Trajectory Approach to Two-State Kinetics of Single Particles on Sculpted Energy Landscapes

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.

Figure 1

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 (infrequent transitions). Bottom: The upper state is more populated; the barrier is small (frequent transitions). The distance between the two potential minima ranges from 200 to 700 nm.

Figure 2

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 $⟨N_B⟩$; right: variance of $⟨N_{ba}⟩$. The dashed lines are the best linear fits; fitting parameters are inset. Each point represents one experimentally observed trajectory. Trajectories were 30 000 $\Delta t$ units long, and errors were calculated for around 600 trajectories. ${\omega }_r$ and ${\omega }_f$ values ranged from $1\times {10}^{-5}$ to $1\times {10}^{-3}$.

Figure 3

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 4

The probability distribution of $N_A/N_B=K_{\mathrm{eq}}$ 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 becomes narrower for increasing trajectory length (right). $N$ denotes the length of each trajectory, each repeated around 100 times. As the length of trajectories increases, the equilibrium constant assumes a delta-function distribution, commensurate with equilibrium assumptions regarding chemical reactions.