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Phys Rev Lett. Author manuscript; available in PMC 2012 February 6.

Published in final edited form as:

Published online 2009 July 31. doi: 10.1103/PhysRevLett.103.050603

PMCID: PMC3273425

NIHMSID: NIHMS331510

David Wu,^{1} Kingshuk Ghosh,^{2} Mandar Inamdar,^{3} Heun Jin Lee,^{1} Scott Fraser,^{1} Ken Dill,^{2} and Rob Phillips^{1,}^{*}

The publisher's final edited version of this article is available at Phys Rev Lett

See other articles in PMC that cite the published article.

We study the trajectories of a single colloidal particle as it hops between two energy wells which are sculpted using optical traps. Whereas the dynamical behaviors of such systems are often treated by master-equation methods that focus on *particles* as actors, we analyze them instead using a *trajectory-based* variational method called maximum caliber (MaxCal). We show that the MaxCal strategy accurately predicts the full dynamics that we observe in the experiments: From the observed averages, it predicts second and third moments and covariances, with no free parameters. The covariances are the dynamical equivalents of Maxwell-like equilibrium reciprocal relations and Onsager-like dynamical relations.

We explore the kinetics of two-state processes, *A* *B*, at the one-particle level. Examples of single-molecule or single-particle dynamical processes that mimic this two-state dynamics include DNA loop formation [1], protein folding oscillations [2], or ion-channel opening and closing kinetics [3]. Two-state fluctuating systems having fixed rates are called *random-telegraph* processes.

One way to understand two-state and random-telegraph processes is through master equations, which are differential equations that are solved for time-dependent probability density functions [4]. For single-particle and few-particle systems, however, other convenient experimental observables are the trajectories themselves rather than the time-dependent populations of the two states. Here we describe an experimental model system to study such single-particle two-state stochastic trajectories. We use these experiments to test a theoretical strategy, called maximum caliber, that provides a way to predict the full trajectory distributions, given certain observed mean values. It has not yet been much tested experimentally; that is the purpose of the present work.

Using dual optical traps, we have “sculpted” various energy landscapes. We can control the relative time the particle spends in its two states and the rate of transitioning between them. Our method follows from earlier work on the dual trapping of colloidal particles that was used to study Kramers reaction rate theory [5]. While these experiments were previously focused on studying average rates, our interest here is in the probability distribution of trajectories.

We trap a 1 *μ*m silica bead in a neighboring pair of optical traps. The laser at 532 nm, 100 mW, provides an inverted double-Gaussian shaped potential. An acousto-optic deflector alternately sets up two traps close together in space, at a switching rate of 10 kHz, which is both much faster than each individual trap's corner frequency [6] and also the fastest bead hopping rate. The strength of each trap and the spacing between them can be controlled in order to sculpt the shape of the potential. A tracking 658 nm red laser at 1 mW was used to determine the position of the bead. The forward scattered light is imaged through a microscope condenser onto a position-sensitive detector [7]. The green trapping laser light at the detector is filtered out by a long-pass filter. The data were recorded at a rate of 20 kHz, which sets the fundamental time step Δ*t* for our analysis. Trajectories were recorded for intervals ranging from 20 minutes to more than 1 hour, depending on the hopping rate. A simple threshold was used to determine states in the trajectories.

First proposed by Jaynes in 1980, maximum caliber (MaxCal) is a variational principle that purports to predict dynamical properties of systems in much the same way that the maximum entropy (MaxEnt) approach predicts the properties of equilibrium systems; both use information theory as a basis [8]. MaxCal has been shown to be a simple and useful way to derive the flux distributions in diffusive systems, such as in Fick's law of particle transport, Fourier's law of heat transport, and Newton's viscosity law of momentum transport [9]. The algorithm of MaxCal is identical to that of MaxEnt. As a reminder, MaxEnt augments the Shannon entropy *S* = Σ* _{s}p_{s}* ln

Examples of trajectories in our system are shown in Fig. 1. By *trajectory*, we mean one individual time sequence of events over which the particle can transition back and forth many times between states *A* and *B*. In our experiments, we divide time into discretized time intervals Δ*t* set by the inverse of our sampling rate. A trajectory has *N* time steps, so it lasts for a total time *N*Δ*t*. We aim to characterize the probability distribution for all trajectories given the known constraints on the space of possible trajectories, as defined by our first-moment experimental observables.

