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Solution Scattering 141 3. Method 3.1. Underlying Theory and Scattering Data Interpretation 3.1.1. The Basic Scattering Equation X-rays and neutrons have the properties of plane waves, and the scattering pattern at small angles results from the constructive interference of secondary waves that are generated when a plane wave interacts with matter. Interference occurs when the secondary waves are additive. This additive property requires the scattering to be elastic (i.e., no energy change between the incident and scattered wave) and coherent (i.e., there is a defined phase relationship between the incident and scattered wave). The scattering profile for a macromolecule, such as a protein, in solution is a maximum at zero-scattering angle and falls off with a rate that depends upon the protein’s size and shape; the larger the protein, the faster the fall off. The coherent, elastic scattering I(Q) from a homogeneous solution of monodisperse proteins can be expressed mathematically as I(Q) ∝ 〈∫|[ρ(r) –ρ s]e –iQ·r dr| 2 〉 (1) The integration is taken over the volume of the protein, and 〈 〉 denotes the average over all orientations. Q is the momentum transfer or scattering vector and its amplitude is 4π(sinθ)/λ, where θ is half the scattering angle and λ is the wavelength of the scattered radiation. ρ(r) and ρ s are the scattering densities for the protein and its solvent, respectively, and the difference between them is the “contrast.” Scattering densities are calculated by summing the scattering amplitudes of each atom within a volume and dividing by that volume. The intensity of the scattering signal is proportional to the square of the molecular weight of the scattering molecules and to their number density (i.e., protein concentration). For a solution containing a mixture of different structures, the measured scattering will be the average of all the structures present weighted by their relative concentrations and the square of their molecular weights. 3.1.2. X-Ray vs Neutron Scattering X-ray sources, even the simple laboratory devices, are many orders of magnitude stronger than neutron sources. Because biological molecules are weak scatterers the relatively low fluxes of neutron sources is a disadvantage. However, neutrons are nonionizing radiation and hence quite benign. More importantly they offer the opportunity for “contrast variation” studies. X-rays are scattered by electrons and hence X-ray scattering amplitudes of atoms increase monotonically with the number of electrons. Neutrons are scattered by atomic nuclei, and neutron scattering amplitudes depend upon the complex properties of the neutron-nucleus interaction showing no systematic dependence on atomic number. Importantly, isotopes of the same element can have very differ-
142 Trewhella and Krueger Table 1 Coherent Neutron Scattering, bcoh, and X-Ray Scattering, fX-ray, Amplitudes for Biologically Relevant Nuclei fX-ray for θ = 0 in electrons Atom Nucleus bcoh (10 –12 cm) (and in units of 10 –12 cm) Hydrogen 1 H – 0.3742 1.000 (0.28) Deuterium 2 H 0.6671 1.000 (0.28) Carbon 12 C 0.6651 6.000 (1.69) Nitrogen 14 N 0.940 7.000 (1.97) Oxygen 16 O 0.5804 8.000 (2.25) Phosphorous 31 P 0.517 15.000 (4.23) Sulfur Mostly 32 S 0.2847 16.000 (4.5) ent neutron-scattering properties. For neutrons, one of the largest differences is between the isotopes of hydrogen ( 1 H = hydrogen, and 2 H = deuterium). Table 1 gives the coherent, elastic X-ray, and neutron-scattering amplitudes for the atoms commonly found in biological systems. Note that the neutron scattering amplitudes for most nuclei, including 2 H, are positive and approximately equal. The exception is the negative scattering amplitude for 1 H that results from a 180° phase shift between neutrons scattered by 1 H compared to the other nuclei. As a consequence, the neutron scattering density of a molecule depends strongly on its mean hydrogen content, and deuterium substitution can be used to manipulate neutron scattering densities. 3.1.2. Neutron Scattering and Contrast Variation Equation 1 shows that the intensity of the scattering from a protein in solution depends upon the difference in scattering density between the particle and the solvent, i.e., its “contrast.” If a complex is made using one deuterated and one nondeuterated protein, the two proteins will have very different neutron scattering densities. Further, by changing the deuterium level in the solvent, the neutron scattering contrast of each protein can be systematically varied. By adjusting the deuteration level in the solvent so that the mean solvent density matches that of either the deuterated or the nondeuterated protein, it is possible to “solvent match” that protein such that it has zero contrast and becomes “invisible” in the neutron experiment. Solvent matching thus provides a means for extracting structural information on the individual components within a complex. A more robust approach to utilizing contrast variation methods with neutron scattering for extracting structural information on the components within the macromolecular complexes is to measure a “contrast series” in which the solvent deuteration level is systematically varied over he widest range possible.
