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# MATH BIOPHYSICS MAP 5485

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This 35 page Class Notes was uploaded by Tremaine Johnson on Thursday September 17, 2015. The Class Notes belongs to MAP 5485 at Florida State University taught by John Quine in Fall. Since its upload, it has received 57 views. For similar materials see /class/205470/map-5485-florida-state-university in Applied Mathematics at Florida State University.

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Date Created: 09/17/15

MAP 54857 Maple Tutorial 2 Use Maple to solve the following problems Write your answers below the ques tions 1 Let A be the matrix 2 3 1 3 2 2 L521 4 12 to two decimal places Let P be the matrix composed of eigenvectors of A and M be diagonal matrix with the corresponding eigenvalues Use Maple to compute that AP PM approximately ii er7 show that AP 7 PM 0 This will work for any matrix A with distinct eigenvalues How do you explain t is 2 A rotation matrix is a matrix A such that 1 AA I and detA 1 Let 1319 1329 and 1339 be rotations an angle of 9 around the my and 2 axes respectively and let A R11i21222 1R1si4i Verify using Maple that the matrix A above satis es 1 and so it is a rotation matrix Remember that prime indicates transpose7 whose Maple command is Transposei Find the eigenvalues of A and the absolute values of the eigenvalues of A to 2 decimal places 3 Let A be the rotation matrix in problem 2 Notice that there is one real eigen vector and two complex ones The real eigenvector corresponds to the eigenvalue 1 and is the axis of the rotation Let wl and 112 be the two complex eigenvectors Notice that wl wig Find 1 7w1w2 andv iwliua 1 2 2 2239 and verify that v1 and v2 are vectors with real coordinates These are just the real and imaginary parts of the complex eigenvectors By using the dot product DotProduct7 show that v1 and v2 are perpendicular 1 0 Find and simplify the partial derivative E 13y 2 81 x7y5 6122 Use the Maple command simplifyi 0 Find x t if t7 xt 0052t1nt earctanxt Use the Maple command map 5 The Frenet frame for the helix ht cos tsin tt is 7 sint 7 cost I sint F 12 cost isint 7 cost b 0 b Where b Plot the spacecurve 0 C ht 5F cosat sinat fort0tot27randwitha5i Alsoplotfora4anda3i 4 GEOMETRY OF SPACE LATTICES AND SPACE GROUPS We have discussed the orthogonal group C71 and the special orthogonal group SOn for n 2 3 Which are groups of rigid mo tions transformations leaving the origin xed The group C71 preserves angles and dis tances The group 8001 also preserves ori entation We can write these in terms of square matrices called orthogonal rnatrices There are also groups of all rigid motions with or Without xed points This is the group of euclidean rnotions EM and the group of orientation preserving euclidean mo tions We describe these groups for n 2 and n 3 2 41 The Euclidean Group The sim plest example of a rigid motion that does not leave any point fixed is a translation A translation is a motion of the form X gt X X0 Translations preserve orientation Assume that all transformations in are of the form X gt AX X0 Where A is an orthogonal matrix Think of this as an orthogonal transformation leav ing the origin xed followed by a transla tion All transformations in En are of the same form Where A is orientation pre serving 3 411 Euclidean transformations with xed points other than the origin We de ned the orthogonal transformations as the rigid motions leaving the origin xed Our Choice of the origin is arbitrary so we can say that given a point the orthogonal transforma tions are the rigid rnotions leaving it xed We can easily go from the origin to any other xed point X0 If A leaves the origin xed then X gt X0 X0 leaves X0 xed If A is a rotation about a line axis L through the origin then this is a rotation about a line through X0 parallel to L 4 412 Euclidean group in dimension 2 Ori entation preserving euclidean transformations in dimension 2 are easy to describe The translations have no fixed points but every other motion does Every element of E2 which is not a translation is rotation about some xed point To show this use complex numbers to de scribe the motion Consider the transforma tion 2 gt ewz 20 which is a rotation an angle of theta about the origin followed by translation by 20 It is easy to solve for a fixed point 20 f 1 ew39 Now the transformation can be written as Heinz aw which is a rotation of an angle 8 about the fixed point f 5 Another type of transformation in dimen sion 2 not sense preserving is a glide re ection A glide reflection is re ection in a line followed by translation in the direction of the line for example 2 gt 2 0 Where we is a real number is a glide re ection in the real axis 6 413 Euclidean group in dimension 3 ln dimension 3 there is a type of transforma tion which is rotation about an axis followed by translation