## Model System

1.2.1. Equilibrium and Kinetics

We choose a simple system of a nondiffusing single molecule oscillating between a strongly fluorescent state and a weakly fluorescent state to characterize the particularities associated with the study of single molecules in small probe volumes. We focus on the problems and opportunities that result from using probe volumes less than or equal to a femtoliter. Equations (1.1a) and (1.1b) describe the equilibrium behavior of two conformations of the same molecule, A and B:

where [A] and [B] are the respective concentrations of each species, k1 and k2 are the rate constants, and Keq is the equilibrium constant. For a small displacement from equilibrium, the differential rate law is

dt dt where A fA (t) and A fB (t) denote the deviation of the concentration from equilibrium for species A and B at time t. For the closed, two-state system modeled here, the sum of [A] plus [B] is constant; thus AfA(t) equals -AfB(t). The solution to Eq. (1.2) is

where A fA is the initial deviation from equilibrium (Cantor and Schimmel, 1980). The probability of finding the system in A or B is k2 k1

where PA and PB are the probabilities for A and B, respectively. The mean fluorescence intensity, (//), is given by

where a and j depend upon the absorption, emission, and detection factors for A and B, and 0A and 0B are the fluorescence quantum yields for A and B.

Single-molecule measurements of the system give ki, k2, IA, and IB; therefore the parameters a and j can usually be determined by calibration of the apparatus allowing one to obtain values for 0A and 0B.

### 1.2.2. Generation of Synthetic Data

The parameters used in the following Poisson simulation of a data stream (Demas, 1983; Matthews and Walker, 1964), chosen to represent a typical photophysical process, are listed in Table 1.1. The conclusions drawn from data synthesized from these particular values may change significantly for a different choice of parameters. Here a cycle is defined as the total of the time period that a molecule spends in each state.

where t1 and t2 are the inverse of k1 and k2, and Tcyc is the cycle time. The probability for the forward reaction in Eq. (1.1), P (t), that A changes its conformation at time t within dt is

The approximation is valid for small dt. For the simulation, a molecule changes states if a uniform random number is less than k1dt. The reaction time is determined by the number of trials needed for a newly generated random number to meet the specified condition above; the number of trials is then multiplied by dt. In the case for simulations of multiple molecules, data from individual molecules were added together. In Figure 1.1, we choose to start all of the molecules in A in order to simulate a perturbation-jump experiment. Alternatively, when a random start is desired, a uniform random number is compared to one of the probabilities in Eq. (1.4) to determine in which state each molecule starts.

1.2.3. Data Analysis

### 1.2.3.1. Single Molecule in the Probe Volume

Figure 1.1, N = 1, is a simulation of the time-dependent fluorescence intensity for a single molecule, fixed in the probe volume, that oscillates between a fluorescent and a weakly fluorescent conformation; thermodynamic driving forces cause the molecule to change conformations. The time that the molecule spends in each conformation is stochastic. Confining the molecule to the probe volume is useful because it extends the range of kinetic parameters that can be studied and improves the accuracy of the parameters extracted from fits of the data to curves derived from the reaction mechanism and eliminates complications due to diffusion (Baldini et al., 2005; Jia et al., 1999; Rhoades et al., 2003; Talaga et al., 2000). In practice, the sampling time is limited to less than 100,000 photon absorptions by photobleaching (Soper et al., 1993) and other considerations (Vazquez et al., 1998; Zander et al., 2002).

We describe three methods for data analysis when only one molecule is in the probe volume: (1) averaging, (2) autocorrelation, and (3) distribution analyses.

1. The time averages of the "on and off" periods are measured for a chosen number of cycles; the averages are then inverted to give the rate constants, k1 and k2. The rate constants then can be summed to give the relaxation time, (k1 + k2)-1, of the system. Here, the assumption is made that differences between states are discernible such that their respective time dependence can be accurately determined (Verberk and Orrit, 2003).