Sculpted energy landscapes (left, averaged 20 minutes) and the corresponding microtrajectories. The trace is raw data; states are assigned after boxcar filtering and threshold finding. Top: The lower state is slightly more populated; there is a high barrier **...**

We take a reduced view of the experimental system which will allows the solution to *p _{i}* to be analytically tractable. For each Δ

The calculation of the *p _{i}*'s is made simple by the use of the trajectory partition function

$${Q}_{d}=\sum _{i}^{2N}\left({\alpha}^{{N}_{aa,i}}{\beta}^{{N}_{bb,i}}{\omega}_{f}^{{N}_{ab,i}}{\omega}_{r}^{{N}_{ba,i}}\right),\phantom{\rule{0.1em}{0ex}}$$

(1)

and the probability of a particular trajectory labeled *i* is given by

$${p}_{i}={Q}_{d}^{-1}{\alpha}^{{N}_{aa,i}}{\beta}^{{N}_{bb,i}}{\omega}_{f}^{{N}_{ab,i}}{\omega}_{r}^{{N}_{ba,i}}.\phantom{\rule{0.1em}{0ex}}$$

(2)

We write the exponentiated Lagrange multipliers as the “statistical weights” *α*, *β*, *ω _{f}*, and

It is readily shown that the average quantities are simply derivatives of the partition function; for example,

$$\langle {N}_{bb}\rangle ={\frac{\partial \phantom{\rule{0.1em}{0ex}}ln{Q}_{d}}{\partial \phantom{\rule{0.1em}{0ex}}ln\beta}|}_{\alpha ,{\omega}_{f},{\omega}_{r}},\phantom{\rule{0.1em}{0ex}}\langle {N}_{ab}\rangle ={\frac{\partial \phantom{\rule{0.1em}{0ex}}ln{Q}_{d}}{\partial \phantom{\rule{0.1em}{0ex}}ln{\omega}_{f}}|}_{\alpha ,\beta ,{\omega}_{r}}\phantom{\rule{0.1em}{0ex}}$$

(3)

and

$$\langle {N}_{aa}\rangle ={\frac{\partial \phantom{\rule{0.1em}{0ex}}ln{Q}_{d}}{\partial \phantom{\rule{0.1em}{0ex}}ln\alpha}|}_{\beta ,{\omega}_{f},{\omega}_{r}},\phantom{\rule{0.1em}{0ex}}\langle {N}_{ba}\rangle ={\frac{\partial \phantom{\rule{0.1em}{0ex}}ln{Q}_{d}}{\partial \phantom{\rule{0.1em}{0ex}}ln{\omega}_{r}}|}_{\alpha ,\beta ,{\omega}_{f}}\phantom{\rule{0.1em}{0ex}}$$

(4)

together define the statistical weights.

The MaxCal strategy is as follows. First, experiments give the four trajectory-averaged quantities such as *N _{bb}*,

In addition, other properties of interest are also readily computed. Let *N _{B}* represent the number of units of time that the system spends in state

Mixed moments and covariances are obtained from mixed derivatives of *Q _{d}*. For example,

$${\frac{{\partial}^{2}\phantom{\rule{0.1em}{0ex}}ln{Q}_{d}}{\partial \phantom{\rule{0.1em}{0ex}}ln{\omega}_{f}\partial \phantom{\rule{0.1em}{0ex}}ln\beta}|}_{\beta ,{\omega}_{r}}={\frac{{\partial}^{2}\phantom{\rule{0.1em}{0ex}}ln{Q}_{d}}{\partial \phantom{\rule{0.1em}{0ex}}ln\beta \partial \phantom{\rule{0.1em}{0ex}}ln{\omega}_{f}}|}_{\alpha ,{\omega}_{r}},$$

(5)

which leads to
${\frac{\partial \langle {N}_{bb}\rangle}{\partial \phantom{\rule{0.1em}{0ex}}ln{\omega}_{f}}|}_{\alpha ,\beta ,{\omega}_{r}}={\frac{\partial \langle {N}_{ab}\rangle}{\partial \phantom{\rule{0.1em}{0ex}}ln\beta}|}_{\alpha ,{\omega}_{f},{\omega}_{r}}=\langle {N}_{ab}{N}_{bb}\rangle -\langle {N}_{ab}\rangle \langle {N}_{bb}\rangle $. Higher derivatives of *Q _{d}* give information about the higher-order fluctuations. Hence, given

A simple way to compute *Q _{d}* is through the matrix propagator

$$\mathbf{\text{G}}=\left(\begin{array}{cc}\alpha & {\omega}_{r}\\ {\omega}_{f}& \beta \end{array}\right),\phantom{\rule{0.1em}{0ex}}$$

