SPECTROSCOPY FOR THE BIOLOGICAL SCIENCESS

SPECTROSCOPY FOR THE BIOLOGICAL SCIENCES Spectroscopy for the Biological Sciences, by Gordon G Hammes Copyright © 2005 John Wiley & Sons, Inc SPECTROSCOPY FOR THE BIOLOGICAL SCIENCES GORDON G HAMMES Department of Biochemistry Duke University A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2005 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic formats For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Hammes, Gordon G., 1934– Spectroscopy for the biological sciences / Gordon G Hammes p ; cm Companion v to: Thermodynamics and kinetics for the biological sciences / Gordon G Hammes c2000 Includes bibliographical references and index ISBN-13 978-0-471-71344-9 (pbk.) ISBN-10 0-471-71344-9 (pbk.) Biomolecules—Spectra Spectrum analysis [DNLM: Spectrum Analysis Crystallography, X-Ray ] I Hammes, Gordon G., 1934– Thermodynamics and kinetics for the biological sciences II Title QP519 S6H35 2005 572—dc22 2004028306 Printed in the United States of America 10 CONTENTS PREFACE ix 1 FUNDAMENTALS OF SPECTROSCOPY Introduction / Quantum Mechanics / Particle in a Box / Properties of Waves / References / 13 Problems / 14 X-RAY CRYSTALLOGRAPHY 17 Introduction / 17 Scattering of X Rays by a Crystal / 18 Structure Determination / 22 Neutron Diffraction / 25 Nucleic Acid Structure / 25 Protein Structure / 28 Enzyme Catalysis / 30 References / 32 Problems / 32 v vi CONTENTS ELECTRONIC SPECTRA 35 Introduction / 35 Absorption Spectra / 36 Ultraviolet Spectra of Proteins / 38 Nucleic Acid Spectra / 40 Prosthetic Groups / 41 Difference Spectroscopy / 44 X-Ray Absorption Spectroscopy / 46 Fluorescence and Phosphorescence / 47 RecBCD: Helicase Activity Monitored by Fluorescence / 51 Fluorescence Energy Transfer: A Molecular Ruler / 52 Application of Energy Transfer to Biological Systems / 54 Dihydrofolate Reductase / 57 References / 58 Problems / 59 CIRCULAR DICHROISM, OPTICAL ROTARY DISPERSION, AND FLUORESCENCE POLARIZATION 63 Introduction / 63 Optical Rotary Dispersion / 65 Circular Dichroism / 66 Optical Rotary Dispersion and Circular Dichroism of Proteins / 67 Optical Rotation and Circular Dichroism of Nucleic Acids / 69 Small Molecule Binding to DNA / 71 Protein Folding / 74 Interaction of DNA with Zinc Finger Proteins / 77 Fluorescence Polarization / 78 Integration of HIV Genome into Host Genome / 80 a-Ketoglutarate Dehyrogenase / 81 References / 84 Problems / 84 VIBRATIONS IN MACROMOLECULES Introduction / 89 Infrared Spectroscopy / 92 Raman Spectroscopy / 92 Structure Determination with Vibrational Spectroscopy / 95 89 CONTENTS vii Resonance Raman Spectroscopy / 98 Structure of Enzyme-Substrate Complexes / 100 References / 101 Problems / 102 PRINCIPLES OF NUCLEAR MAGNETIC RESONANCE AND ELECTRON SPIN RESONANCE 103 Introduction / 103 NMR Spectrometers / 106 Chemical Shifts / 108 Spin-Spin Splitting / 110 Relaxation Times / 112 Multidimensional NMR / 115 Magnetic Resonance Imaging / 121 Electron Spin Resonance / 122 References / 125 Problems / 125 APPLICATIONS OF MAGNETIC RESONANCE TO BIOLOGY 129 Introduction / 129 Regulation of DNA Transcription / 129 Protein-DNA Interactions / 132 Dynamics of Protein Folding / 133 RNA Folding / 136 Lactose Permease / 139 Conclusion / 142 References / 142 MASS SPECTROMETRY Introduction / 145 Mass Analysis / 145 Tandem Mass Spectrometry (MS/MS) / 149 Ion Detectors / 150 Ionization of the Sample / 150 Sample Preparation/Analysis / 154 Proteins and Peptides / 154 Protein Folding / 157 Other Biomolecules / 160 145 viii CONTENTS References / 161 Problems / 161 APPENDICES Useful Constants and Conversion Factors / 163 Structures of the Common Amino Acids at Neutral pH / 165 Common Nucleic Acid Components / 