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* Cielo Institute, Asheville, North Carolina 28804; Department of Psychiatry and Behavioral Sciences, Emory University School of Medicine, Atlanta, Georgia 30322; and Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33136
Correspondence: Address reprint requests to Karen A. Selz, E-mail: selz{at}cieloinstitute.org.
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ABSTRACT |
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 TOP ABSTRACT A BRIEF DESCRIPTION OF...MODE SIMILARITIES OF PEPTIDE...SEQUENTIAL MODE ANALYSES IN...DESIGN OF M1ACHR-TARGETED...LOCATING PUTATIVE ALLOSTERIC...ACKNOWLEDGEMENTSREFERENCES |
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A BRIEF DESCRIPTION OF THE CONTEXT FOR THE DEVELOPMENT OF ALGORITHMS FOR RECEPTOR-TARGETED PEPTIDE DESIGN |
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TOPABSTRACTA BRIEF DESCRIPTION OF...MODE SIMILARITIES OF PEPTIDE...SEQUENTIAL MODE ANALYSES IN...DESIGN OF M1ACHR-TARGETED...LOCATING PUTATIVE ALLOSTERIC...ACKNOWLEDGEMENTSREFERENCES |
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; Mandell et al., 1987, 1997a,b,c, 2000a,b; Selz et al., 1998) and derivative membrane protein-targeted peptide design methods (Mandell et al., 1998a, 2003). These methods have historical roots in Sanger's observation that the uniqueness of polypeptides and proteins lie in their one-dimensional sequence of amino acids (Sanger, 1952), and Anfinsen's conclusion that all the information determining the structure and function of proteins is encoded in the potentially quantifiable physical and chemical properties of their amino acid sequences (Anfinsen, 1973). Our one-dimensional signal processing approach to physical property-transformed amino acid sequences includes the use of wavelet transformations, maximum entropy power spectral transformations, multiple "ergodic" entropic measures, and linear decompositions of lagged autocovariance matrices.
Since the introduction of this work, a number of other laboratories have begun to use similar methods of analyses of physical property-transformed amino acid sequences (Giuliani et al., 2003, 2000, 2002; Hirakawa et al., 1999; Lio and Vannucci, 2000; Murray et al., 2002; Rackovsky, 1998). Related matrix methods for sequential pattern finding have been extended to predictions of transmembrane protein structure from nucleotide sequences (Krogh et al., 2001). No others have exploited these techniques for receptor-targeted peptide design. We have recently been granted a United States patent for these and related algorithmic receptor-targeted peptide design methods (Algorithmic Design of Peptides for Binding and/or Modulation of the Functions of Receptors and/or Other Proteins, United States Patent No. 6,560,542, May 6, 2003).
The use of one-dimensional signal processing techniques to reveal patterns in apparently disordered numerical sequences has a long history in fields as diverse as econometrics and meteorology (Bendat, 1958; Box and Jenkins, 1970; Wold, 1965; Yaglom, 1962). More recently this approach has included finding indices of order in apparently disordered, deterministic, chaotic series of quantitative observations (Broomhead et al., 1987; Broomhead and King, 1986; Ott et al., 1994), including biological applications (Guevara et al., 1981; May, 1974), a field with which we have had some experience (Mandell and Russo, 1981; Mandell and Selz, 1997; Russo and Mandell, 1984; Selz and Mandell, 1991; Selz et al., 1995).
This work uses signal processing techniques in the analyses and design of peptide ligands targeting the human m1 cholinergic, G-protein coupled, seven-transmembrane receptor. These forms of pattern analyses require the transformation of amino acid sequences into numerical series. To do so, we must first choose among quantifiable amino acid physical properties relevant to peptide-protein, noncovalent interactions in aqueous solution. Candidate physical properties include van der Waals forces, potential for interactions between charged groups, possible relations between the hydrogen bonds of polar groups, and the availability for interactions between nonpolar groups in water, called hydrophobic aggregation (Israelachvilli, 1992).
Choice of a physical quantity for amino acid sequence transformation
Van der Waals interactions occur between aqueous polypeptide groups as well as between these groups and water. The two components are estimated to contribute about equally, and therefore, do not produce a significant net effect in the stabilization of aqueous polypeptide-polypeptide interactions, as in peptide ligand binding or the initial phases of protein folding (Chalaskinski and Szczesniak, 1994; Makhatadze and Privalov, 1993; Privalov, 1987). Amino acid charged groups are statistically relatively rare (Creighton, 1993), and their interactions are shielded by the high dielectric constant, aqueous surround (Bashford and Case, 2000; Ptitsyn et al., 1995; Stillinger, 1977). They are, therefore, assumed to play only a small role in the stabilization of polypeptide interactions (Privalov, 1989; Privalov and Gill, 1988). However, they may play some role in spatially directing the initial phases of protein folding and, analogously, peptide-receptor aggregation (Shoemaker et al., 1997).
Hydrogen bonding occurs between protein polar groups and between these groups and water molecules, leading to only a small net contribution to polypeptide interactional stability (Caflisch and Karplus, 1994; Yang and Honig, 1995). A study of 80 crystallographically characterized protein-ligand complexes, many binding within the nanomolar range, demonstrated no hydrogen bonds between the protein and the ligand. There was, however, a good correlation between lipophioic contact surface in angstroms squared (Å2) and the negative logarithm of their Ki values (Bohm and Klebe, 1996). In addition, using our design algorithms, we have found that the physiologically active, m1-targeted peptides appear to require a restricted range of average amino acid side-chain pKa values of 
7.2–7.8 (Table 4, bottom). In addition, our recent work indicates that the probability of producing physiologically active autocovariance eigenvector template-generated, de novo peptides can be increased by considering the sequential pattern of receptor sequence's amino acid side-chain pKas (as represented in the relevant pKa autocovariance eigenvector) in the choices of amino acid assignment within its hydrophobic group to the corresponding four-partitioned hydrophobic free-energy eigenvector template (see below for explanatory details).
