{\frac{1}{s+k_{1}}-\frac{1}{s+k_{3}+k_{9}}}{-k_{1}+k_{3}+k_{9}}-\frac{\frac{1}{s+k_{2}+k_{7}}-\frac{1}{s+k_{3}+k_{9}}}{-k_{2}-k_{7}+k_{3}+k_{9}})\\ & & +\frac{k_{1}k_{2}k_{3}k_{4}}{k_{7}-k_{1}+k_{2}}\left( \frac{\frac{\frac{1}{s+k_{1}}-\frac{1}{s+k_{4}}}{k_{4}-k_{1}}-\frac{\frac{1}{s+k_{3}+k_{9}}-\frac{1}{s+k_{4}}}{-k_{9}-k_{3}+k_{4}}}{k_{9}-k_{1}+k_{3}}-\frac{\frac{\frac{1}{s+k_{2}+k_{7}}-\frac{1}{s+k_{4}}}{k_{4}-k_{7}-k_{2}}-\frac{\frac{1}{s+k_{3}+k_{9}}-\frac{1}{s+k_{4}}}{-k_{9}-k_{3}+k_{4}}}{k_{9}-k_{7}-k_{2}+k_{3}}\right)\\ & & +\frac{k_{1}k_{7}}{k_{2}-k_{1}+k_{7}}(\frac{1}{s+k_{1}}-\frac{1}{s+k_{2}+k_{7}})\tilde{f}_{T_{N}(s)}. \end{array} $$

(A.3)

Therefore,

$$ \tilde{f}_{T_{N}}(s) = \frac{(k_{9}s+k_{4}k_{9}+k_{3}k_{4})k_{1}k_{2}}{(s^{2}+sk_{2}+sk_{7}+sk_{1}+k_{1}k_{2})(s+k_{4})(s+k_{3}+k_{9})}. $$

(A.4)

The probability of off-time less than t is

$$\begin{array}{@{}rcl@{}} P(t_{off}<t) \!& =&\! \frac{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}}{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}+P_{A*}k_{4}}P(T_{k_{7}}<T_{k_{2}})P(T_{k_{6}}<T_{k_{8}})P(T_{k_{6}}\,+\,T_{k_{1}}+T_{k_{7}}+T_{N}<t)\\ && +\frac{P_{A*}k_{4}}{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}+P_{A*}k_{4}}P(T_{k_{5}}<T_{k_{10}})P(T_{k_{6}}<T_{k_{8}})\\&&\times P(T_{k_{5}}+T_{k_{6}}+T_{k_{1}}+T_{k_{7}}+T_{N}<t) \end{array} $$

(A.5)

where P _I∗, P _A∗ are the probabilities of the protein being in states I* and A*, and $T_{k_{i}}$ is the time it takes to complete the transition with rate k _i . The distribution of the off-time can be shown as:

$$\begin{array}{@{}rcl@{}} f_{t_{off}}(t) & =& \frac{dP(t_{off}(t)<t)}{dt}\\ & = &\frac{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}}{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}+P_{A*}k_{4}}\\ & & \times{{\int}_{0}^{t}}{\int}_{0}^{t_{1}}{\int}_{0}^{t_{2}}dt_{1}dt_{2}dt_{3}e^{-k_{2}(t_{1}-t_{2})}e^{-k_{8}t_{3}}k_{6}e^{-k_{6}t_{3}}k_{1}e^{-k_{1}(t_{2}-t_{3})}k_{7}e^{-k_{7}(t_{1}-t_{2})}\\&&\times f_{T_{N}}(t-t_{1})\\ & & +\frac{P_{A*}k_{4}}{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}+P_{A*}k_{4}}{{\int}_{0}^{t}}{\int}_{0}^{t_{1}}{\int}_{0}^{t_{2}}{\int}_{0}^{t_{3}}dt_{1}dt_{2}dt_{3}dt_{4}e^{-k_{10}t_{4}}e^{-k_{8}(t_{3}-t_{4})}e^{-k_{2}(t_{1}-t_{2})}\\ && \times k_{5}e^{-k_{5}t_{4}}k_{6}e^{-k_{6}(t_{3}-t_{4})}k_{1}e^{-k_{1}e^{-k_{1}(t_{2}-t_{3})}}k_{7}e^{-k_{7}(t_{1}-t_{2})}f_{T_{N}}(t-t_{1}). \end{array} $$

(A.6)