in a direction parallel to the axis This is called a screw translation and the axis of the rotation the xed line is called the screw axis An example is 1 0 0 a 0 0 cos6 sin6 y 0 0 sine cos6 z 0 The vector moi is the translation vector and the 1 axis is the screw axis It can be shown in analogy to the situ ation in dimension 2 that every orientation preserving motion which is not a translation has a xed line Every element of E3 which is not a translation is a screw translation about some axis The axis of the screw translation with A in 803 can be found using a method called Chasles formula 7 Another type of transformation in dimen sion 3 is a glide re ection re ection in a plane followed by translation by a vector in the same plane An example is 100 x 0 010 yyo 001 z 20 which is a glide re ection in the 32 plane followed by translation in the direction yo j 20 k The translation vector is in the my plane See pictures of glide re ections 8 42 Lattices The simplest type of rigid motion is a translation A lattice is a group of translations Each translation can be thought of as a vector A lattice in dimension 2 or 3 can be thought of as combinations of inte ger scalar rnultiples a set 2 or 3 resp vectors called basis vectors We can view a lattice as a set of points in space the endpoints of all the translation vectors from the origin The parallelograrn formed by a set of basis vectors is called a unit cell Maple derno 9 421 2D lattices Lattices are classi ed into different types depending on their symmetry properties A symmetry of the lattice is a euclidean transformation leaving the lattice xed The symmetry of the lattice is often seen by looking at the unit cell Also it helps to sketch all the xed lines and xed points of symmetries of the lattice Note that the xed line of a re ection is called a mirror axis Here is the classi cation in dimension 2 LATrICEs FOR PERIODIC PLANE PATTERNS 7 Parallelogram Rectangular 7 Square Hexagonal Equilateral Triangles CHART 1 The lattice units outlined are those chosen by crystallographers for purposes of classification The centered cell containing 2 units is shown in dotted outline on the rhombic lattice FIGURE 1 Types of 2 dimensional lattices The 5 types of lattices are c parallelogram o rectangular o rhornbic a square 0 hexagonal Both rectangular and rhornbic lattices have two perpendicular re ection axes of symme try What is the difference See answer 11 422 3D lattices The classi cation in di mension 3 is called the Bmmm39s classi ca tion There are 14 types and as in dimen sion 2 the classi cation has to do With the symmetry properties of the lattice The 14 types of 3D lattices are c tricliriic P o monocliriic P and C o orthorhombic P C l and F o tetragorial P and l 0 cubic P l and F o trigorial P o hexagonal P We will not try to describe these here but these are described in books on crystallog raphy We will look at some examples later 12 43 Packing density One concept often used in the study of proteins is packing den sity It is often remarked that the pack ing density of proteins is high The concept of packing density is important not just to help Visualize molecule in space but the spe cial lattices involved are often used in lattice models of proteins It is also a highly con troversial mathematical topic because of the Kepler conjecture 13 431 Packing density in 2D To help un derstand the concept of packing density in 2D we look at two lattices circle packings Which of the following two 2D lattice pack ings is the densest the square lattice circle packing or the hexagonal lattice circle pack ing 88888 FIGURE 2 Square Lattice Circle Packing FIGURE 3 Hexagonal Lattice Circle Packing 37mm Ml te Segment N 62 small while 41 FIGURE 4 Segments of hex and square packings The density is de ned to be the ratio of space covered by circles to all of space To compute the density it is easiest to look at a segment of the packing which can be used as a tile to form the whole packing 15 432 Square and hexagonal lattice circle packing densities To compute the density of the hexagonal and the square 2D lattice packings it is easiest to look at a segment of the packing which can be used as a tile to nd the whole packing g 4 For the square packing we have a square tile of area 1 and the portion covered by the circle of radius 12 has area 7r4 so the density of square packing 7r4 78 For the hexagonal packing we have an equi lateral triangular tile of area Zl and the portion covered by the circle segments of ra dius 12 has area 7r8 so the 7r density of hex packing 2 N 91 The above shows that the hexagonal pack ing has greater density the the square pack ing 16 433 Packing density in 3D To help un derstand the concept of packing density in 3D look at sphere packings with sphere cen ters on the cubic lattice and the face centered cubic lattice The cubic sphere packing has centers of spheres at all points 