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; Israelachvili, 1992; Tanford, 1980). This is consistent with more than a half century of work, beginning with Edsall's 1935 observation of an anomalous increase in the molar heat capacity of water in the presence of nonpolar solutes (Edsall and Wyman, 1958), and Franks and Evans (1945) related thermodynamic observations that nonpolar solutes increase the order (decrease the entropy) of water (Franks, 1972). The decrease in entropy that occurs with the transfer of nonpolar groups into water is thermodynamically unfavorable, such that the aqueous system tends to expel these molecules. The expelling action of water on nonpolar groups leads them to aggregate, minimizing their surface area, and this came to be regarded as the general definition of hydrophobic interactions between nonpolar molecules (Kauzmann, 1959; Tanford, 1980). Hydrophobic attraction as measured by atomic force spectroscopy remains significant at distances 
20 nanometers, and decays more slowly than van der Waals forces with distance (Israelachvilli, 1992; Tsao et al., 1993). These considerations led naturally to our choice of relative amino acid side-chain hydrophobicity as the amino acid parameter for numerical transformation of the targeted G-protein coupled receptor's (GPCR's) sequences, their analyses and the design of GPCR-targeted de novo peptides (Mandell et al., 1998a, 2003).
Choices among and some physical correlates of hydrophobicity scales
There are a large number of experimentally derived numerical scales quantitatively ordering the 20 essential amino acids' relative hydrophobicity in an aqueous environment. Among them are a variety of binary solvent partitions (Cornette et al., 1987; Fauchere and Pliska, 1983; Manavalen and Ponnuswamy, 1978; Nozaki and Tanford, 1971) and relative vapor 
water concentrations (Kyte and Doolittle, 1982; Radzicka and Wolfenden, 1988) (see http://pref.etfox.hr./split/scales.html for a relatively complete reference listing with methods used and physical properties quantified leading to 88 hydrophobicity scales). Our computations have demonstrated that interscale correlations ranged from r = 0.579 to r = 0.918, using a variety of representative solvent partition and vapor pressure determined hydrophobicity scales. Another aqueous solvent-dependent amino acid side-chain property, accessible surface area, in Å2, correlated r = 0.56 with the (Eyring-Tanford-Zimmerman) Tanford hydrophobicity scale (Manavalen and Ponnuswamy, 1978) used in these studies. Accessible surface area may be an important physical property with respect to the problem of polypeptide aggregation and binding (Chothia, 1974; Dyson and Wright, 2002a; Huntly et al., 2003). Partial specific volume, in Å3, (Zamyatnin, 1984) and van der Waals volume (volume enclosed by van der Waal radius), in Å3, (Creighton, 1993) correlated with the Tanford hydrophobicity scale as r = 0.67 and r = 0.72, respectively. The covariation of these properties with the Tanford hydrophobicity scale suggests that the hydrophobic scale, as we have used it, may also partially address one or more of these hydrophobicity-related properties. Whereas the magnitude of the hydrophobic effect, as measured from the surface area dependence of the solubilities of hydrocarbons in water, has generally been reported to be 
25 calories/mol/Å2, more recent work taking surface tension at a hydrocarbon-water interface into consideration (Sharp et al., 1991a,b) results in higher estimates of 47 calories/mol/Å2.
Our research program has used the Tanford scale from the beginning (Mandell, 1984) (Manavalen and Ponnuswamy, 1978) due to its historical precedence, relatively high correlation with most other hydrophobicity scales as well as with other amino acid physical properties, and our previous successes using it in our peptide design algorithms (Mandell et al., 2003). Particularly useful is the Tanford scale's nonarbitrary, four-part partition of the values it gives to the 20 essential amino acids (Manavalen and Ponnuswamy, 1978). These fall into correspondence with the four-partitioned autocovariance eigenvector templates, key elements in our peptide design algorithms.
Proline, which individually partitions with the Tanford scale's most hydrophobic group, behaves like glycine with respect to its influence on secondary structure in x-ray studies of amino acid sequences (Deber et al., 1990; Gunasekaran et al., 1998). Glycine is a member of the Tanford scale's least hydrophobic group. Both are known to disrupt membrane protein secondary structure (Nilsson and von Heijne, 1998; von Heijne, 1991), including turn propensity (Monne et al., 1999) and loops (Duerson et al., 1993; Fetrow et al., 1998). For this reason, we code proline like glycine (0.0 kcal/mol) in the lowest of the natural clusters constituting the four amino acid hydrophobicity groups.
The amino acid sequence of the human m1 muscarinic cholinergic receptor was obtained from the SWISS-PROT protein sequence data bank. The amino acid hydrophobic free energies in kcal/mol used in these studies divide naturally into four groups (with the above noted proline "secondary structure breaking" correction): 1), G = 0.0, P = 0.0, Q = 0.0, S = 0.07, T = 0.07, N = 0.09; 2), D = 0.66, E = 0.67, R = 0.85, A = 0.87, H = 0.87; 3), C = 1.52, K = 1.65, M = 1.67, V = 1.87; and 4), L = 2.17, Y = 2.76, F = 2.87, I = 3.15, W = 3.77, providing the natural four partitions mentioned above (Manavalen and Ponnuswamy, 1978). These values will be referred to as the Tanford scale.
