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* School of Chemistry, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel; and Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892
Correspondence: Address reprint requests to Ophir Flomenbom, E-mail: flomenbo{at}post.tau.ac.il.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONFROM TRAJECTORIES TO KINETIC...DISCUSSION AND CONCLUDING...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONFROM TRAJECTORIES TO KINETIC...DISCUSSION AND CONCLUDING...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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), great advances have been made in our ability to look at complex systems on the single-molecule level (Moerner and Orrit, 1999; Weiss, 1999; Nie et al., 1994; Shera et al., 1990; Mets and Rigler, 1994; Ha et al., 1999; Schuler et al., 2002; Yang et al., 2003; Rhoades et al., 2003; Wennmalm et al., 1997; Bokinsky et al., 2003; Lu et al., 1998; Edman et al., 1999; Edman and Rigler, 2000; Velonia et al., 2005; Flomenbom et al., 2005; Kasianowicz et al., 1996). In an important class of such experiments, the output is a time series (trajectory) of on-off events (Fig. 1, A and B). For example, in patch-clamp measurements (Neher and Sakmann, 1976), one records the ion current through a membrane pore under an applied electric field for a long time. The fluctuations between two values of the current are attributed to conformational changes that result in opening and closing the membrane pore. From the two-state current trajectory, one wishes to learn about the dynamics of conformational changes of the membrane pore. In single enzyme activity measurements (Lu et al., 1998; Edman et al., 1999; Edman and Rigler, 2000; Velonia et al., 2005; Flomenbom et al., 2005), one monitors photon counts as a function of time, and collects the counts into bins giving rise to the trajectory. A two-state trajectory is obtained when either the enzyme itself switches between a fluorescent state and a nonfluorescent state (Lu et al., 1998), or a nonfluorescent substrate is transformed into a fluorescent product (Edman et al., 1999; Edman and Rigler, 2000; Velonia et al., 2005; Flomenbom et al., 2005). By studying this system, one wishes to deduce the mechanism of the enzymatic activity.
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Such a two-state trajectory contains information about the underlying mechanism, which we describe by a kinetic scheme in which each substate belongs either to the on state or to the off state. The kinetic scheme may have a large number of substates (Fig. 2, A–F), and a net flow at steady state along some of the connections (Fig. 2, G–I) (i.e., a nonequilibrium steady state), when an external source of energy is present (Hill, 1985). The goal is to learn as much as possible about the underlying kinetic scheme.
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FROM TRAJECTORIES TO KINETIC SCHEMES |
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TOPABSTRACTINTRODUCTIONFROM TRAJECTORIES TO KINETIC...DISCUSSION AND CONCLUDING...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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and of the off state, 
These functions, which cannot be found from bulk experiments, can be calculated for any kinetic scheme (Cao, 2000). Clearly, any proposed kinetic scheme must reproduce 
and However, when and 
are multiexponentials, several models will fulfill this requirement, and their number increases with the complexity of the waiting time PDFs (the trajectories on Fig. 1, A and B, have the same waiting time PDFs, but were produced from different kinetic schemes). Can one discriminate between kinetic schemes that lead to the same and by looking at the trajectory in more detail?
A trajectory is completely described by and only when waiting times along the trajectory are uncorrelated. Therefore, kinetic schemes that lead to uncorrelated trajectories with the same and cannot be distinguished by the trajectory analysis. This means that the trajectory from such a kinetic scheme does not contain information about the connectivity of substates within each of the two states, which, as shown below, is a consequence of a specific connectivity between substates of different states. We say that such schemes are "reducible" to a two-state semi-Markovian (TSSM) scheme (Fig. 2 J). A TSSM process is one where the on [off] waiting times are drawn randomly and independently out of a nonexponential []. In the literature, the term non-Markovian is often used for any process with nonexponential waiting time PDFs. However, here we reserve this term to describe a trajectory of correlated waiting times.
The most straightforward test for correlation in the trajectory is based on the two successive waiting times PDFs, 
A trajectory shows no correlations when 
can be written, for every x and y, as a product of the individual waiting time PDFs, 
and 
 | (1) |
When all two successive waiting times PDFs are factorized, higher order successive waiting times PDFs, e.g., 
will also be factorized. Since higher order successive waiting times PDFs determine all the statistical properties of the trajectory and these factorize when Eq. 1 is fulfilled, it follows that for uncorrelated trajectories and contain all the information in the time series.
Kinetic schemes are reducible (i.e., Eq. 1 is fulfilled) regardless of the system parameters if and only if after every transition from the on state to the off state, the off substates are populated with the same initial probabilities, and vice versa. This occurs only for a very specific connectivity between the on and the off substates, and we now give a full characterization of the reducible schemes. When only reversible connections between substates are present, a scheme is reducible when the on and the off regions are connected through one substate (Fig. 2, A–F), called a gateway substate. In general, there are two types of gateway substates. A type 1 gateway substate is one where all the transitions from the other state enter it (the on substate 1 in Fig. 2 G). A type 2 gateway substate is one where all the transitions to the other state originate from it (the on substate 2 in Fig. 2 G). Thus, for a reducible scheme with only reversible connections, the gateway substate is of both types simultaneously. For a kinetic scheme with a nonequilibrium steady state, there are three combinations of gateway substates that lead to a reducible scheme: a), two gateway substates of different types in the same state (Fig. 2 G), and b) and c), two gateway substates of the same type, either type 1 (Fig. 2 H) or type 2 (Fig. 2 I), in different states. Note that the above requirements are the minimal ones and a reducible scheme can possess more than two gateway substates. Because our argument relies only on the connectivity of the scheme, the reducible schemes can be characterized by any substate waiting time PDFs and not just the Markovian (exponential) one. Finally, it should be pointed out that other less general schemes can fulfill Eq. 1, thus are reducible, because of symmetry for special choices of the transition rates.