167 INDEX 169 PREFACE This book is intended as a companion to Thermodynamics and Kinetics for the Biological Sciences, published in 2000 These two books are based on a course that has been given to first-year graduate students in the biological sciences at Duke University These students typically not have a strong background in mathematics and have not taken a course in physical chemistry The intent of both volumes is to introduce the concepts of physical chemistry that are of particular interest to biologists with a minimum of mathematics I believe that it is essential for all students in the biological sciences to feel comfortable with quantitative interpretations of the phenomena they are studying Indeed, the necessity to be able to use quantitative concepts has become even more important with recent advances, for example, in the fields of proteomics and genomics The two volumes can be used for a one-semester introduction to physical chemistry at both the first-year graduate level and at the sophomore-junior undergraduate level As in the first volume, some problems are included, as they are necessary to achieve a full understanding of the subject matter I have taken some liberties with the definition of spectroscopy so that chapters on x-ray crystallography and mass spectrometry are included in this volume This is because of the importance of these tools for understanding biological phenomena The intent is to give students a fairly complete background in the physical chemical aspects of biology, although obviously the coverage cannot be as complete or as rigorous as a traditional two-semester course in physical chemistry The approach is more conceptual than traditional physical chemistry, and many examples of applications to biology are presented I am indebted to my colleagues at Duke for their assistance in looking over parts of the text and supplying material Special thanks are due to Professors ix x PREFACE David Richardson, Lorena Beese, Leonard Spicer, Terrence Oas, and Michael Fitzgerald I again thank my wife, Judy, who has encouraged, assisted, and tolerated this effort I welcome comments and suggestions from readers Gordon G Hammes APPENDIX STRUCTURES OF THE COMMON AMINO ACIDS AT NEUTRAL pH Spectroscopy for the Biological Sciences, by Gordon G Hammes Copyright © 2005 John Wiley & Sons, Inc 165 166 APPENDIX COO- COO+ H3N C Aliphatic + H3N H COO+ H C H3N CH CH3 H3C CH3 H3N Nonpolar C H H C H H3C C H CH2 CH3 CH3 Leucine (Leu) (L) COO+ H3N CH Valine (Val) (V) COO+ COO+ CH2 H3C Alanine (Ala) (A) H C H2N C H H2C CH2 Isoleucine (Ile) (I) COO+ H3N CH2 C H COO+ H3N C CH2 CH2 SH CH2 H S CH3 Proline (Pro) (P) Glycine (Gly) (G) COO+ Aromatic H3N H C H3N CH2 C N H H3N C C CH2 NH2 C H H C OH COO+ H3N C H C H OH CH3 NH2 Serine (Ser) (S) COO+ H3N C Threonine (Thr) (T) COO- COO+ H H3N CH2 CH2 CH2 CH2 CH2 CH2 N H C NH2 + Tryptophan (Trp) (W) H CH2 NH3 C N CH H H3N Glutamine (Gln) (Q) COOH H CH2 COO+ C C Asparagine (Asn) (N) Charged H3N Tyrosine (Tyr) (Y) H CH2 H3N C H CH2 COO+ O + C OH CH2 O H3N Phenylalanine (Phe) (F) COOH3N C H C COO+ CH NH Histidine (His) (H) Polar + CH2 C H + COO- COO+ Methionine (Met) (M) Cysteine (Cys) (C) C H + H3N C O C H CH2 CH2 CH2 O- C O O- + NH2 Lysine (Lys) (K) Arginine (Arg) (R) Aspartate (Asp) (D) Glutamate (Glu) (E) APPENDIX COMMON NUCLEIC ACID COMPONENTS H N N O N H N Guanine (G) O O H NH N H H O H H OH H OH H N H Adenine (A) N H H O Uracil (U) OH O N H NH Thymine (T) HOCH2 N H NH2 Cytosine (C) CH3 N NH H N H H NH2 O NH2 b-D-ribofuranose HOCH2 H OH O H H H OH H b-D-2-deoxyribofuranose Spectroscopy for the Biological Sciences, by Gordon G Hammes Copyright © 2005 John Wiley & Sons, Inc 167 INDEX Absorption spectra 36 Active transport 139 