17 k Where i j and k are integers The centers are the integer lattice The radius of each sphere is 12 gure 5 FIGURE 5 The cubic lattice packing The face centered cubic fcc packing has centers at each point on the lattice gener ated by the three vectors 011 101 and 110 Figure 6 The fcc lattice is the set of points in the integer lattice Whose coordinates sum to an even integer Points 17 in the fcc lattice are a distance from the neighboring point so we take spheres Whose radius is half this distance Here is a Maple demo of the two packings FIGURE 6 The face centered cubic lattice packing 18 Now we can compute the densities of these two packings How do you compute the den sity of a lattice sphere packing You would like to compute the ratio of volume covered by spheres to the volume of space Since both of these are in nity you have to cut space up into nite tiles and compute the density of the spheres in one of the tiles 19 434 Density of cubic lattice For the cu bic integer lattice 23 1x2x3 l E Zz 12 3 a sphere packing can be made by putting a sphere of radius 12 and volume 7r 6 cen tered at every lattice point Z denotes the set of integers Space can be divided into cubes of side 1 and volume 1 each contain ing one sphere So the density is 7r 6 w 52 Recall that the volume of a sphere of radius 7 is 47 3 3 435 Density of face centered cubic lat tiee The face centered cubic lattice is the set of all lattice points in Z3 such that x1x2x3 is an even integer A sphere packing can be made from this lattice by putting spheres of radius 2 and volume 27T3 cen tered at every lattice point Now space can be divided into cubes of side 2 and volume 8 each containing 1 sphere and 12 quarter spheres see gure 7 Thus the density is ws e 74 Hansen sphere packing FIGURE 7 Cross section of the face centered cubic lattice packing 21 SO the fee lattice packing is denser that the cubic lattice packing In fact it is the densest sphere packing in three dimensions 22 44 Space groups A lattice is the origin and its image under a group of translations Lattices are simple patterns They remain unchanged if you translate them in certain directions When you take more complex euclidean motions and apply them to a patterns in stead of just a point you get more complex patterns These are a good introduction to the study of crystallography 23 441 2D space groups Again it is best to start with 2D The study of in nitely re peating 2D patterns of symmetry has shown that there are only 17 possible types These patterns are usually associated with wallpa per Here is a list of 2D space groups with pictures from the paper of Schattschneider Note that each pattern is constructed by applying a few euclidean motions the gen erators repeatedly to just one part of the pattern To construct your own patterns have fun with java kali or download the Kali program to your computer It will con struct a wallpaper pattern from your own small pattern The patterns in 2 dimensions are called wallpaper groups Note that each group is associated with a lattice symmetry type from the list 1 p1 192 pm 199 cm pmm mm 1999 6mm 0 2222 XX X gtk2222 22 22X 222 parall parall rect rect rhomb rect rect rect rhomb p4 p4m p49 p3 p3m1 p31m p6 p6m 442 gtk442 42 333 gtk333 33 632 gtk632 square square square hexag hexag hexag hexag hexag The 14 2D space groups given in crystallographic and Conway notation Also indicated is the lattice type associated With each group 442 3D space groups Patterns correspond ing to space groups in 3D are a lot harder to picture There are 230 of them each as sociated with a 3D lattice symmetry type Information on them can be found in crys tallography tables Here is an online table of crystallographic groups 25 443 One example of a 3D space group The space group P212121 is commonly seen in protein crystals for example in a protein kinase pdb number 1AQ1 This space group is related to the plane group pmg It is generated by three screw rota tions 1 mag7 2 gt 12y7 Z mag7 2 gt 537 mag7 2 gt 127 y7 Here is a gure showing molecule crystal lized in that space group See gure 8 FIGURE 8 A molecule crystallized in space group P2l2121i Axes of screw rotations are shown with dotted lines and arrows Answer to question in caption to gure 1 The xed lines for symmetries of the rhorn bic lattice all go through the lattice points Also for the rectangular lattice there are four centers of order two symmetries in a unit cell For the rhornbic lattice there are two 4i GEOMETRY OF SPACE LATTICES AND SPACE GROUPS We have discussed the orthogonal group On and the special orthogonal group SO for n 2 3 which are groups of rigid motions transformations leaving the origin xed The group On preserves angles and distances The group SOn also preserves orientationi We can write these in terms of square matrices called orthogonal matricesi There are also groups of all rigid motions with or without xed points