2. The fluorescence autocorrelation function for a simple first-order kinetic process, undergoing fluctuations around the equilibrium concentration, exhibits an exponential decay with a decay rate that is the sum of the rate constants (k1 + k2). This type of analysis is called fluorescence correlation spectroscopy (FCS). Autocorrelation requires an analysis time long enough for both states to be sampled such that their average is independent of the time—that is, constant within statistical variation. Note that the time to reach a stationary system is independent of the number of molecules in the system. The fluorescence intensity autocorrelation function is defined as

where 8I are the zero-mean intensity fluctuations, 8I(t) = I(t) - (I}, at time t and t + t, respectively (Aragon and Pecora, 1976, Elson, 1985, Krichevsky and Bonnet, 2002, Zander et al., 2002).

3. Distribution analyses include both binning and cumulative distribution functions. In the binning method, "on and off" time periods are binned separately and plotted individually in histogram format; rate constants are determined from each histogram. For the simulations here, a minimum of 10 cycles is needed to define a rudimentary histogram. Alternatively, the data can be fit to the cumulative distribution function for the probability density function that describes the data. The cumulative distribution method has an advantage over binning in that as few as two to three points are needed. However, for cases more complicated than considered here, the cumulative distribution function may be difficult to compute analytically.

### 1.2.3.2. Multiple Molecules in the Probe Volume

In the study of biological systems within a fixed probe volume (cells, lipid vesicles, cell membranes, etc.), it is often of interest to determine the number of a particular species in the probe volume. The ability to measure the number of molecules and determine individual molecular properties leads to new insights regarding basic intracellular functions. Here, two limits of multiple molecules in the probe volume are examined: a bulk system, N > 10, where discrete contributions from individual molecules are not apparent, and a system where discrete contributions from individual molecules are obvious, 2 < N < 10, herein referred to as a "prebulk" system.

Figure 1.1 illustrates the change from single-molecule system to prebulk system to bulk system as N is increased from 1 to 5000. This corresponds to a concentration range from 1 nM to 10 |xM for a probe volume of 1 fL. Although each molecule is in either A or B, the system is not constrained to a number normalized intensity of either 1 or 0.06 because the molecules get out of phase. For example, in the case where the number of molecules is five, all of the molecules must be in B for a normalized intensity of 0.06; this is improbable, approximately one out of a thousand, for the simulation parameters used here. For five molecules, six normalized intensities are possible with frequencies determined by probability theory. The corresponding plateaus are clearly visible for N equals 2 and 5, and occasionally 10 in Figure 1.1. Discrete steps are not evident for N greater than 10.

Bulk Systems and Synchronous Starts. Consider the simulations in Figure 1.1 for the bulk case. Here, at t = 0 all of the molecules are in A and relax to the equilibrium distribution in less than the average of one cycle. As stated previously, the simulations are analogous to rapid perturbation experiments—for example, optical pumping, temperature jump, pH jump, or the addition of an appropriate chemical. For this type of experiment, the time dependence of the signal relaxation gives the sum of the rate constants, k1 + k2, whereas the long-time limit of the signal gives the average fluorescence intensity as given in Eq. (1.5). In the special case where one of the states is "dark," the equilibrium constant is determined from the ratio of the intensity at time zero to the intensity at times greater than the chemical relaxation time. Depending on the type of experiment, bulk measurements give the equilibrium fluorescence intensity and the sum of the rate constants; individual values of IA, IB , k1, and k2 are typically difficult to measure.

Prebulk System. For the prebulk system we refer to two distinct states of the system: all on and all off. (There are a total of N + 1 distinct states; the number of configurations is (A + B)N.) Here all on refers to the case where every molecule in the probe volume is A concurrently; correspondingly, all off indicates that every molecule is B at the same time. Assuming that all the fluorescing molecules in the probe volume are identical, probability theory predicts that the individual rate constants can be determined by the average of the period when all the molecules are either all on or all off; the average period multiplied by the number of molecules gives the inverse of the corresponding rate constant (Riley et al., 1997). This measurement depends on explicit knowledge of the number of molecules in the probe volume. Figure 1.2 gives an example for 2 to 10 molecules. Conversely, if the rate constants are known and assuming identical molecules, the average of either the all on or all off periods can be used to determine the number of molecules in the probe volume. If neither the rate constant nor the number of molecules is known, the ratio of the average lifetime of the all off state to the all on state gives the equilibrium constant. For all types of measurements in a prebulk system the system state must be known—that is, whether the system is in the all on or all off state. There are two possible experimental procedures that

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