Sequential patterns, 
-1, in hydrophobicity and secondary structures
Helical turns and ß-strands and turns are the best established hydrophobic rotations based on x-ray and NMR evidence of secondary structure, with modes of -1 = 3.6 amino acids (aa), 2.2 aa, and 2.0 aa, respectively (Eisenberg et al., 1984; Irback et al., 1996; Irback and Sandelin, 2000; Lazovic, 1996; Milner-White and Poet, 1987; Penel et al., 1999; Rose, 1978; Rose and Wetlaufer, 1977; Schiffer and Edmundson, 1967) (see Table 1).
The autocovariance eigenvector modes represent, in effect, a hierarchy of secondary structural, noisy, semiperiodically recurrent, hydrophobic variational wavelengths, -1. Our studies of neuropeptides and their receptors have demonstrated dominant hydrophobic wavelengths ranging from -1 = 2.06 in porins, connexin, and other proteins dominated by up and down antiparallel ß-structures (Selz et al., 1998), and -1 = 2.18 aa in corticotropin releasing hormone, to -1 = 13.6 aa in acid fibroblastic growth factor in hormonal and neuropeptides (Mandell et al., 1987) to -1 50–70+ aa in subsequences containing the transmembrane segments of GPCRs (Mandell et al., 1997c). Other examples of dominant hydrophobic wavelengths in protein sequences include: the AIDS viral coat protein manifesting a waxing and waning of hydrophobicity with -1 = 7–9 aa, a pattern which was conserved across several mutations (Mandell et al., 1987); eigenmodes of representative Ig 
-, 
-, and 
-chains of -1 = 12.5–13.7 aa (Mandell et al., 1997c); peptide antagonists of the estrogen receptor (Norris et al., 1999) and the ER
estrogen receptor that share a dominant -1 = 8.3 aa indexed mode (Mandell et al., 2000a); the chaperone GroEL of ß-lactamase (Gething, 1997) and ß-lactamase, both of which contain leading hydrophobic eigenmode wavelengths of -1 = 16.25 aa, 4.53 aa, and 2.11 aa (Mandell et al., 2000a). Differences in physical length, as translations of computationally derived hydrophobic variations as wavelengths, are implied by the range of values for the average "Translation per mode in Å" seen in the last column in Table 1. These values were computed from amino acid "Residues per turn" and "Translation per residue" from x-ray crystallographic data (Creighton, 1993).
The role of sequential hydrophobic patterns in peptide-receptor interactions has been demonstrated in studies showing that complete substitution of hydrophobically equivalent amino acids in peptide hormones and neurotransmitters maintains and sometimes increases their peptide-receptor mediated physiological potency (Blanc et al., 1983; Fukushima et al., 1980; Kaiser and Kezdy, 1983; Lau et al., 1983). Self-organizing helical secondary structures of differing average rotational length can be designed with sequences of amino acids that have been binary partitioned into high and low hydrophobicities, independent of the specific amino acids being chosen within either hydrophobicity class (Kamtekar et al., 1993).
Intermolecular signaling by -1-matched hydrophobic sequence attractive aggregation
With respect to the physical chemistry of the eigenmode-matched peptide-protein attraction and aggregation, the hydrophobic attractive force between hydrophobic moieties is a function of surface area and radius of curvature, and can be measured directly using techniques such as atomic force microscopy (Israelachvili and Wennerstrom, 1996). The hydrophobic attractive forces are generally two orders of magnitude greater than those predicted from van der Waals theory, and extend spatially in a slower-than-exponential decay to beyond 20 nm (Pashley et al., 1985). The role of hydrophobic mode matches in attractive aggregation has been observed in the ß-strands of interleukin 1ß, which bind together and initiate protein folding (Gronenborn and Clore, 1994) in a dynamic process called the "hydrophobic zipper" (Dill et al., 1993). Two long helical secondary structures with congruent hydrophobic oscillatory wavelengths bind to form the central "hydrophobic knot" that stabilizes the tertiary structure of phospholipase A2 (Lumry, 1995). The binding of extracellular domains of the growth hormone receptor by polyclonal antibodies to bovine growth hormone is apparently mediated by common sequential hydrophobic variational modes in helical, loop, and disordered secondary structures (Beattie et al., 1996b).
Most of today's approaches to rational peptide ligand design, beyond the high throughput screening of randomly generated peptide libraries (Zysk and Baumbach, 1998), are dominated by pharmacophores, three-dimensional geometric models of proteins' putative active or regulatory binding sites (Guner, 1999; Hruby and Agnes, 1999; Takeuchi et al., 1998). The accelerating increase in the availability of x-ray and NMR characterized three-dimensional structures of proteins might seem to make the use of one-dimensional sequential pattern analyses irrelevant for characterizing peptide-targeted proteins. Polypeptide-protein and protein-protein interactions of physically characterized structures are explored using similar databases and docking algorithms (Janin, 1995; Makino and Kuntz, 1997; Sandak et al., 1998; Stahl and Bohm, 1998). However, an increasing number of studies (Romero et al., 1998, 2001) have demonstrated conformationally disordered polypeptide sequences, particularly relevant to the extracellular loops and juxta-transmembrane subsequences of GPCRs (the targets of our hydrophobic mode targeting peptide design) (Dyson and Wright, 2002a,b; Wright and Dyson, 1999). These subsequences are without stable tertiary structure, as evidenced by missing electron densities in x-ray crystallographic studies, sharp peaks in NMRs, and/or the absence of secondary structural nonlinear Overhauser effects and/or circular dichroism spectroscopic examination with low intensity signals from 210 to 240 nm (Romero et al., 2001). Many of these disordered subsequences become ordered upon ligand binding, going through a disorder-order transition, achieving x-ray and NMR-demonstrable stable tertiary structures (Dyson and Wright, 2002a; Kriwacki et al., 1996; Wright and Dyson, 1999).