This is the group of euclidean motions En and the group of orientation preserving euclidean motions E4r We describe these groups for n 2 and n 3 4i The Euclidean Group The simplest example of a rigid motion that does not leave any point xed is a translation A translation is a motion of the form XHXX0i Translations preserve orientationi Assume that all transformations in are of the form X A AX X0 where A is an orthogonal matrixi Think of this as an orthogonal transformation leaving the origin xed followed by a translation All transformations in E are of the same form where A is orientation preservingi 4ilili Euclidean transformations with xed points other than the origin We de ned the orthogonal transformations as the rigid motions leaving the origin xedi Our choice of the origin is arbitrary so we can say that given a point the orthogonal transformations are the rigid motions leaving it xed We can easily go from the origin to any other xed point Xoi If A leaves the origin xed then XgtAX7X0X0 leaves X0 xed If A is a rotation about a line axis L through the origin then this is a rotation about a line through X0 parallel to L 412 Euclidean group in dimension 2 Orientation preserving euclidean transfor mations in dimension 2 are easy to describe The translations have no xed points but every other motion doesi Every element of E2 which is not a translation is rotation about some xed point To show this use complex numbers to describe the motion Consider the trans formation 2 A eiez 20 which is a rotation an angle of theta about the origin followed by translation by 20 It is easy to solve for a xed point f Now the transformation can be written as 2H i92ff which is a rotation of an angle 9 about the xed point Another type of transformation in dimension 2 not sense preserving is a glide re ectioni A glide reflection is re ection in a line followed by translation in the 1 direction of the line for example 2 A 2 10 where 10 is a real number is a glide re ection in the real axis 413 Euclidean group in dimension 5 ln dimension 3 there is a type of trans formation which is rotation about an axis followed by translation in a direction parallel to the axis This is called a screw translation and the axis of the rotation the xed line is called the screw axis An example is 1 0 0 z 10 0 cos 9 7 sin 9 y 0 0 sin 9 cos 9 2 0 The vector zoi is the translation vector and the z axis is the screw axisi It can be shown in analogy to the situation in dimension 2 that every orientation preserving motion which is not a translation has a xed line Every element of E3 which is not a translation is a screw translation about some axis The axis of the screw translation X A AX X0 with A in SO3 can be found using a method called Chasles formulai Another type of transformation in dimension 3 is a glide re ection re ection in a plane followed by translation by a vector in the same plane An example is 71 0 0 z 0 0 1 0 y yo 7 0 0 1 2 20 which is a glide re ection in the yz plane followed by translation in the direction yoj 20 k The translation vector is in the my plane See pictures of glide re ectionsi 42 Lattices The simplest type of rigid motion is a translation A lattice is a group of translations Each translation can be thought of as a vector A lattice in dimension 2 or 3 can be thought of as combinations of integer scalar multiples a set 2 or 3 respi vectors called basis vectors We can view a lattice as a set of points in space the endpoints of all the translation vectors from the origin The parallelogram formed by a set of basis vectors is called a unit cell Maple demoi 42L 2D lattices Lattices are classi ed into different types depending on their symmetry properties A symmetry of the lattice is a euclidean transformation leaving the lattice xed The symmetry of the lattice is often seen by looking at the unit celli Also it helps to sketch all the xed lines and xed points of symmetries of the lattice Note that the xed line of a re ection is called a mirror axis Here is the classi cation in dimension 2 The 5 types of lattices are parallelogram rec tan gu ar rhomb ic uare hexagonal 0000 m riAuAAV LUVAALVUALVU LVvuuAvAlu LVALVVU VAAU V View avannvvvav AI Aggv w llvvlvvva O va v v 39 r quot39 v v of all isometries which map the pattern onto itself The classification of periodic patterns according to their symmetry groups is the twodimensional counterpart of the system used by crystallographers to classify crystals Hence these groups are also termed the ovadimensional crystallographic groups The symmetry group of a periodic pattern necessarily maps a lattice associated to the pattern onto itself Since centers of rotation of a pattern are mapped by translations to new centers of rotation having the same order only rotations of order 2 3 4 or 6 can occur as isometries of a periodic design This is often referred to as the crystallographic restriction If a pattern has