Acridine orange-DNA binding 71 Adenine spectrum 41 Adenylic acid (poly) 70 Aldolase, folding 159 a-helix CD and ORD 68 structure 28 a-ketoglutarate dehydrogenase 81–83 fluorescence anisotropy 82 Amino acids spectra 38 structures 165–166 Anisotropy 79, 80, 82 Anharmonic oscillator 91 Beer-Lambert law 36 b sheet CD and ORD 68 structure 28 Bravais lattice 18 Chemical shifts 108–110 Chloroplast coupling factor 55–57 Circular dichroism (CD) 64, 66 differential extinction coefficient 67 DNA-zinc fingers 77–78 ellipticity 67 molar ellipticity 67 nucleic acids 69–70 P protein 75–76 proteins 67–69 COSY (correlated spectroscopy) 116 l cro repressor protein 118 Cotton effect 67 Coupled equilibria 77 Crystal systems 19 Cyclic-AMP response element 129–132 Cytochromes 43 Cytosine spectrum 41 de Broglie wavelength Difference spectroscopy 41 Diffraction Laue 23–24 neutron 25 x rays 21 Spectroscopy for the Biological Sciences, by Gordon G Hammes Copyright © 2005 John Wiley & Sons, Inc 169 170 INDEX Diffusion facilitated 139 rotational 80 Dihydrofolate reductase 57–58 resonance Raman 100–101 DNA A-DNA structure 26 acridine orange binding 71–73 B-DNA structure 25 melting 42 protein interactions 71 transcription 129, 132 Z-DNA structure 26 DNA polymerase enzyme catalysis 30 processive reaction 31 proofreading 31 structure 23 Electromagnetic radiation polarized 63 wavelength designations Electron spin resonance (electron paramagnetic resonance) 122–125 hyperfine splitting 122 lactose permease 140–142 nitroxide radical 123 Flavins 44 Fluorescence 35, 47 anisotropy 79, 80, 82 energy transfer 52–54 lifetime 53 microscopy 58 polarization 78–79 Förster energy transfer 52–54 Fourier transform 11, 13 infrared 92 ion cyclotron resonance 148 NMR 107 x ray 22 Franck-Condon principle 89 Free induction decay (FID) 107 FUSE binding protein 132 Glycosylation 160 Guanine spectrum 41 Gyromagnetic ratio 104–105 Harmonic oscillator 90–91 energy levels 90 frequency 90 Helicase 51 Heme 29–42 Hemoglobin resonance Raman 99 spectrum 12, 42 structure 30 HIV genome 80–81 Hydrogen exchange 157–159 aldolase 159 EX1 158 EX2 158 Hyperchromicity 41 Hypochromicity 41 HSQC (heteronuclear single quantum correlation) 118 ATP synthase c subunit 120 c-AMP response element 131 Infrared spectroscopy 92 Fourier transform 92 group frequencies 96 protein secondary structure 95–97 Interference constructive 10, 20 destructive 10, 20 Ionization electron 150 electrospray (ESI) 153 fast ion bombardment (FAB) 151 MALDI 151 Isomorphous replacement 23 Jablonski diagram 48 Lactose permease 139–142 nitroxide spin label 140 structure 141 l repressor 133–135 Larmor frequency 106 MAD (multiple wavelength anomalous dispersion) 22 Magnetic resonance imaging 121 Mass analysis 145–147 resolution 146 INDEX Mass spectra hydrogen exchange 159 lipids 160 monoclonal antibody 152 myoglobin 153 oligonucleotides 160 peptides 154–156 proteins 154–156 Mass spectrometers double focusing magnetic sector 147 ion cyclotron resonance 148 quadrupole 147 tandem 149 time-of-flight 148 Myoglobin mass spectrum 153 resonance Raman 98 structure 29 Multidimensional NMR 115–120 COSY 116 HSQC 118 NOESY 116 protein structure 120 two dimensional 115 Nitroxide radical 123–124 lactose permease 140 NOESY (nuclear Overhauser effect spectroscopy) 116 c-AMP response element 129–132 DNA-distamycin A 119 NOE walk 136 RNA structure 136–138 Normal mode vibration 90 Nuclear energy levels 104 Nuclear g factor 104 Nuclear magnetic resonance 103–120 chemical shift 108–110 Larmor frequency 106 multidimensional 115–120 relaxation times 112–115 spectrometers 106 spin-spin splitting 110–112 Nuclear magneton 104 Nucleic acid CD and ORD 69–70 component structures 167 structure 25 ultraviolet