Disordered protein subsequences may play significant roles in polypeptide-polypeptide and protein-protein interactions, making them logical targets in rational peptide design (Dunker et al., 1998, 2001; Dunker and Obradovic, 2001). The disordered loop subsequences of globular proteins, 7 transmembrane segment GPCRs, and 12 transmembrane segment membrane transporters (Buck and Amara, 1995; Nirenberg et al., 1997) participate in intramacromolecular signaling as active, allosteric, and antibody binding sites (Bruccoleri et al., 1988; Howl and Wheatley, 1996; Lin et al., 1998; Milner-White and Poet, 1987; Qu et al., 1999; Rondard and Bedouelle, 2000). They may also serve as signal-invoked "switches", modulating access to active sites (Branden and Tooze, 1999; Ulloa-Aguirre and Conn, 2000).
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MODE SIMILARITIES OF PEPTIDE-TARGETED SUBSEQUENCES OF MUSCARINIC CHOLINERGIC RECEPTOR SUBTYPES AND HYDROPHOBIC MODE-MATCHED MUSCARINIC SNAKE TOXINS |
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TOPABSTRACTA BRIEF DESCRIPTION OF...MODE SIMILARITIES OF PEPTIDE...SEQUENTIAL MODE ANALYSES IN...DESIGN OF M1ACHR-TARGETED...LOCATING PUTATIVE ALLOSTERIC...ACKNOWLEDGEMENTSREFERENCES |
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; Wang et al., 2000). In contrast, the muscarinic cholinergic receptors (mAChR) are G-protein coupled, seven transmembrane receptors GPCRs. mAChRs have a 10-fold higher concentration than nAChRs in the brain, and play important roles in attention, arousal, cognition, learning, and memory (Bartus et al., 1982; Flicker et al., 1985; Dickinson-Anson et al., 1998; Fisher et al., 1993; Damasio et al., 1985; Davis et al., 1992).
mAChR's consist of five distinct subtypes, mi, i=1,2, ... 5, categorized by dissimilarities in their amino acid sequences, cholinergic drug-affinities (Bonner, 1989; Bonner et al., 1987; Buckley et al., 1989; McKinney and Coyle, 1991), and differential regional and subcellular distributions in the brain (Adem et al., 1997; Levey, 1996; Mrzljak et al., 1998). Positional alignment of the amino acid sequences of the five mAChR subtypes demonstrates that the greatest differences in segment length occur in the extramembranous amino acids, as well as the terminal carboxy and third intracellular loop, i3. (Bonner et al., 1987; Liao et al., 1989; Numa et al., 1988; Peralta et al., 1987). The greatest variability in amino acid composition and side-chain sequential physical properties across mAChR subtypes also occurs in the extracellular loops (e1, e2, and e3) (Hulme, 1990; Hulme et al., 1991). These are the sequence locations of computationally and experimentally demonstrable and differential hydrophobic mode specificities, reflected in the autocovariance eigenvector-generated peptides targeting m1AChR (and theoretically the mode-equivalent m4AChR, but not m2AChR, m3AChR, or m5AChR that evidence different dominant hydrophobic modes).
The highly conserved, negatively charged aspartic acid residue in the third transmembrane segment, TM3 (e.g., D-105 in human m1AChR and D-111 in human m4AChR) (Fraser et al., 1989; Hulme et al., 1995) is speculated to bind to the characteristic positively charged headgroups of active site-directed muscarinic ligands (Ehlert and Delen, 1990; Kurtenbach et al., 1990). The hydrophobic, low dielectric constant, membrane interior location of the putative active site on TM3 is permissive of charge-charge interactions, unlike the high dielectric constant, charge shielded aqueous environments of the ei and intracellular loops, ii (Israelachvili and Wennerstrom, 1996; Israelachvilli, 1992; Tsao et al., 1993). We and others have speculated that aqueous environments facilitate the hydrophobic interactions of allosteric and/or indirectly acting peptide ligands with extramembranous portions, ei, of seven transmembrane receptors (Leckband et al., 1992, 1994; Mandell et al., 2003, 1997c, 2000a; Richards and Richmond, 1977; Tucek and Proska, 1995). Similarly, extracellular hydrophobic interactions with muscarinic GPCRs' ei are thought to position three-fingered muscarinic snake toxins for their subsequent actions (Jolkkonen et al., 1995; Olianas et al., 1999, 2000).
Noncompetitive modulators of mAChRs exert their positive or negative effects on receptor affinity and/or efficacy via interactions with away-from-the-active-site locations (Birdsall et al., 1995; Jakubik et al., 1996; Lazareno and Birdsall, 1996; Proska and Tucek, 1995). All of the mAChR subtypes are subject to kinetic modulation by a multiple and diverse group of ligands (Birdsall et al., 1997, 1983; Jakubik et al., 1997; Lanzafame et al., 1997; Lee and el-Fakahany, 1991; Proska and Tucek, 1995). Studies using point mutations and chimeral exchanges have indicated that modulatory sites are subject to influence by hydrophobic interactions that involve subsequences in ei and/or the extramembranous vicinity of the TMi, i= 1...7 (Gnagey et al., 1999; Matsui et al., 1995).
