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Mathematical Approaches to Molecular Structural Biology This pageintentionallyleftblank Mathematical Approaches to Molecular Structural Biology Subrata Pal AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1650,SanDiego,CA92101,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom Copyright©2023ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronic ormechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem, withoutpermissioninwritingfromthepublisher.Detailsonhowtoseekpermission,further informationaboutthePublisher’spermissionspoliciesandourarrangementswithorganizationssuch astheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions. 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ISBN:978-0-323-90397-4 ForInformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:AndreG.Wolff AcquisitionsEditor:MichelleFisher EditorialProjectManager:TracyI.Tufaga ProductionProjectManager:SwapnaSrinivasan ElsevierCoverDesigner:VickyPearson CoverImageDesigner:SharmishthaPal TypesetbyMPSLimited,Chennai,India Dedication To my parents This page intentionallyleftblank Contents Aboutthe author.......................................................................................................xi Preface...................................................................................................................xiii Acknowledgments...................................................................................................xv Table of symbols..................................................................................................xvii CHAPTER 1 Mathematical preliminaries.........................................1 1.1 Functions........................................................................................1 1.1.1 Algebraic functions.............................................................1 1.1.2 Trigonometric functions......................................................3 1.1.3 Exponentialandlogarithmic functions...............................5 1.1.4 Complex number andfunctions..........................................6 1.2 Vectors............................................................................................8 1.2.1 Concept ofvectorinphysics..............................................9 1.2.2 Vector asan ordered set ofnumbers................................11 1.2.3 Mathematicalviewpointofvector....................................13 1.3 Matrices anddeterminants...........................................................14 1.3.1 Systems oflinear equations..............................................14 1.3.2 Matrices.............................................................................20 1.3.3 Determinants.....................................................................26 1.4 Calculus........................................................................................29 1.4.1 Differentiation...................................................................29 1.4.2 Integration.........................................................................34 1.4.3 Multivariate function........................................................38 1.5 Series andlimits...........................................................................40 1.5.1 Taylor series......................................................................41 1.5.2 Fourier series.....................................................................42 Further reading............................................................................45 CHAPTER 2 Vector spaces and matrices.......................................47 2.1 Linear systems..............................................................................47 2.2 Sets and subsets............................................................................48 2.2.1 Set......................................................................................49 2.2.2 Subset................................................................................51 2.3 Vector spacesand subspaces.......................................................53 2.3.1 Vector space......................................................................53 2.3.2 Vector subspaces...............................................................55 2.3.3 Nullspace/rowspace/column space.................................57 vii viii Contents 2.4 Liner combination/linear independence.......................................60 2.5 Basis vectors.................................................................................64 2.6 Dimension and rank.....................................................................68 2.7 Inner productspace......................................................................73 2.8 Orthogonality................................................................................78 2.9 Mappingand transformation........................................................90 2.10 Changeof basis............................................................................98 Further reading..........................................................................101 CHAPTER 3 Matrix decomposition...............................................103 3.1 Eigensystemsfrom different perspectives.................................103 3.1.1 Astable distributionvector.............................................103 3.1.2 System oflinear differential equations...........................105 3.2 Eigensystem basics.....................................................................107 3.3 Singular value decomposition....................................................129 Further Reading.........................................................................136 CHAPTER 4 Vector calculus.........................................................137 4.1 Derivatives of univariatefunctions............................................137 4.2 Derivatives of multivariate functions........................................139 4.3 Gradients ofscalar-and vector-valued functions......................144 4.4 Gradients ofmatrices.................................................................149 4.5 Higher-order derivates (cid:1)Hessian.............................................150 4.6 Linearization and multivariate Taylorseries.............................153 Further Reading.........................................................................157 CHAPTER 5 Integral transform.....................................................159 5.1 Fourier transform........................................................................159 5.2 Dirac delta function....................................................................162 5.3 Convolution and deconvolution.................................................166 5.4 Discrete Fourier transform.........................................................168 5.5 Laplacetransform.......................................................................170 Further reading..........................................................................172 CHAPTER 6 Probability and statistics..........................................173 6.1 Probability—definitions andproperties.....................................173 6.1.1 Probabilityfunction........................................................173 6.1.2 Conditional probability...................................................175 6.2 Randomvariables anddistribution............................................177 6.2.1 Discreterandomvariable................................................177 Contents ix 6.2.2 Continuous random variable...........................................181 6.2.3 Transformation ofrandomvariables..............................185 6.2.4 Expectation andvariance................................................186 6.3 Multivariate distribution.............................................................189 6.3.1 Bivariate distribution......................................................189 6.3.2 Generalized multivariate distribution.............................194 6.4 Covariance andcorrelation........................................................195 6.5 Principal component analysis.....................................................202 Further reading..........................................................................210 CHAPTER 7 X-ray crystallography................................................211 7.1 X-ray scattering..........................................................................211 7.1.1 Electromagnetic waves...................................................212 7.1.2 Thomson scattering.........................................................214 7.1.3 Compton scattering.........................................................216 7.2 Scattering by anatom................................................................218 7.3 Diffraction froma crystal (cid:1)Laue equations............................221 7.3.1 Lattice andreciprocal lattice..........................................223 7.3.2 Structure factor................................................................224 7.3.3 Bragg’s law.....................................................................225 7.4 Diffraction andFourier transform..............................................226 7.5 Convolution anddiffraction.......................................................227 7.6 The electron density equation....................................................228 7.6.1 Phase problem and the Patterson function.....................229 7.6.2 Isomorphous replacement...............................................230 7.6.3 Electron density sharpening............................................230 Further reading..........................................................................232 CHAPTER 8 Cryo-electron microscopy ........................................235 8.1 Quantum physics........................................................................235 8.1.1 Wave(cid:1)particle duality....................................................235 8.1.2 Schro¨dingerequation......................................................236 8.1.3 Hamiltonian.....................................................................237 8.2 Wave optics ofelectrons—scattering........................................239 8.3 Theory ofimage formation........................................................242 8.3.1 Electrodynamics oflens system.....................................242 8.3.2 Image formation..............................................................243 8.4 Image processing by multivariate statistical analysis—principalcomponent analysis....................................245 8.4.1 Hyperspace anddata cloud.............................................245