spectra 40 171 Optical rotation 64 molar rotation 66 specific rotation 66 Optical rotary dispersion 65 proteins 67–69 nucleic acids 69–70 P-protein 74–77 ligand binding 75–77 Particle in a box energy 7, probability wave function Peptide sequencing, mass spectrometry 155 peptide ladders 155–156 Photoelectric effect 3–4 Phase angle 11 Phenylalanine spectrum 40 Phosphorescence 47, 51 Phosphorylation 160 Planck relationship Polarization 78–79 Polarized light circularly 64 linear 63 planar 63 Prosthetic groups 41 heme 42 flavins 44 Protein(s) a helix 28 b hairpins 28 b sheet 28 CD 67–69 DNA interactions 71, 132 infrared spectra 95 NMR structure 120 ORD 67–69 primary structure 28 quaternary structure 30 random coil 68 secondary structure 28 tertiary structure 29 ultraviolet spectra 38 Protein folding 74–77 aldolase 159 hydrogen exchange 157–159 l repressor dynamics 133–135 172 INDEX P protein 74–77 Proteomics 155 Proto-oncogenes 132 Quantum mechanics Quantum yield 49 Raman scattering 92–95 anti-Stokes line 93 dihydrofolate reductase 100–101 hemoglobin 99 myoglobin 98 Rayleigh line 93 resonance Raman 94, 98–100 Stokes line 93 Relaxation times chemical 114, 134–135 spin-lattice 113 spin-spin 113 Ribonuclease A difference spectra 44–46 nucleotide binding 44–46 structure 29 Ribonuclease P 74–77 Ribozyme 27 hammerhead structure 27 ribonuclease P 74 RNA folding 136–138 tRNA structure 26 Rotational correlation time 80 Scattering Bragg 19 Raman 92–95 x rays 18 Selection rules 36 Serum albumin spectrum 39 Singlet electronic state 48, 51 Spin quantum number electrons 103 nuclei 103–104 Spin-spin splitting 110–112 Karplus equation 112 Structure factor 22 Thymine spectrum 41 Triplet electronic state 48, 51 Tryptophan fluorescence 49–50 ultraviolet spectrum 40 Tyrosine spectrum 40 Uncertainty principle Uracil spectrum 41 Wave equation 5, Wave number, definition 92 Waves 2, 9, 20 X rays absorption spectroscopy 46 diffraction 21 scattering 18 Zinc fingers 77 cyclic-AMP response element 129–132 DNA interaction 77–78 DNA transcription 129 COLOR PLATES Figure 2-5 Stick-figure representation of the B form of the DNA double helix The planes of the hydrogen-bonded bases can be seen, as well as the twisting of the chains to form a double helix The pale yellow spheres are the phosphorous atoms of the sugar phosphate backbone on the outside of the helix The bases are color coded, and the two grooves in the structure are labeled Copyright by Professor Jane Richardson Reprinted with permission [From G G Hammes, Thermodynamics and Kinetics for the Biological Sciences, Wiley-Interscience, New York, 2000.] Spectroscopy for the Biological Sciences, by Gordon G Hammes Copyright © 2005 John Wiley & Sons, Inc COLOR PLATES Figure 2-6 Structure of yeast phenylalanine t-RNA with the backbone in cyan (Protein Data Bank entry 1EHZ) The L-shaped structure can be seen and the hydrogen bonding of the side chains to form helical structures The anticodon loop is at the bottom of the long arm of the L Copyright by Professor David C Richardson Reprinted with permission Kinemage graphics, then rendered in Raster3D COLOR PLATES Figure 2-8 The a-helix found in many proteins The yellow arrow follows the righthanded spiral of one helical turn The hydrogen bonds between backbone peptide bonds are brown lines, the oxygens are red, and the nitrogens blue The hydrogen bonds are formed between the ith carbonyl and i + NH in the peptide backbone Copyright by Professor Jane Richardson Reprinted with permission [From G G Hammes, Thermodynamics and Kinetics for the Biological Sciences, Wiley-Interscience, New York, 2000.] COLOR PLATES Figure 2-9 The b-pleated sheets found in many proteins Both parallel and antiparallel strand-strand