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Methods in Molecular BiologyTM Biol
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M E T H O D S I N M O L E C U L A R
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© 2002 Humana Press Inc. 999 River
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Preface Calcium plays an important
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Contents Dedication ...............
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Contents xi 26 Enzymatic Assays to
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xiv Contents of Companion Volume 15
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xvi Contributors LESLIE D. HICKS
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20 Dean, Kelsey, and Re
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2 Yazawa
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4 Yazawa Fig. 1. Schematic represen
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6 Yazawa Fig. 2. Shop drawing for t
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8 Yazawa DuPont-NEN and the molar c
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10 Yazawa Fig. 4. Examples of the C
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12 Yazawa 5. Plot the calculated Ca
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14 Yazawa 12. Starovasnik, M. A., D
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16 Fig. 1. Molecular structures and
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18 Linse 7. 0.1 M EDTA. Dissolve 37
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20 Linse iterate the other paramete
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22 Linse tive to small alterations
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24 Linse References 1. Tsien, R. Y.
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26 Haiech and Kilhoffer to use the
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28 Haiech and Kilhoffer where γ is
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30 Haiech and Kilhoffer reporter gr
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32 Haiech and Kilhoffer tein with i
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34 Haiech and Kilhoffer Calmodulin
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36 Haiech and Kilhoffer Fig. 3. Mec
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38 Haiech and Kilhoffer We may sugg
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40 Haiech and Kilhoffer 29. Stewart
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42 Haiech and Kilhoffer 62. Becking
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44 Martin and Bayley Fig. 1. Absorp
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46 Martin and Bayley evaporation. C
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48 Martin and Bayley 4. Set the tem
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50 Martin and Bayley Fig. 2. Near-U
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52 Martin and Bayley discussed by M
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54 Martin and Bayley References 1.
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20 Dean, Kelsey, and Reik
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58 Fabian and Vogel are equipped wi
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60 Fabian and Vogel The penetration
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62 Fabian and Vogel perature, ionic
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64 Fabian and Vogel Fig. 3. IR spec
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66 Fabian and Vogel Fig. 4. Lower t
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68 Fabian and Vogel These correlati
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70 Fabian and Vogel Fig. 5. Amide I
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72 Fabian and Vogel 4. ATR-FTIR spe
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74 Fabian and Vogel 25. Georg, H.,
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76 Weljie and Vogel Fig. 1. Simplif
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78 Weljie and Vogel Fig. 3. Steady-
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80 Weljie and Vogel VIS spectrophot
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82 Weljie and Vogel 4. A series of
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84 Table 1 Advanced Fluorescence Me
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86 Weljie and Vogel References 1. L
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Page 121 and 122: 102 Johnson and Tikunova 6. Johnson
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Page 143 and 144: 124 Lopez and Makhatadze 2.5 mL are
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Page 149 and 150: 130 Hicks et al. Fig. 2. Debye plot
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Page 153 and 154: 134 Hicks et al. Fig. 3. Sedimentat
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Page 157 and 158: 138 Trewhella and Krueger This chap
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Page 203 and 204: 184 Brokx and Vogel Table 1 An Over
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192 Brokx and Vogel 8. Weljie, A. M
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196 Berliner Fig. 1. Aqueous X-band
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198 Berliner in the Bruker instrume
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200 Berliner Fig. 3. ESR spectra of
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202 Berliner Fig. 5. Low-temperatur
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204 Berliner 14. Musci, G., Reed, G
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206 Clarke and Vogel cal calcium-bi
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208 Clarke and Vogel Fig. 1. 113 Cd
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214 Clarke and Vogel The linewidth
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218 Drakenberg sands, of Hertz broa
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220 Drakenberg 1. Bloch equations m
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222 Drakenberg Fig. 2. Measurement
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224 Drakenberg Fig. 3. (A) 43 Ca NM
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226 Drakenberg Fig. 5. 43 Ca NMR sp
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228 Drakenberg NMR at sub-mM concen
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230 Drakenberg 30. Shimizu, T., Hat
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232 Weljie and Heringa Table 1 Amin
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234 Weljie and Heringa Table 2 Webs
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236 Weljie and Heringa diction meth
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238 Weljie and Heringa these databa
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240 Weljie and Heringa lineages. Th
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242 Weljie and Heringa compared, an