The polypeptide "fingers" of the muscarinic three-fingered snake toxins bind in the same locations as the active site-directed ligands, such as atropine and diphenylactoxy-N-methylpiperidine (Jolkkonen et al., 1995; Olianas et al., 1999, 2000), but with much higher subtype specificity (Kukhtina et al., 2000). The small differences in the dissociation constants in –log(Kd) for atropine studied in murine muscarinic receptor transfected fibroblasts have been reported as: m1 = 9.29, m2 = 9.00, m3 = 9.28, m4 = 9.66, and m5 = 9.43, and for diphenylactoxy-N-methylpiperidine, 4-DAMP, m1 = 8.92, m2 = 8.21, m3 = 9.19, m4 = 9.15, and m5 = 8.92 (Kashihara et al., 1992). The failure of these active site-targeted small molecule cholinergic agents to discriminate among muscarinic receptor subtypes contrasts with the high selectivity of the m1 and m4 muscarinic receptor polypeptide "three-finger" snake toxins. The values of their inhibition constants, Ki = IC50/(1+[L]/Kd) in nM of [3H]-N-methylscopoline binding in human mAChR-transfected Chinese hamster ovary (CHO) cells are: m1 = 0.2–48.0 and m4 = 2–120 vs. m2, m3, and m5 = >1000–20,000, the members of this snake toxin family being the most m1 and m4 receptor selective ligands known. (Adem and Karlsson, 1997).
The amino acid sequence analyses and GPCR-targeted, modulatory peptide designs presented here focus on the sequential hydrophobic patterns that are statistically dominant after the hydrophobic variation due to the transmembrane segments is computationally removed. m1AChR and m4AChR were found to share "hydrophobic modes" that differed from those evidenced in the m2, m3, and m5 subtypes (see Table 3). The modes shared by the m1 and m4 muscarinic cholinergic receptors were also dominant in the family of three-fingered muscarinic snake toxins which bind with specificity to those receptor subtypes (see Table 2).
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SEQUENTIAL MODE ANALYSES IN THE HYDROPHOBICALLY TRANSFORMED AMINO ACID SEQUENCE OF M1ACHR: BACKGROUND, METHODS AND FINDINGS |
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TOPABSTRACTA BRIEF DESCRIPTION OF...MODE SIMILARITIES OF PEPTIDE...SEQUENTIAL MODE ANALYSES IN...DESIGN OF M1ACHR-TARGETED...LOCATING PUTATIVE ALLOSTERIC...ACKNOWLEDGEMENTSREFERENCES |
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; Chothia and Janin, 1975; Chou and Fasman, 1974a,b; Cornette et al., 1987; Rose, 1978; Rose and Roy, 1980; Rose and Wetlaufer, 1977; Schiffer and Edmundson, 1967). These patterns have been characterized using a variety of descriptions of sequential hydrophobic variation, sometimes including smoothing by nearest neighbor averaging of amino acid hydrophobicities (Kyte and Doolittle, 1982). Among the best known of the hydrophobic mode descriptors are Edmondson's amphipathic helical wheel (Kaiser and Kezdy, 1983; Schiffer and Edmundson, 1967), Eisenberg's hydrophobic moments (Eisenberg et al., 1984), Rose's chain turns (Rose, 1978), Skolnick's ß-turns (Godzik and Skolnick, 1992), and Fourier and wavelet transformations in hydrophobic wavelengths (Eisenberg et al., 1984; Giuliani et al., 2000, 2002; Hirakawa et al., 1999; Lazovic, 1996; Lio and Vannucci, 2000; Mandell, 1984; Mandell et al., 1987, 1997a,b,c; Murray et al., 2002; Rackovsky, 1998; Selz et al., 1998).
Our findings suggest hydrophobic mode-selective attractive aggregation between polypeptide ligands and ei, juxta-TMi receptor, and other subsequences containing matching modes (e.g., chaperone-protein binding) (Mandell, 1984; Mandell et al., 1998a, 1987, 1997c, 2000a,b). As noted above, these phenomena appear analogous to the self-aggregation of sequential, amphipathically matching amino acid sequences in a process called "hydrophobic zippering" (Dill et al., 1995, 1993; Gronenborn and Clore, 1994), a process hypothesized to initiate folding in many proteins. Similarly, helical bundle formation is known to occur along the long axes of amphipathic helices with matching hydrophobic modes (Gibney et al., 2000; Park et al., 2000; Shu et al., 2000).
Characteristic wavelengths of hydrophobic recurrence (modes) in polypeptides
Generalizing the above suggestive relationships between patterns of hydrophobic variation and x-ray determined secondary structures in proteins, and exploiting the evolutionary statistical stationarity of polypeptide and protein sequence structure that justify the use of signal processing techniques on short series lengths, our laboratory used methods including Fourier and moving window Fourier transformations to elucidate familial mode commonalities in hormonal and neural peptides (Mandell, 1983, 1984, 1986, 1987; Mandell et al., 1987). For instance, a family of trophic peptide hormones with <50% sequence homology, growth hormone releasing factor, glucagon, vasoactive intestinal peptide, PHI, somatoliberin, and the A-chain of insulin, were found to share a dominant hydrophobic mode of -1 = 4.0 aa wavelength (Mandell et al., 1987). Attempts to find mode matches by applying these techniques to the >400-aa-long sequences of their and other membrane receptors were only occasionally successful; for example, there appeared to be mode matches between several nicotinic cholinergic toxins and nicotinic cholinergic receptors in the electric organ of the Torpedo ray. Generally, however, even the 50-residue windowed moving Fourier power spectra of the hydrophobic transformed sequences of receptor sequences were ambiguous. The spectra were dominated by the high amplitude, long wavelength, -1 50–70+ aa variation of the highly hydrophobic transmembrane segments. The spectra were noisy, with difficult-to-localize broadband peaks, manifesting end effects and often multiperiodic. All of these problems made an examination for peptide ligand and receptor hydrophobic mode matches difficult.
