Linear order

Most reactions in cells are carried out by enzymes [1], In many instances the rates of enzyme-catalysed reactions are enhanced by a factor of a million. A significantly large fraction of all known enzymes are proteins which are made from twenty naturally occurring amino acids. The amino acids are linked by peptide bonds to fonn polypeptide chains. The primary sequence of a protein specifies the linear order in which the amino acids are linked. To carry out the catalytic activity the linear sequence has to fold to a well defined tliree-dimensional (3D) stmcture. In cells only a relatively small fraction of proteins require assistance from chaperones (helper proteins) [2]. Even in the complicated cellular environment most proteins fold spontaneously upon synthesis. The detennination of the 3D folded stmcture from the one-dimensional primary sequence is the most popular protein folding problem. 

Much of protein engineering concerns attempts to explore the relationship between protein stmcture and function. Proteins are polymers of amino acids (qv), which have general stmcture +H3N—CHR—COO , where R, the amino acid side chain, determines the unique identity and hence the stmcture and reactivity of the amino acid (Fig. 1, Table 1). Formation of a polypeptide or protein from the constituent amino acids involves the condensation of the amino-nitrogen of one residue to the carboxylate-carbon of another residue to form an amide, also called peptide, bond and water. The linear order in which amino acids are linked in the protein is called the primary stmcture of the protein or, more commonly, the amino acid sequence. Only 20 amino acid stmctures are used commonly in the cellular biosynthesis of proteins (qv). 

Finally, a point can be made about using the present analysis to calculate a trajectory. Most of the above analysis invoked a small-x expansion and kept terms up to linear order. This restricts the analysis to times not too long, xx(x, x) x. If one has the entire correlation function, then one can predict... 

Hence to linear order in the differences, in the intermediate regime this is... 

Figure 5.11 Individual transferrin receptor domains. Ribbon diagrams for domain I, the protease-like domain (a) domain II, the apical domain (b) and domain III, the helical domain (c). Secondary structure elements are labelled and are referred to in the text first with respect to the domain number, then with respect to the linear order of the elements within the domain. For example aI-3 refers to the third helix in the first domain. In (a), the two grey spheres indicate the positions that would be occupied by Zn2+ in an authentic protease. Reprinted with permission from Lawrence et ah, 1999. Copyright (1999) American Association for the Advancement of Science.

DNA sequence The relative linear order of base pairs, whether in a fragment of DNA, a gene, a chromosome, or an entire genome. 

Of course, these steps usually are not processed in a linear order. Rather there are loops and iterations. The acquired data might lead to new insights in the problem... 

A yours is less so than your friend s, then mine must be less curvy than hers. These proper-, ties are quite important in some contexts, for example if they are to be sorted into a unique linear order. 

A sorted list keeps its items in a uniquely determined linear order. You can make sorted lists of almost any type of object provided that it has a comparison operator with the Tota-... 

What happens if you try to model a sorted list for something like project Tasks, which should not really be totally ordered based on must-precede The TotalOrdering properties would be imposed on Task, something that (1) is probably not what you intended, because it imposes a linear order on all tasks, and (2) could be inconsistent with the definitions of the must-precede operator itself. 

Trend tests are generally described as -sample tests of the null hypothesis of identical distribution against an alternative of linear order, i.e., if sample I has distribution function F i = 1 then the null hypothesis... 

Fig. 1. The four types of linear protein domain relationships. They have been ordered by degree of complexity in a manner analogous to the Chomsky finite automaton language hierarchy. Only the first three have been identified in nature thus far. This is no doubt that, in the fourth case, multiple simultaneous mutations would be required. If the linear order of the amino acid sequence is viewed as generated by a finite automaton, there is an equivalence with the Chomsky hierarchy. 

Fig. 2. An example of a complex multidomain protein that includes both domain concatenation and intercalation. (A) See color insert. RASMOL view of phosphotransferase pyruvate kinase (pdb entry lpkn) colored to show the three identifiable domains. Blue is the j3 barrel regulatory domain, orange is an eightfold a/fi barrel, the catalytic substrate binding domain, and green is a central /3, a/(B nucleotide binding domain. Not displayed is the leader subsequence composed of a random coil and short helix. (B) Linear order along the sequence of these components.

Type of Values Linearly-Ordered Descriptive Phrases Values ... 

Eor 1-D sampling, one dimension in space dominates the lot completely as all increments extend fully over the other two physical dimensions, and there is a natural linear order between increments. Conveyor belts and process pipelines transporting a material flux are the two most important examples of 1-D lots. 

Amorphous polymers with irregular bulky groups are seldom crystallizable, and unless special techniques are used even ordered polymers are seldom 100% crystalline. The combination of amorphous and crystalline structures varies with the structure of the polymer and the precise conditions that have been imposed on the material. For instance, rapid cooling often decreases the amount of crystallinity because there is not sufficient time to allow the long chains to organize themselves into more ordered structures. The reason linear ordered polymers fail to be almost totally crystalline is largely kinetic, resulting from an inability of... 

We assume that, in the creeping flow limit, the Cartesian drag force on bead p may be expanded to linear order in U and Vv, as... 

By expanding the basis vector a (r ) to linear order in the displacement A (r ) from its value at t = 0, Fixman showed that... 

Here, we construct approximations for the random quantities AR and A,- to linear order in ARj. Because A, appears multiplied by At throughout this analysis, we define a variable... 

Here and in what follows, we use super- and subscripts of (1) to indicate approximations valid to linear order in AR. The required linear approximation for A, is obtained by expanding the change in d to linear order in AR, as 0 = Ac lo obtain... 

Taylor-expanding — PyP(t ) to linear order in AR (t ) around its value at f = 0 within the integrand, we find that the contribution involving the value of this tensor at f = 0 vanishes, but that the next term in the expansion yields... 

Near equilibrium, we expect that the dynamics will be controlled by the energetics of small variations of the step positions in Eq. (2). By definition, the change in to linear order induced by a small variation Ax (y) in the position of the th step is... 

In the linearly-coupled version of the INM theory, which corresponds to the expansion of the dynamical variable of interest to linear order in INMs, G i) is related by Fourier transformation to the solvation influence spectrum. 

Then, to the linear order for 5Ja r,t), the mean-field Hamiltonian (4) can be decomposed into static and time-dependent response parts... 

Identification of multiple input signals and several effector proteins underlines the high complexity of signal transduction via the Ras protein. The Ras pathway cannot be seen as a linear ordering of signal elements, by which information is conducted verti-... 

Linear polymers prepared by step reaction polymerization, such as nylon 66, and linear, ordered polymers prepared by the chain polymerization of symmetrical vinylidene monomers, such as polyvinylidene chloride (PVDC), can usually be crystallized because of symmetry and secondary-bonding. Isotactic polymers, such as isotactic polypropylene (PP), usually crystallize as helices. 

In a protein the amino acids are joined together in a linear order by amide linkages, also known as peptide bonds. The sequence of the covalently linked amino acids is referred to as the primary level of structure for the protein. In writing the sequence of amino acids in a chain, it is conventional to orient the chain so that the amino acid on the left is the one with a free amino group on its a-carbon, while the last amino acid on the right is the one whose a-carbon carboxylate is free. In other words, the amino- or N-terminus of the peptide chain is written on the left, and the carboxyl-or C-terminus is written on the right. One more convention the term backbone for a protein refers to the series of covalent bonds joining one a-carbon in a chain to the next a-carbon. 

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