N 1 /NFraction
7.9 Summary and Comments
We saw earlier thatL12 equalsL21 and hence do not need to evaluate it separately. To obtainL22, we use equation 7.61:
L22= ∂hJei
∂λe
0,0
=2p(1−p)(m1+m2). (7.72)
Thus we have obtained the values of the kinetic coefficients for our simple hot-dog model.
Chapter 8
The Dynamics of Two-state Systems
In the previous chapter we introduced Jaynes’ principle of maximum caliber and discussed how it can be utilized to model systems far from equilibrium. When we have a classical (or a quantum) system with a well defined Hamiltonian, the maximum caliber formalism gives us the time-evolution of macro-variables. It would have been extremely gratifying if that were the end of the story. On the other hand, the major drawback of writing down an exact description (classical or quantum) is that not only is it almost impossible to analytically solve the problem, but the exact solution of even an approximate model is rare. Fortunately, we saw in Chapter 6 that the maximum caliber principle need not be restricted to such full blown description. In fact, we can develop our own simple models to describe a particular system of interest.
Consider, for example, the case of the Ising model in equilibrium statistical mechanics, which was introduced to describe magnetic phase transition [26]. Although the relation of this model with the actual quantum model describing the evolution of quantum spins is evident, the Ising model is still a coarse-grained approximation to the underlying reality. Despite the drastic coarse-graining, the Ising model has been truly instrumental in modeling critical phenomena and has provided an insightful description of disparate systems such as the lattice gases and binary mixtures [26]. This particular argument is just to drive home the point that simple coarse-grained models are capable of providing insight into problems. The lack of generality of these models is made up for by the interesting physics they produce for specific cases [147]. Like the Ising model, such models are also capable of generating analogies with seemingly unrelated problems. Our goal is to capitalize on this observation and develop simple coarse-grained models for a few representative non-equilibrium systems. Once we specify the underlying microscopic dynamics and the macroscopic constraints, as explained in Chapter 6, the maximum caliber principle provides us with a recipe to obtain the probabilities of the micro-trajectories. A concern that needs to be answered is about the appropriateness of the model and the underlying microscopic dynamics. Also, since we coarse-grain our model, the constraints
that may have been obvious in the complete classical (or quantum) setting may have a very different coarse-grained form. But this is not too much of a price to pay, considering the fact that we obtain nice closed-form solutions from these coarse-grained models. In this chapter, we will study a few problems with coarse-grained microscopic dynamics.
1
0
time
1
0
time
1
0
time
Figure 8.1: Description of the microtrajectories for the two-state systemA↔B. For mathematical convenience, the time is discretized into time steps of ∆t, which can be thought of as the time- resolution of the corresponding experiment.
Certain physical systems have dynamics with a signature quantity, which assumes a set of discrete values as a function of time. The discreteness of these values is intrinsically related to the internal states of the system. For example, in the case of ion channels, the ion channels are either closed or open. The ion-current flowing through the channel either has a finite value or is zero when the channel is open or closed, respectively. The ion-current hence represents the state of the ion channel.
We can represent the temporal states of such physical systems with such finite-state time trajectories (see Fig. 8.1). The principle of maximum caliber can then be applied to these trajectories in order to obtain their probabilities. These probabilities in turn can be used to quantify the dynamics of the physical system they represent.
The outline of this chapter is as follows. In Section 8.1 we discuss the class of two state trajectories in which the system stays in state-1 for a certain time period and then permanently switches to state-2. In particular, we discuss the application of the maximum caliber principle to this class of trajectories in order to explain the dynamics of processive molecular motors. In Section 8.2 we discuss another class of two-state trajectories where the system continuously switches between states- 1 and 2. We then apply the maximum caliber principle over this set of trajectories to elucidate the
dynamics of opening and closing of ion channels when the applied voltage can be time-dependent.
Section 8.2.1 discusses an experiment, currently in progress in our lab, on the diffusion of a Brownian particle in dual-laser traps. The experiment is expected (as a test ground) to check our ideas on the application of the maximum caliber principle to the class of two state trajectories.
Optical Trap Optical Bead
Myosin-V
Actin
(a) (b) (c)
Figure 8.2: Measurements of myosin-V motor dynamics. (a)The molecular motor myosin-V moves over actin by consuming ATP. An optical bead is attached to the myosin molecule. The bead is then held in an optical trap and a force is exerted on myosin. (b) The motion of myosin-V occurs in a series of steps of around 38 nm, an experimental observable. The transition from 1 to 0, as described in Fig. 8.3, corresponds to the step taken by the motor. (c) Myosin-V waits for a certain amount of timeτ before taking a step. The waiting time distribution is a way of summarizing the behavior of different separate trajectories and transitions. The stepping behavior depends on the ATP concentration and on the applied force (figure taken from Rief et al. [136]).