interactions are shown, as indicated by the yellow arrows Again, the hydrogen bonds between backbone peptide bonds are shown as brown lines between the oxygens and nitrogens Copyright by Professor Jane Richardson Reprinted with permission [From G G Hammes, Thermodynamics and Kinetics for the Biological Sciences, Wiley-Interscience, New York, 2000.] Figure 2-10 Ribbon structure of sperm whale myoglobin (Protein Data Bank entry 1A6M) The large amount of a-helical structure is apparent The space filling structure in cyan is the heme Copyright by Professor David C Richardson Reprinted with permission Kinemage graphics, then rendered in Raster3D Figure 2-11 Schematic structure of ribonuclease A with uridine vanadate (magenta) bound to the active site (Protein Data Bank entry 1RUV) Both a-helical and b-sheet structure can be seen, as well as structure that does not have a regular array of hydrogen bonds The disulfide linkages are in yellow, and the green residues are histidines 12 and 119, which are essential for catalytic activity Copyright by Professor David C Richardson Reprinted with permission Kinemage graphics, then rendered in Raster3D COLOR PLATES Figure 2-12 Schematic representation of the R (bottom, pink) and T (top, blue) forms of the hemoglobin a2b2 tetramer The hemes where the oxygen binds can be seen in the structure The yellow side chains form salt bonds in the T structure that are broken tin the R structure One pair of the a-b subunits also rotates with respect to the other by about 15° in the interconversion of R and T forms Copyright by Professor Jane Richardson Reprinted with permission [From G G Hammes, Thermodynamics and Kinetics for the Biological Sciences, Wiley-Interscience, New York, 2000.] COLOR PLATES Figure 2-13 Structure of a catalytic fragment of DNA polymerase from Bacillus stearothermophilus The “thumb” and “fingers” of the structure are labeled, and the template and primers can be seen as stick structures Copyright by Professor L S Beese, Duke University Reproduced with permission Figure 7-2 NMR structure of the TAZ2 (CH3) domain of CBP A superposition of 20 structures are shown The orange balls are Zn2+, the yellow ball-and-stick representations are cysteines, and the purple and blue structures are imidazoles from histidines The four a-helices can be easily discerned PDB entry 1F81 Copyright by Professor David C Richardson Reprinted with permission Kinemage graphics, then rendered in Raster3D COLOR PLATES Figure 7-3 Structure of a KH3-DNA complex determined with NMR The DNA (10 mer) is shown in magenta as a ball-and-stick model, and the protein backbone is in cyan PDB entry 1J5K Copyright by Professor David C Richardson Reprinted with permission Kinemage graphics, then rendered in Raster3D [...]... shown is the relationship between the sine wave and the wavelength of the light The useful wavelength of radiation for spectroscopy extends from x-rays, l ~ 1–100 nm, to microwaves, l ~ 105–106 nm For biology, the most useful radiation for spectroscopy is in the ultraviolet and visible region of the spectrum The entire useful spectrum is shown in Figure 1-2, along with the common names for the various... the potential energy, and En is the energy associated with the wave function Yn Since the potential walls are infinitely high, the solution to this equation outside of the box is easy—there is no chance the particle is outside the box so the wave function must be 0 Inside the box, U = 0, and Eq 1-5 can be easily solved The solution is y n = A sinbx (1-6) where A and b are constants At the ends of the. .. spaced for a macroscopic particle Similarly, the smaller the box, the more widely spaced the energy levels For a tennis ball being hit