Orthogonal mode decomposition augments power spectral techniques in elucidating hydrophobic modes in GPCRs
The evolutionary stationarity of what would otherwise be considered too-short GPCRs aa series length for statistical treatment, the Wiener-Khinchine relations (Bendat and Piersol, 1986) associating the correlation and spectral density functions (with spectra already found promising for characterizing hydrophobic modes in shorter hormonal and neuropeptide series), and the need to deal with the sources of poor mode definition listed above, led naturally to the selection of techniques derived from principle component analyses (Fukunaga, 1990). These methods involve the linear decomposition of correlation matrices and allow the computational removal of the dominant transmembrane hydrophobic pattern, the discrimination of the remaining leading modes from the noise floor, and the isolation of the leading orthogonal modes from each other in potentially multimodal patterns. Being without end effects and other potential limitations of fixed transforms (e.g., of Fourier or Bessel), these transformations individually tailor the "shape" of the mode representation to the specific data series (in the form of the leading eigenvectors, Xi, of the series' lagged autocovariance matrices) (Brillinger, 1981; Broomhead et al., 1987; Broomhead and King, 1986). In the following, please notice that the computationally derived hydrophobic mode(s) of GPCRs are represented in most detail by the leading eigenvectors, Xi, singular or summed, of the lagged autocovariance matrix transformation of the original hydrophobic series (Broomhead et al., 1987; Broomhead and King, 1986; Golub and Van Loan, 1993) (see below). They are represented more generally by an approximating index, -1 in aa, by the maximum entropy, all-poles power spectral transformations (which picks out one or two leading -values) (Press et al., 1988) of the leading Xi or of a constructed new smoothed function formed by the serial composition of Xi with the original amino acid hydrophobic series to form an eigenvector-weighted nearest neighbor-averaged autocovariance matrix eigenfunction, 
i (Broomhead et al., 1987; Broomhead and King, 1986).
Methods of lagged autocovariance matrix formation, eigenvector extraction, and autocovariance matrix eigenfunction construction from m1's hydrophobic series
The hydrophobically transformed amino acid sequences were used to generate an M-lagged data matrix from which M x M covariance matrices, CM, were computed. These CM were decomposed into sets of l orthogonal eigenfunctions, l(j), l = 1...M, j = 1...n - M+1 (Broomhead et al., 1987; Broomhead and King, 1986; Golub and Van Loan, 1993).
More specifically, from the sequentially lagged data column vectors (T 
transpose) 
and where K = n - M + 1, the sequence-averaged dyadic product, 
is used to obtain the autocovariance matrix, a Hermitean M x M matrix, 
was chosen for the m1 muscarinic cholinergic receptor to minimize the least-squares error of m1's leading 1, representing the pattern of the highly hydrophobic transmembrane segments, and the m1 receptor's nearest neighbor-smoothed hydropathy plot, which is also dominated by the transmembrane pattern (Kyte and Doolittle, 1982). We compute the ordered eigenvalues, {
i}i = 1...M, and the associated ordered eigenvectors, Xi(j), of CM , where i = 1...M and labels the eigenvector, and j = 1...M and refers to the jth components of the eigenvector Xi. The {i}i = 1...M are ordered from largest to smallest and constitute the eigenvalue spectrum of CM. The similarly ordered associated Xi(j) are convolved with original hydrophobic series into their associated eigenfunctions l(j), where l = 1...M labels the eigenvector and j = 1...n - M indexes the eigenfunction's jth component. That is, the convolution of each of the leading eigenvectors with the hydropathy series is carried out by computing the sums of the scalar products of the M length eigenvector with an M length of the hydrophobic series to produce a point in the eigenfunction; this process is translated down the data series by one step and repeated to generate each of the sequential points of the eigenfunction that corresponds to its ordered eigenvalue-associated eigenvector in the computation. In those studies, the l of M are plotted as a function of the M lag-reduced sequence position.
Intuitively, CM scans for hydrophobic modes across a range of autocorrelation lengths from 1 to M, the range of the lags defined in the original data matrices and reflected in the autocovariance matrices. Because CM is real, symmetric and normal 
its {i}i = 1...M are real, nonnegative, and distinct, and its associated Xi(j) constitute a natural basis for orthonormal projections on m1's hydrophobic series (Golub and Van Loan, 1993). The set of l can be regarded as orthonormally decomposed sequences of eigenvector-weighted, moving average values (Broomhead et al., 1987; Broomhead and King, 1986).