As an example of reducible kinetic schemes, consider the two schemes shown in Fig. 2, B and C, each containing n off substates and one on substate. Both schemes are reducible because there is only one substate in the on state. Even though they reflect very different mechanisms, it is possible to find transition rates that make and of the two schemes the same (e.g., by equating coefficients of the powers of the Laplace variable s of 
and 
from the two schemes (
), and solving the resulting set of equations relating the transition rates of the two models). The trajectories generated from the two schemes will then be identical (in a statistical sense). Contrary to our results, in the context of enzyme kinetics, it has been previously suggested that it is possible to distinguish between schemes, Fig. 2, B and C, using more sophisticated analyses of the trajectory (Edman and Rigler, 2000). The simplest equivalent reducible schemes are shown in Fig. 2, D–F. Recently, Witkoskie and Cao (2004) pointed out that counter to intuition two of those schemes (Fig. 2, E and F) can be made indistinguishable using similarity transformation arguments.
For irreducible kinetic schemes is not factorized for at least one combination of 
In these cases, functions other than the waiting time PDFs contain additional information. Such functions are: i), 
itself (Lu et al., 1998; Cao, 2000; McManus et al., 1985; Colquhoun et al., 1996), as used in the pioneering work of Xie and collaborators (1998), and calculated for any kinetic scheme by Cao (2000); ii), the x-y propagator for stationary processes, which is the probability density to be in state y at time t given that the process was in state x at time 0 (Lu et al., 1998; Edman et al., 1999; Edman and Rigler, 2000; Flomenbom et al., 2005; Schenter et al., 1999; Boguñá et al., 2000), and determines the normalized state-correlation function, which is the bulk relaxation function; iii), higher order state propagators (Edman and Rigler, 2000; Schenter et al., 1999; Wang and Wolynes, 1995), or the corresponding higher order state-correlation functions; iv), higher order successive waiting times PDFs, e.g. 
Note that the functions in i, iii, and iv can be obtained only from single-molecule experiments.
Which of these functions is the most useful in differentiating among irreducible schemes is still an open question. In practice, a function that involves many arguments will be noisy due to the limited number of events in the time series. We have found that the PDF of the sum of (or, binned) successive waiting times, e.g., 
can not only be more accurately obtained from finite trajectories, but is more discriminatory than the equal successive waiting times PDF (Supplementary Material), e.g. 
(Cao, 2000). 
can be easily constructed from the trajectory by building the histogram of the random variable 
obtained from all adjacent waiting times in the time series. One can also calculate, in addition to the functions themselves, the difference between them and the product of the individual waiting time PDFs, e.g., 
and 
where 
These differences vanish for reducible schemes.
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DISCUSSION AND CONCLUDING REMARKS |
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TOPABSTRACTINTRODUCTIONFROM TRAJECTORIES TO KINETIC...DISCUSSION AND CONCLUDING...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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and 
one should immediately determine whether the underlying kinetic scheme is reducible using Eq. 1. Due to the finite length of the trajectory, the moments of can be more accurately calculated than the PDF, and should be compared to the corresponding products of the moments of 
and 
Another test compares the bulk relaxation function (the state-correlation function) obtained directly from the trajectory, with the corresponding theoretical result for a TSSM process (Flomenbom et al., 2005). The expression for the bulk relaxation function for a stationary TSSM is known, in Laplace space, for arbitrary waiting time PDFs (Cox, 1962; also see Eq. 3.15 in Boguñá et al., 2000), so one can plug in the Laplace transforms of the experimental and into this expression, and invert the result, either analytically or numerically, back into the time domain. If the experimental bulk relaxation function and the theoretical one for a TSSM process with the experimental and coincide, the scheme is reducible, and no further analysis is required. Another simple and informative analysis method involves the trajectory of the waiting times as a function of the occurrence index. Correlations between waiting times can be detected more easily from this trajectory than the on-off trajectory (compare Figs. 1 and 3), and used to learn about the scheme transition rate values (see the caption of Fig. 3).
Finally, we note that some of the fundamental concepts presented in this work have already been used in the analyses of the catalytic activity of individual lipase molecules (Flomenbom et al., 2005). In this case, the (immobilized) enzyme converted a nonfluorescent substrate molecule into a fluorescent product molecule. The two-state trajectories were constructed from the photon count trajectories. The off state was associated with the conversion substrate
product, whereas the on state was associated with the diffusion of the product molecule away from the enzyme and its vicinity. The off waiting time PDF was best fitted to a stretched exponential. This functional behavior was interpreted as stemming from a spectrum of active enzymatic conformations. The bulk relaxation function test was then applied, and the kinetic scheme was shown to be irreducible. Additionally, clusters of fast events were detected in the ordered off waiting times trajectory (similar to Fig. 3 A), indicating that single lipase molecules display correlations in their activity. These findings were combined to build a kinetic scheme that involves reaction and conformational changes simultaneously, and to extract some of the conformational and reaction rate values.
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SUPPLEMENTARY MATERIAL |
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TOPABSTRACTINTRODUCTIONFROM TRAJECTORIES TO KINETIC...DISCUSSION AND CONCLUDING...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONFROM TRAJECTORIES TO KINETIC...DISCUSSION AND CONCLUDING...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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Submitted on November 8, 2004; accepted for publication March 3, 2005.
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TOPABSTRACTINTRODUCTIONFROM TRAJECTORIES TO KINETIC...DISCUSSION AND CONCLUDING...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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is the transition rate from substate i to j), 
and 
for the irreducible one, and 
and 
for the reducible one. These values are found by comparing 
and 
and the average number of successive fast and slow off waiting times, given by, 
and 