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244 Weljie and Heringa sequences as
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246 Weljie and Heringa much larger
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248 Weljie and Heringa ment require
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250 Weljie and Heringa 10. Kawasaki
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252 Weljie and Heringa 44. Thompson
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254 Weljie and Heringa
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256 Li et al. In order for these ex
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258 Li et al. 7. Mineral Mixture (s
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260 Li et al. 3. When the 45% D 2O/
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262 Li et al. sion system works eff
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264 Li et al. reliably produce high
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268 Mal et al. NMR data, X-PLOR (11
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270 Mal et al. Table 1 Simulated An
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272 Mal et al. 3.1.2.1. AMBIGUOUS D
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274 Mal et al. molecular dynamics a
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276 Mal et al. Assuming a rigid bod
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278 Mal et al. assign (segid A and
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280 Mal et al. References 1. Drenth
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282 Mal et al. 33. Jeener, J., Meie
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286 Werner et al. Our research has
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288 Fig. 1. (A) T 1, T 2 and the he
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290 Werner et al. tion, using gradi
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292 Werner et al. Fig. 3. (A) Rotat
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294 Werner et al. Fig. 4. (A) Line-
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296 Werner et al. Table 2 Models of
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298 Werner et al. 9. Reinhardt, D.
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300 Werner et al. 39. Press, W. H.,
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302 Boyd et al. utilize residual di
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304 Boyd et al. Fig. 1. (A) The sol
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306 Boyd et al. or significant peak
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308 Boyd et al. Fig. 3. Histogram o
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310 Boyd et al. total and NOE energ
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312 Boyd et al. 2. When studying mu
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314 Boyd et al. 4. Downing, A. K.,
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316 Boyd et al. 35. Schulte-Herbrug
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318 Yap et al. image representation
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320 Yap et al. modified EF-hand is
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322 Table 1 Angle and Distance Outp
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324 Yap et al. 2. Ikura, M. (1995)
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326 Kobayashi that S-100A1 and S-10
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328 Kobayashi 3.2. Coupling of Crom
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330 Kobayashi Fig. 2. Tricine/SDS/P
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332 Kobayashi 0.1% TFA at a flow ra
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334 Kobayashi Both recombinant S-10
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336 Kobayashi Fig. 6. Affinity chro
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340 Walsh et al. verts CaM from an
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342 Walsh et al. 11. Bovine serum a
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344 Walsh et al. Fig. 1. CaM-depend
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348 Walsh et al. 3.5. NOS Reduction
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354 Walsh et al. 3. Cho, M. J., Vag
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356 Hughes et al. The electroporati
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358 Hughes et al. 4. 1 M Na 2CO 3.
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360 Hughes et al. range, repeat ste
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362 Hughes et al. Table 1 Examples
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366 Persechini of the different CaM
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368 Persechini Fig. 1. Schematic re
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370 Persechini 2. 1 L of Terrific b
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372 Persechini Fig. 4. Titration wi
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374 Persechini Table 1 Parameters f
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376 Persechini Fig. 6. The [Ca 2+ -
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378 Persechini determined if the di
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380 Persechini relatively insensiti
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382 Persechini 18. Chafouleas, J. G
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384 Török et al. actions in the c
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386 Török et al. 3. The reaction
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388 Török et al. Fig. 2. Electros
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390 Török et al. Fig. 3. Peptides
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392 Török et al. Table 1 Peptide
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394 Török et al. Fig. 7. Nanospra
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396 Török et al. Fig. 9. Nanospra
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398 Török et al. Fig. 11. Electro
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400 Török et al. 3 µM FL-calmodu
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402 Török et al. Fig. 14. Localiz
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404 Török et al. Fig. 16. Indicat
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406 Török et al. the significantl
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408 Török et al.
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410 Index substitutes, see Fluoresc
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412 Index Stern-Volmer plot, 78, 81
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414 Index Phenothiazines, see Calmo
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METHODS IN MOLECULAR BIOLOGY TM •
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