All-poles maximum entropy power spectral transformation of the leading eigenfunctions
The l were transformed into their dominant hydrophobic modes (inverse frequencies, -1) using all-poles maximum entropy power spectra, S(l) (Madan, 1993; Press et al., 1988)). Generally, 
such that the zeros of the denominator result in peaks ("poles") marking the dominant hydrophobic mode(s) of the l. This form of the Fourier transformation generates relatively well-resolved spectral peaks in finite series in which only a small number, k, of the (auto)covariance coefficients, ck, are known. In these studies, k 
8 to avoid "splitting" S() into spurious modes. The Fourier coefficients, ak , can be calculated from the set of data-derived ck, chosen so that the entropy of the spectral estimate, 
is maximal. Beyond the limited information of the small set of data-derived autocorrelation-matched Fourier coefficients, called the "correlation matching property", the process is extended into a Gaussian process such that h is maximized. It is known that the Gaussian is the function that maximizes h under the constraints of a finite number of second order ck, as in our data (Madan, 1993). Intuitively, the all-poles maximum entropy S() yields results like an autoregressive, maximum likelihood spectral estimate in that it is not model-dependent, but rather its ak values follow from the ck values that are derived directly from the data (Priestly, 1981). This technique acts as a filtering process, yielding the moduli of the one or two leading complex poles of discrete hydrophobic variational frequency in minimally distorted form from the leading l of the lagged autocovariance matrix of the m1 sequence (Mandell et al., 1998a, 1997a,b,c; Selz et al., 1998).
The transmembrane mode as an example: the leading eigenfunction, 1, of the m1 lagged autocovariance matrix, CM, and its all-poles power spectral representation, S(1)
To exemplify the results of computational hydrophobic mode extraction, we compare the standard m1 smoothed hydropathy plot with the X1 determined (convolved with the original sequence) leading eigenfunction, 1, and its all-poles power spectral transformation, representing the transmembrane mode of the m1 muscarinic cholinergic receptor (Giuliani et al., 2002; Mandell et al., 1998a, 1997c). Fig. 1 A is a plot of the hydrophobically transformed amino acid series of m1 using the Tanford scale as listed above (Manavalen and Ponnuswamy, 1978). Fig. 1 B is the all-poles power spectral transformation of the hydropathy plot of Fig. 1 A evidencing its noisy floor, broadband peaks, and multimodality, all of which make the isolation and characterization of its hydrophobic modes not possible. Fig. 1 C was computed by a nearest-neighbor, three-point moving average of the hydrophobically transformed m1 amino acid sequence, iterated 11 times and plotted as a function of the m1-11 length adapted from the approach of Kyte and Doolittle (1982). Five-point moving averages iterated 11 times were used to test the stability of the pattern of its hydrophobic oscillations. Fig. 1 D is a graph of the all-poles power spectral transformation of Fig. 1 C demonstrating its long wavelength of -1 = 71.43 aa. Fig. 1 E is a graph of m1's 1 that very closely resembles the pattern of hydrophobic oscillations of the transmembrane mode in the graph of Fig. 1 C. Fig. 1 F is a graph of the all-poles maximum entropy power spectral transformation of m1's 1, indicating its long hydrophobic wavelength, -1 = 73.15 aa. Note that the spectrum of the leading eigenfunction differs by <2% in the frequency-wavelength domain from that of the nearest neighbor-averaged hydropathy plot (Fig. 1 D).
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, 1997c; Selz et al., 1998). As described above, the maximum lag described in the autocovariance matrix, here 15, determines the length of the eigenvalue-weighted eigenvector(s) used as templates for amino acid assignment in peptide design (see below). This value is chosen from a range of values so as to maximize the correlation between the leading eigenfunction, 1 (Fig. 1 E), and the nearest neighbor-averaged hydropathy plot (Kyte and Doolittle, 1982) (Fig. 1 C) (Mandell et al., 1997c).
Hydrophobic mode hierarchies in the family of muscarinic cholinergic receptors, miAChR: characteristic m1 and m4 modes
Using the methods illustrated above for the isolation and characterization of the m1 transmembrane mode, Table 3 lists the all-poles power spectral hydrophobic mode wavelengths -1 of the ordered leading eigenfunctions, i, of the five human miAChR subtypes. The m1 and m4 subtypes share hydrophobic eigenfunction modes of -1 7.8 aa and 2.4 aa, neither of which are present in the other three mAChR subtypes. Note that the closest to these values of these three is the 7.94 aa mode of the m5 subtype. This m5 mode demonstrates a difference from the m1 target of >2% in the frequency-wavelength domain (compare with <2% mode similarity comparing the nearest neighbor-averaged and eigenfunction representations of the transmembrane mode). Two percent or less difference is our approximate criteria for declaring mode frequency-wavelength similarities and differences. Power spectral modes of the eigenfunctions associated with the 8th through the 15th ordered eigenvectors are not shown. The receptors' eigenvalue spectra decayed quickly, and values of 7 through 15 were very small, accounting for very little of the receptors' variance in hydrophobic free energy.
When a cell in Table 3 contains two entries—for example, the power spectral peaks of 6 in m4—it is because a single eigenfunction, here 6, evidenced the indicated two well-defined peaks that were of nearly equivalent power. The other kind of redundancy sometimes demonstrated within a single receptor subtype results from isomodal eigenfunctions that are out of phase and therefore orthogonal. In those cases, we see two eigenfunctions with approximately the same statistical wavelength and usually with similar eigenvalues and relative spectral power. An example of this is also seen in Table 3, where the second and third eigenfunctions of m1AChR (2 and 3) manifest an isomodal, out-of-phase redundancy. Generally in such cases, the eigenfunction associated with the slightly larger eigenvalue is chosen in the construction of the receptor-targeted peptide template.
