Z-domain

Andrew Braisted and J.A. Wells prepared phage containing Z domain helices 1 and 2 and restored Fc binding of this 38 residue minidomain in three iterative stages (see Figure 17.10). The truncated peptide was first randomized at four hydrophobic residues which contact helix 3 in the complete Z domain. The consensus sequence from this library maintained the wild-type residues lie 17 and Leu 23 while the hydrophobic residues Leu 20 and... 

Figure 17.10 Construction of a two helix truncated Z domain, (a) Diagram of the three-helix bundle Z domain of protein A (blue) bound to the Fc fragment of IgG (green). The third helix stabilizes the two Fc-binding helices, (b) Three phage-display libraries of the truncated Z-domaln peptide were selected for binding to the Fc. First, four residues at the former helix 3 interface ("exoface") were sorted the consensus sequence from this library was used as the template for an "intrafece" library, in which residues between helices 1 and 2 were randomized. The most active sequence from this library was used as a template for five libraries in which residues on the Fc-binding face ("interface") were randomized. Colored residues were randomized blue residues were conserved as the wild-type amino acid while yellow residues reached a nonwild-type consensus, [(b) Adapted from A.C. Braisted and J.A. Wells,...

EMPl, selected by phage display from random peptide libraries, demonstrates that a dimer of a 20-residue peptide can mimic the function of a monomeric 166-residue protein. In contrast to the minimized Z domain, this selected peptide shares neither the sequence nor the structure of the natural hormone. Thus, there can be a number of ways to solve a molecular recognition problem, and combinatorial methods such as phage display allow us to sort through a multitude of structural scaffolds to discover novel solutions. 

Show that (PqjD is partially correct with respect to A and B by first generating the verification conditions for (P, I) and this choice of inductive assertions Aa and Ag and of input predicate B, and then checking the verification conditions. You may be reasonably informal in establishing that the verification conditions hold for all x,yl,y2 z domain of 1 ... 

In recent years a number of commercial programs have been developed that produce root locus plots (and provide other types of analysis tools). These software packages can speed up controller design. Some of the most popular include CC, CONSYD, and MATRIX-X. We will refer to these packages again later in the book since they are also useful in the frequency and z domains, as well as for handling multivariable systems. /... 

To analyze systems with discontinuous control elements we will need to learn another new language. The mathematical tool of z transformation is used to desigii control systems for discrete systems, z transforms are to sampled-data systems what Laplace transforms are to continuous systems. The mathematics in the z domain and in the Laplace domain are very similar. We have to learn how to translate our small list of words from English and Russian into the language of z transforms, which we will call German. 

In Chap. 18 we will define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary) and develop transfer functions in the z domain. These fundamentals are then applied to basic controller design in Chap. 19 and to advanced controllers in Chap. 20. We will find that practically all the stability-analysis and controller-design techniques that we used in the Laplace and frequency domains can be directly applied in the z domain for sampled-data systems. 

We will find later in this chapter that a (z — 1) in the denominator of a transfer function in the z domain means that there is an integrator in the system, just like the presence of an s in the denominator in the Laplace domain told us there was an integrator. 

D. EXPONENTIAL MULTIPLIED BY TIME. In the Laplace domain we found that repeated roots l/(s -I- a) occur when we have the exponential multiplied by time. We can guess that similar repeated roots should occur in the z domain. Let us consider a very general function ... 

We sometimes want to invert from the z domain back into the time domain. The inversion will give the values of the functiononly at the sampling instants. 

We know how to find the z transformations of functions. Let us now turn to the problem of expressing input-output transfer-function relationships in the z domain. Figure 18.9a shows a system with samplers on the input and on the output of the process. Time, Laplace, and z-domain representations are shown. G(2, is called a pulse transfer function. It will be defined below. 

Defining in this way permits us to use transfer functions in the z domain [Eq. (18.57)] just as we use transfer functions in the Laplace domain. G,, is the z transform of the impulse-sampled response of the process to a unit impulse function <5( . In z-transforming functions, we used the notation =... 

By z-transforming this equation, using Eq. (18.59), the output in the z domain is... 

The term in parentheses is the Laplace transformation of the impulse-sampled response of the total combined process to a unit impulse input. We will call this (GiG,)S, in the Laplace domain and (Gi G2)(i) in the z domain. 

In the calculation above, we went through the time domain, getting by inverting and then z-transforming g. The operation can be represented more concisely by going directly from the Laplace domain to the z domain. 

A specific example will illustrate how the output of the closedloop system can be obtained in the z domain. 

The order of the system in the z domain (the highest power of z in the denominator) increases with increasing deadtime or k. 

When the deadtime in a process is an integer multiple of the sampling period, the function in the Laplace domain converts easily into z in the z domain, where dead time D = kT. When the dead time is not an integer multiple of the sampling period, we can use modified z transforms to handle the situation. 

Find the pulse transfer functions in the z domain (HBGm z) for the systems is... 

The azide can also react with phosphine derivatives through the Staudinger ligation. Azidophenylalanine was incorporated into the Z-domain protein in E. coli or into peptides displayed on phage, and was labeled with fluorescein-derived phosphines in phosphate buffer at room temperature (Figure 9(b)). ... 

Figures Selective protein modification using a keto amino acid, p-acetyl-L-phenylalanine. (a) Labeling of fluorescein hydrazide to the Z domain protein. Only the mutant protein containing p-acetyl-L-phenylalanine was labeled and became fluorescent, (b) A general method for preparing glycoprotein mimetics with defined glycan structure.

The steady-state stagnation-flow equations represent a boundary-value problem. The momentum, energy, and species equations are second order while the continuity equation is first order. Although the details of boundary-condition specification depend in the particular problem, there are some common characteristics. The second-order equations demand some independent information about V,W,T and Yk at both ends of the z domain. The first-order continuity equation requires information about u on one boundary. As developed in the following sections, we consider both finite and semi-infinite domains. In the case of a semi-infinite domain, the pressure term kr can be determined from an outer potential flow. In the case of a finite domain where u is known on both boundaries, Ar is determined as an eigenvalue of the problem. 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

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