The Empty System

The system consists of two subunits, each of which can attain one of two states, denoted by L and H, having energies and Efj, respectively. In addition, we have intersubunit interactions, which we denote by E j - Ejj, and depending 

Let US first examine the equilibrium properties of this system in the absence of ligands. The GPF of a single system is 

Note that this is actually the canonical partition function for a single empty system. It is also the limit of the GPF of the system [Eq. (4.7.16)] obtained for A, — 0. 

The four states of the polymer are LL, LH, HL, and HH (see the first row of Fig. 4.18) with corresponding probabilities, or mole fractions, 

3) the superscript zero indicates the empty system, i.e., the absence of ligands. Note that is the probability of finding, say, the rhs subunits in the H state and the Ihs subunits in the L state. The mole fraction of the system such that any one of its subunits is in the L state and the other in the H state is the sum of and which in our case is simply 2X2 .,. 

We shall first examine some aspects of the behavior of the empty system and then proceed to the study of the system with ligands. 

FIGURE 3.14. Eight possible configurations of the empty three-subunit system. 

FIGURE 3.15. All possible configurations of the polymer and ligands. The configurations of the subunits are shown in the first row only. Subsequent rows correspond to increasing occupation number. 

We have seen in section 3.3 that the deviation of j] from unity is a measure of the extent of communication between the subunits. 

If rj =, then the subunits are effectively independent. In that case we write 

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Note that whenever we sum over all possible states of one variable, we eliminate the notation of that variable from the resulting probability. In Eq. (1.5.3), we refer to P(0,0) as the probability of the event site a and site b are empty. This is obtained by summing over all possible states (here, L and H) of the empty system. 

At equilibrium, the probability of finding the empty system in one of the two states is the same as the equUibrium mole fractions, and these can be read from the PF,... 

Thus, in the empty system the equilibrium constant q is determined only by the interaction energies Condition (4.7.9) is equivalent to the condition q = 1. We shall see in Section 4.7.3 that the equilibrium constant q is also responsible for transmitting information between the two ligands across the boundary between the two subunits. 

For example, let us consider the reaction mechanism with sf — Ai, A2, A3, A4 and reactions Ai + A2 2A3, Ai + A2 A3 + A4, A3 A4 and A4 A3. There are no hanging components, but two hanging reactions, A3 A4 and A4 A3. After deletion of these two reactions, two hanging components appear, A3 and A4. After deletion these two components, we get two hanging reactions, Al + A2 0 and Ai + A2 0 (they coincide). We delete these reactions and get two components Aj and A2 without reactions. After deletion these hanging components we obtain the empty system. The reaction network is solvable. 

We solve this problem in three steps. First, we discuss the equilibrium condition for the empty system of L and H. Then we examine the system with a ligand (or a solute) s. And finally we calculate the solvation quantities. 

These are also the probabilities of finding a specific site in the L or H state for the empty system. 

To = practical process time (the average time for one job to traverse the empty system)... 

Figure 53a displays typical noise power spectra of the Euq 4QSrp spin glass at two different temperatures above and below T = 1.54 K, compared to the noise of the detection system alone (Reim et al. 1986). While the empty-system noise is white above 1 Hz and shows some 1/tv component below 1 Hz, the noise level of the sample is up to three orders of magnitude larger and displays a 1/tv" behavior... 

Fig. 53. Magnetic noise for the spin glass Eu . jpSro oS (a) as function of frequency at two different temperatures above and below r,= 1.54 K, and (b) as function of temperature at five different frequencies 0.01, 0.1, 1, 10, and 50 Hz (at zero applied field). The data in (b) are obtained after subtracting the noise of the empty system at each temperature (the one at T = 1.42 K is shown in (a)) (from Reim et al. 1986).

These are also the probabilities of finding a specific site in the Lov H state for the empty system. (The term empty refers here to the system of absorbent molecules in the absence of ligands.)... 

Each term in (3.5.12) corresponds to one row in Fig. 3.15. The first term is simply (0), for the empty system. The second term corresponds to all of the 24 states with one ligand. There are eight configurations of the subunits, and for each of these we can place a ligand on three different sites. Similarly, we have 24 terms for the polymer with two ligands and eight terms for the fully occupied polymer. Altogether there are 64 terms. 

Note that the average of Qxp(-pBG ) is taken with respect to the probability distribution of the empty system. 

As we have seen in section 3.3, it is quite difficult to fulfill the mathematical requirement of the sequential model. In this model we require that in the empty system all the subunits be in the L state, and each occupied subunit is turned from L io H but leaves the empty subunits in the L state. 

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