on a tennis court, the ball is sufficiently heavy and the court (box) sufficiently big so that the energy levels would be a continuum for all practical purposes The uncertainty in the momentum and position of the ball cannot be blamed on quantum mechanics in this case! The. .. middle of the box, that is, between L/4 and 3L/4 for the lowest energy state is 3L 4 Ú L 4 3L 4 y 12 dx = (2 L) Ú sin 2 (p x L)dx L 4 Evaluation of this integral gives a probability of 0.82 .The probability of finding the particle within the middle part of the box is independent of L, the size of the box, but does depend on the value of the quantum number, n For the second energy level, n = 2, the probability... levels for n = 1, 2, and 3 b What is the wavelength of light emitted when the electron moves from the energy level with n = 2 to the energy level with n = 1? PROBLEMS 15 c What is the probability of finding the electron between 12.5 and 37.5 Å for n = 1 1.6 Sketch the graph of I versus t for sine wave radiation that obeys the relationship I = I0sin (wt + d) for d = 0, p/4, p/2, and p Plot the sum of the. .. The scattering intensity depends on the scattering effectiveness of the individual atoms and the phase of the wave from each scattering source The structure factor, F, for each plane can be defined as the sum of the structure factors for individual atoms, fi, times a phase factor, ai, for each atom: F = Â f ja i (2-2) i The intensity of scattered radiation is proportional to the absolute value of the. .. plates) The L-shaped molecule is typical of tRNAs The hydrogen bonding network is shown in this structure The acceptor stem (upper right-hand corner) is where the amino acid is linked to form the aminoacyl-tRNA The amino acid is transferred to the growing protein chain during protein synthesis The anticodon, which specifies the amino acid to be added to a protein during synthesis, is at the end of the. .. cases at the lowest cost With this brief introduction to the underlying theoretical principles of spectroscopy, we are ready to proceed with consideration of specific types of spectroscopy and their application to biological systems 13 Amplitude REFERENCES Time Figure 1-9 The upper part of the figure shows sine waves of four different frequencies, and the lower part of the figure is the sum of the sine... straightforward from the standpoint of the mathematics, but the problem is that the actual structure factors contain both amplitudes and phases Only the amplitude, or to be more precise, its square, can be directly derived from the measured intensity of the diffracted beam The phase factor must be determined before a structure can be calculated Two methods are commonly used to solve the phase problem for. .. length L The particle is confined to the box by setting the potential energy equal to 0 inside the box and to • outside of the box the box, the potential energy of the system is 0, whereas outside of the box, the potential energy is infinite This is depicted in Figure 1-4 The Schrödinger equation in one dimension is - h2 d 2y n + U = En yn 8 mp 2 dx 2 (1-5) where Yn is the wave function, x is the position .. .SPECTROSCOPY FOR THE BIOLOGICAL SCIENCES Spectroscopy for the Biological Sciences, by Gordon G Hammes Copyright © 2005 John Wiley & Sons, Inc SPECTROSCOPY FOR THE BIOLOGICAL SCIENCES... shown in the figure, the electric component of the wave is perpendicular to the magnetic component Also shown is the relationship between the sine wave and the wavelength of the light The useful... middle part of the box is independent of L, the size of the box, but does depend on the value of the quantum number, n For the second energy level, n = 2, the probability is 0.50 The probability