We rewrite (A.6) as:

$$ f_{t_{off}}(t) = \frac{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}}{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}+P_{A*}k_{4}}k_{1}k_{6}k_{7}I_{1}+\frac{P_{A*}k_{4}}{P_{I*}\frac{{k_{9}^{2}}}{k_{9}+k_{3}}+P_{A*}k_{4}}k_{5}k_{6}k_{1}k_{7}I_{2} $$

(A.7)

where

$$\begin{array}{@{}rcl@{}} I_{1} & =& {{\int}_{0}^{t}}{\int}_{0}^{t_{1}}{\int}_{0}^{t_{2}}dt_{1}dt_{2}dt_{3}e^{-k_{2}(t_{1}-t_{2})}e^{-k_{8}t_{3}}e^{-k_{6}t_{3}}e^{-k_{1}(t_{2}-t_{3})}e^{-k_{7}(t_{1}-t_{2})}f_{T_{N}}(t-t_{1})\\ & =& {{\int}_{0}^{t}}dt_{1}\frac{\frac{e^{-k_{6}t_{1}-k_{8}t_{1}}-e^{-k_{2}t_{1}-k_{7}t_{1}}}{-k_{6}-k_{8}+k_{2}+k_{7}}-\frac{e^{-k_{1}t_{1}}-e^{-k_{2}t_{1}-k_{7}t_{1}}}{k_{2}+k_{7}-k_{1}}}{k_{1}-k_{6}-k_{8}}f_{T_{N}}(t-t_{1}) \end{array} $$

(A.8)

and

$$\begin{array}{@{}rcl@{}} I_{2} & =& {{\int}_{0}^{t}}{\int}_{0}^{t_{1}}{\int}_{0}^{t_{2}}{\int}_{0}^{t_{3}}dt_{1}dt_{2}dt_{3}dt_{4}e^{-k_{10}t_{4}}e^{-k_{8}(t_{3}-t_{4})}e^{-k_{2}(t_{1}-t_{2})}\\ &&\times e^{-k_{5}t_{4}}e^{-k_{6}(t_{3}-t_{4})}e^{-k_{1}e^{-k_{1}(t_{2}-t_{3})}}e^{-k_{7}(t_{1}-t_{2})}f_{T_{N}}(t-t_{1})\\ &&\times {{\int}_{0}^{t}}dt_{1}\frac{\frac{e^{-k_{10}t_{1}-k_{5}t_{1}}-e^{-k_{2}t_{1}-k_{7}t_{1}}}{-k_{10}-k_{5}+k_{2}+k_{7}}-\frac{e^{-k_{1}t_{1}}-e^{-k_{2}t_{1}-k_{7}t_{1}}}{k_{2}+k_{7}-k_{1}}}{(-k_{10}-k_{5}+k_{6}+k_{8})(-k_{10}-k_{5}+k_{1})}f_{T_{N}}\\ &&+ {{\int}_{0}^{t}}dt_{1}\frac{\frac{e^{-k_{6}t_{1}-k_{8}t_{1}}-e^{-k_{2}t_{1}-k_{7}t_{1}}}{-k_{6}-k_{8}+k_{2}+k_{7}}-\frac{e^{-k_{1}t_{1}}-e^{-k_{2}t_{1}-k_{7}t_{1}}}{k_{2}+k_{7}-k_{1}}}{(-k_{10}-k_{5}+k_{6}+k_{8})(-k_{6}-k_{8}+k_{1})}f_{T_{N}}(t-t_{1}). \end{array} $$

(A.9)

The Laplace transform of the off-time distribution yields:

$$\begin{array}{@{}rcl@{}} \tilde{f}_{t_{off}}(s) & =& \frac{(s{k_{9}^{2}}+k_{5}{k_{9}^{2}}+{k_{9}^{2}}k_{10}+k_{3}k_{5}k_{9}+{k_{3}^{2}}k_{10}+{k_{3}^{2}}k_{5}+k_{3}k_{9}k_{10})(sk_{9}+k_{4}k_{9}+k_{3}k_{4})}{(s+k_{4})(s+k_{3}+k_{9})(s^{2}+k_{2}s+k_{7}s+k_{1}s+k_{1}k_{2})(s+k_{6}+k_{8})(s+k_{1})}\\ &&\times \frac{1}{(s+k_{5}+k_{10})(s+k_{2}+k_{7})}\\&&\times\frac{{k_{1}^{2}}k_{2}k_{5}k_{6}k_{7}}{k_{3}k_{5}k_{9}+{k_{3}^{2}}k_{5}+k_{3}k_{9}k_{10}+{k_{3}^{2}}k_{10}+k_{5}{k_{9}^{2}}}. \end{array} $$

(A.10)