N 1 /NFraction
8.2 Microscopically Reversible Two-state Systems
8.2.1 Two State Trajectories Using Dual Laser Traps
Earlier we saw the complete derivation for the probability distribution of the trajectories for the ion channels. On the other hand, we had a problem with interpreting the Lagrange multipliers and relating them to the experimental observables. In this section we will describe an experiment, currently underway in our lab, that can shed some light on these issues.
The predictions obtained for ion channels are generic for any two-state system and can be applied
s
1s
2ε
1ε
2A
A
B
B
Figure 8.12: Schematic of the energy-landscape experienced by the Brownian particle. When the bead is in the proximity of energy well A (B) it is taken to be in state-1 (state-2). The energy barrierfor the dual-trap can be controlled by changing the spacing between the traps.
to the diffusion of a Brownian particle in dual-optical traps (see Fig. 8.7). A single optical trap is obtained by focusing one laser beam using a microscope objective(see Fig. 8.7a). Depending on the intensity and the frequency of the laser light, the Brownian particle is trapped in an approximately harmonic landscape. If instead of focusing just one laser beam we focus on two laser beams very close to each other, the Brownian particle is trapped in a combined energy landscape with two energy wells [98, 151] (see Figs. 8.7b, and 8.12b). The Brownian particle then hops between the two energy wells describing a trajectory shown in Fig. 8.7b [151]. The energy wells represent the state of the Brownian particle. The Brownian particle is taken to be in state-1 and 2 when it is in energy wells AandB, respectively.
In order to switch from state-1 to state-2, the Brownian particle has to hop over the energy barrier by expending an amount of energy (Fig. 8.12). This seems to be the equivalent to the switching of the ion channels from the open state to close state, or vice-versa. We can thus relate the energy barrierto the parameterαin our formalism (see Eq. 8.55). The energy barrier between the traps can be controlled by controlling the spacing between the traps (see Fig. 8.12) and the intensity of the laser light. The barriers can also be made time-dependent by changing the spacing between the traps with time (see Fig. 8.12). The time-dependent energy barrier (t) can then be related to the time-dependent Lagrange multiplierα(t) (see Eq. 8.55).
Let us now make a very simple, experimentally verifiable prediction to check the relation of the Lagrange multiplierα with the energy barrier between the energy wells. If there is no external driving force on the particleµshould be equal to zero. This scenario can be experimentally achieved
when the two laser beams are identical and the spacing between the traps is constant, providing an energy barrier of. We can now obtain the theoretical probability distribution,p(Nswitch), for the number of switches using Eq. 8.55. In Eq. 8.55, ifsΓ(t) is not equal to sΓ(t), we say that a switch has occurred. Sincescan be either +1 or−1, the productsΓ(t)sΓ(t+ 1) is−1 in the case of a switch and +1 otherwise. As a result, the sumPN−1
t=1 sΓ(t)sΓ(t+ 1) can be written as
N−1
X
t=1
sΓ(t)sΓ(t+ 1) = −Nswitch+ [(N−1)−Nswitch] and
= (N−1)−2Nswitch. (8.56)
The partition function Eq. 8.54, hence, can be written as
Z = X
Γ
exp
"
α 2
N−1
X
t=1
sΓ(t)sΓ(t+ 1)
# ,
= X
Γ
exp(α(N−1)/2−αNswitch), and (8.57)
=
N−1
X
Nswitch=0
Ω(Nswitch) exp(α(N−1)/2−αNswitch), (8.58)
where Ω(Nswitch) is the total number of microtrajectories with the number switching events equal to Nswitch. Ω(Nswitch) can be easily evaluated by observing the following. A trajectory Γ is determined by the time history of s, i.e., Γ ≡ s1, s2, ...., sN. If s1 is given, then the trajectory is uniquely determined by a sequence of stst+1. Hence, if the total number of switches is Nswitch, the total possible arrangements of all these switches is given by NN−1
switch
. Since s1 can take two values, ±1, Ω(Nswitch) is
Ω(Nswitch) = 2
N−1 Nswitch
. (8.59)
Hence, the partition function is given by,
Z =
N−1
X
Nswitch=0
Ω(Nswitch) exp(α(N−1)/2−αNswitch),
= 2 exp(α(N−1)/2)
N−1
X
Nswitch=0
N−1 Nswitch
exp(−αNswitch),
= 2 exp(α(N−1)/2)(1 + exp(−α))(N−1). (8.60)
Time x
Figure 8.13: Typical data for the position of the Brownian bead versus time. The data can then be converted into binary form, where the Brownian particle is either in state 1 or state 0.
The probabilityp(Nswitch) is hence given by
p(Nswitch) = 1
ZΩ(Nswitch) exp(α(N−1)/2−αNswitch),
=
N−1 Nswitch
exp(−αNswitch)
(1 + exp(−α))N. (8.61)
Thus we have a closed form expression forp(Nswitch).
On the experimental front, we can observe the diffusion of the Brownian particle in the dual-laser traps via video microscopy. The camera can take images at around 60 frames per second, so ∆t corresponding to our Caliber expressions is equal to 1/60 seconds. We can then analyze the images to obtain the positionxof the Brownian particle as a function of time (as shown in Fig. 8.7b and 8.13).
Two things need to be (and can be) done. First, the histogram for the position of the beads can be obtained, i.e., the probability distributionp(x) can be experimentally obtained. But, the probability is related to the energy landscape by
p(x)∼exp
−E(x) kBT
. (8.62)
Thus by knowingp(x) the energy landscapeE(x) can be obtained. By knowing the energy landscape the barriercan be easily evaluated.
Knowing the position history x(t) of the beads, the data can be converted into the binary form of 1s and 0s (see Fig. 8.13). If the bead is near the energy wellA(B), it can be classified as being in state-1 (0). From this modified data, one can obtain the probability distribution p(Nswitch) for
the number of switching events. The experimentalp(Nswitch) can then be fitted to Eq. 8.61 usingα as a parameter. By changing the spacing between the traps and modifying the laser intensity, one can obtain different values for. The same analysis can be repeated to obtainαas a function of. Once we know howαis related tofurther experiments with time varying(t) can be done to check the predictions obtained by using the method of maximum caliber. D. Wu and E. Seitaridou in our lab are currently involved in the dual-laser trap experiment in our lab, and we hope to shortly have the results from the experiment.
In addition to changing the barrier, we can also also add a time-dependent external loading and, hence, can emulate the theoretical use of the time-varying Lagrange multiplierµ(t). Getting a closed form expression in the presence of both µandαis difficult, but we can always resort to numerical calculations to compare experimental results with the theoretical predictions.