(6)

where each element of **G** represents the statistical weight of transitioning from some initial state during each time step. We consider here only stationary processes, for which the statistical weights are time-independent, but the MaxCal method itself is not limited to such simple dynamics. We can express *Q _{d}* = (11)

Thus the probability of being in state *A* at time *t* is given by

$$P\left(A,t\right)=\frac{\left(10\right){\mathbf{\text{G}}}^{N-1}{\left({a}_{0}\phantom{\rule{0.1em}{0ex}}{b}_{0}\right)}^{\mathbf{\text{T}}}}{\left(11\right){\mathbf{\text{G}}}^{N-1}{\left({a}_{0}\phantom{\rule{0.1em}{0ex}}{b}_{0}\right)}^{\mathbf{\text{T}}}}.$$

(7)

Again, *N* = *t*/Δ*t*; this is another form of Eq. (2). For random-telegraph processes, this form of the probability of a trajectory reproduces the conventional result from master equations [4]. In MaxCal, the higher moments of the observables are easily accessed by taking higher derivatives of *Q _{d}*; it is unclear how to compute similar quantities from the traditional approach [11].

Functional similarities between microscopic models in statistical mechanics and equations of state in thermodynamics allows assignment of undetermined Lagrange multipliers in the MaxEnt formalism to physically realizable quantities, such as *β* ↔ *T*^{−1} [12]. We now make similar correspondences between the MaxCal-derived statistical weights with probabilities. The four (exponentiated) Lagrange multipliers *α*, *β*, *ω _{f}*, and

$$\left(\begin{array}{cc}1-{\omega}_{f}& {\omega}_{r}\\ {\omega}_{f}& 1-{\omega}_{r}\end{array}\right),$$

and the master equation follows immediately. The advantage of the MaxCal approach is that *Q _{d}* readily provides information about trajectory observables not obviously accessible from master equations.

We now show tests of the MaxCal predictions. Given the first-moment averages observed for the trajectories, MaxCal predicts the second moments. Figure 2 shows that two such predicted second moments are in good agreement with the experimental data.

Second moment of the trajectory distribution. The *x* axes give the predicted second moments from the MaxCal approach, based on the known first moments. The *y* axes give the experimental values of the second moments. Left: Variance of *N*_{B}; **...**

Figure 3 compares one experimental third cumulant with the predicted value from caliber obtained from the measured first moments. The first moments are easy to measure with good accuracy from short trajectories, so one virtue of the caliber approach is that all of the higher predicted moments are noise-free compared to higher moments extracted from data: Predicted moments are dependent on first moments only, whereas data-based higher moments are contaminated by noise from every other lower moment. See the supplementary information [13] for more detail.

Experiments vs theory for the covariance and third cumulant. Left: One covariance quantity. Right: The third cumulant of *N*_{ba}. The dashed lines are the best linear fits; fitting parameters are inset.

Figure 3 also shows the quantity *N _{B}N_{ab}* −

We can compute the ratio *N _{A}*/

The probability distribution of *N*_{A}/*N*_{B} = *K*_{eq} as a function of time. We obtain the dashed line from Monte Carlo simulation of *Q*_{d} and corresponding columns from experimental data. The distribution of time spent in *A* vs *B* is broad for short times (left) but **...**

In summary, we have studied a single colloidal particle undergoing a two-state process *A* *B* with stationary rates. By measuring short trajectories, we obtain first-moment observables *N _{bb}*,

We are grateful to Dave Drabold, Jane Kondev, Keir Neuman, and Dan Gillespie for helpful comments. K.D. appreciates the support of NIH Grant No. GM 34993 and a UCSF Sandler Blue Sky grant. D. W. acknowledges the support of a NIH UCLA-Caltech MD-PhD grant. This work was also supported by the NIH Director's Pioneer grant to R. P.

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13. See EPAPS Document No. E-PRLTAO-103-018933 for a description of our efforts to reduce the standard error of Figs. 2 and and33 by checking the MaxCal algorithm against Brownian dynamics generated trajectories and a demonstration of the convergence properties of predicting higher order moments of trajectories using MaxCal by plotting the theoretical predictions against simulated moments as a function of trajectory length; in short, we show that the MaxCal predictions converge to the true value of the higher order moments as fast or faster than simulated ones. For more informationon EPAPS, see http://www.aip.org/pubservs/epaps.html.

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