As in our previous work (Mandell et al., 2003, 1997a,b,c, 1998b; Selz et al., 1998), the dominant wavelengths of the power spectral modes of m1AChR's i values were confirmed and localized in the receptor sequence with reference to the dominant mother wavelet scales and their sequence locations using continuous, one-dimensional Morlet wavelet transformations (Farge et al., 1993; Strang, 1993; Wickerhauser, 1994). These demonstrated that the sequence locations of the mother wavelet scale density centroids corresponding to the 7.8 aa power spectral mode of 2 (and isomodal but phase-advanced 3) were in the sequence vicinity of the putative e1 and e3 extracellular loops. (See Fig. 3 B and associated discussion below.) This finding is relevant to the modulatory role of hydrophobic mode matches in these sequence locations because alanine mutagenesis of Tyr-101 and Tyr-404, in the e1 and e3 loops, respectively, resulted in the m1 allosteric ligands, gallamine and himbacine, failing to bind and/or modulate the m1 muscarinic receptor (Matsui et al., 1995). The ei are known to be common locations for ligand-mimetic binding of antibodies raised to transmembrane receptors (Beattie et al., 1996a,b). Molecular dynamics simulations of protein motions show coupling between peptide chain loops (Watanabe et al., 1997), and such modulation of loop movements can serve as "molecular switches" (Ma and Karplus, 1997) initiating conformational transitions in some proteins (Lazaridis and Karplus, 1997).
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; Jerusalinsky and Harvey, 1994). These toxins have been shown to be selectively specific for the m1 and m4, but not m2, m3, or m5 AChR subtypes. They manifest long-lasting "pseudoirreversible" allosteric modulation of muscarinic antagonist binding (Adem et al., 1997; Adem and Karlsson, 1997). The leading hydrophobic modes of the all-poles power spectral transformation of these relatively short, 64–65 residue hydrophobically transformed amino acid series are listed in Table 2. Because of their short lengths and the absence of TM associated modes, the toxins did not require eigendecomposition of their autocovariance matrices. MT-1 through MT-7 are m1AChR and m4AChR subtype specific muscarinic toxins isolated and sequenced from the venom of the green mamba (Adem and Karlsson, 1997; Olianas et al., 2000). MT- and MT-ß were isolated and sequenced from the venom of the black mamba snake (Adem and Karlsson, 1997; Jerusalinsky et al., 1997; Olianas et al., 2000). Note how the two dominant hydrophobic modes of the m1 and m4 specific snake toxins approximate the dominant modes found in the posttransmembrane hydrophobic modes of the m1 and m4 receptors (Table 3). We note that these hydrophobic modes are not generic for all three-fingered toxins in that all of the members of the three-fingered nicotinic acetylcholine receptor-targeted toxins of sea kraits demonstrate dominant hydrophobic wavelengths of 2.0 and 4.15 aa (Housset and Fontecilla-Camps, 1996). |
DESIGN OF M1ACHR-TARGETED PEPTIDES USING LEADING EIGENVECTOR MODES AS TEMPLATES FOR POSITIONAL ASSIGNMENT OF AMINO ACIDS BY HYDROPHOBIC GROUP |
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TOPABSTRACTA BRIEF DESCRIPTION OF...MODE SIMILARITIES OF PEPTIDE...SEQUENTIAL MODE ANALYSES IN...DESIGN OF M1ACHR-TARGETED...LOCATING PUTATIVE ALLOSTERIC...ACKNOWLEDGEMENTSREFERENCES |
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; Takeuchi et al., 1998; Zysk and Baumbach, 1998). These models are composed of three-dimensional arrangements of molecular constituents with potential noncovalent interactions, including hydrogen bond donors and acceptors, ionizable groups, and ring or chain hydrophobic aggregation. Models emphasize modular components found to be common among similarly active agents. These are often discovered by high through-put screening of large peptide libraries (Guner, 1999).
In contrast, our sequence approach to the hydrophobic mode characterization of GPCRs is uniquely relevant to membrane protein subsequences that may be without stable tertiary structure (Romero et al., 1998). As described above, these conformationally disordered polypeptide sequences evidenced in x-ray crystallographic studies by missing electron densities, nuclear magnetic resonance graphs with sharp peaks, and/or the absence of evidence for secondary structure in nuclear Overhauser effects and circular dichroism studies with low-intensity signals from 210 to 240 nm (Dyson and Wright, 2002a,b; Romero et al., 2001; Wright and Dyson, 1999). Such subsequences characteristically become ordered upon ligand binding, going through a disorder-order transition and achieving x-ray and/or NMR demonstrable stable tertiary structure (Dyson and Wright, 2002a,b; Kriwacki et al., 1996; Wright and Dyson, 1999).
Recall that disordered loop sequences of globular proteins, GPCRs, and 12 transmembrane segment transporters (Buck and Amara, 1995; Nirenberg et al., 1997) have been shown to play significant roles in polypeptide-polypeptide and protein-protein interactions, participating in allosteric and antibody binding sites (Rondard and Bedouelle, 2000) and modulatory "switches" (Ulloa-Aguirre and Conn, 2000). This makes them promising targets in rational peptide ligand design (Dunker et al., 1998, 2001; Dunker and Obradovic, 2001). We address these targets, using the leading hydrophobic eigenvector(s), Xi, in mode-matched polypeptide ligand design. In this way, we have successfully targeted "away from the active site" GPCR modulatory mechanisms in the D2 dopamine receptor (Mandell et al., 2003; Naylor et al., 1995; Simpson et al., 1999; Vaughan et al., 2001) and the nerve growth factor tyrosine kinase receptor (Mandell et al., 2001). After computational removal of the transmembrane mode (Fig. 1), the leading eigenvector hydrophobic modes tend to be concentrated in putative extracellular loop and juxta-transmembrane polypeptide subsequences in membrane receptors (Mandell et al., 1998a, 2003, 2001, 1997c