no rotational symmetry but re ections or glide re ections are in its symmetry group then the lattice must have parallel rows of points at right angles to each other These restrictions imply that there are five distinct types of lattice which can occur as the most general lattice possible for a plane symmetry group For each lattice type there are conventionally chosen lattice units for purposes of classification Chart 1 shows the five types of lattice and for each a lattice unit LATrICEs FOR PERIODIC PLANE PATTERNS 7 Parallelogram Rectangular Rhombic Square Hexagonal Equilateral Triangles CHART 1 The lattice units outlined are those chosen by crystallographers for purposes of classification The centered cell containing 2 units is shown in dotted outline on the rhombic lattice FIGURE 1 Types of 2 dimensional lattices Both rectangular and rhombic lattices have two perpendicular re ection axes of symmetry What is the difference See answer 422 3D lattices The classi cation in dimension 3 is called the Bravais classi ca tion There are 14 types and as in dimension 2 the classi cation has to do with the symmetry properties of the lattice The 14 types of 3D lattices are triclinic P monoclinic P and C orthorhombic P C I and F tetragonal P and I cubic P I and F trigonal P hexagonal P We will not try to describe these here but these are described in books on crystallography We will look at some examples later 43 Packing density One concept often used in the study of proteins is packing density It is often remarked that the packing density of proteins is high The concept of packing density is important not just to help visualize molecule in space but the special lattices involved are often used in lattice models of proteins It is also a highly controversial mathematical topic because of the Kepler conjecture 431 Packing density in 2D To help understand the concept of packing density in 2D we look at two lattices circle packings Which of the following two 2D lattice packings is the densest the square lattice circle packing or the hexagonal lattice circle packing FIGURE 2 Square Lattice Circle Packing FIGURE 3 Hexagonal Lattice Circle Packing The density is de ned to be the ratio of space covered by circles to all of space To compute the density it is easiest to look at a segment of the packing which can be used as a tile77 to form the whole packing 432 Square and hexagonal lattice circle packing densities To compute the density of the hexagonal and the square 2D lattice packings it is easiest to look at a segment of the packing which can be used as a tile to nd the whole packing g 4 For the square packing we have a square tile of area 1 and the portion covered by the circle of radius 12 has area 7r47 so the density of square packing 7r4 m 78 For the hexagonal packing we have an equilateral triangular tile of area zl and the portion covered by the circle segments of radius 12 has area 7r8 so the it densit of hex ackin 7 m 91 y p g 2 The above shows that the hexagonal packing has greater density the the square packing gimme 4 fine Seymen t 7 7Z magnum1 a ag Se 12712 1 FIGURE 4 Segments of hex and square packings 413131 Packing density in 3D To help understand the concept of packing density in 3D look at sphere packings with sphere centers on the cubic lattice and the facecentered cubic lattice he cubic sphere packing has centers of spheres at all points ij k where ij and k are integers The centers are the integer lattice The radius of each sphere is 121 gure 5 The face centered cubic fcc packing has centers at each point on the lattice generated by the three vectors 011 101 and 110 Figure 6 The fcc lattice is the set of points in the integer lattice whose coordinates sum to an even integeri Points in the fcc lattice are a distance from the neighboring point so we take spheres whose radius is half this distance Here is a Maple demo of the two packingsi Now we can compute the densities of these two packingsi How do you compute the density of a lattice sphere packing You would like to compute the ratio of volume covered by spheres to the volume of spacer Since both of these are in nity you have to cut space up into nite tiles and compute the density of the spheres in one of the tiles 413141 Density of cubic lattice For the cubic integer lattice Z3 zlzgzg l I 6 Zi123 FIGURE 5 The cubic lattice packing FIGURE 6 The face centered cubic lattice packing a sphere packing can be made by putting a sphere of radius 12 and volume 7r6 centered at every lattice pointi Z denotes the set of integers Space can be divided into cubes of side 1 and volume 17 each containing one spherei So the density is 7r6 m 52 Recall that the volume of a sphere of radius 7 is ANTS3 4 33 Density offace centered cubic lattice The face centered cubic lattice is the set of all lattice points in Z3 such that 11 12 13 is an even integeri A sphere packing can be made from this lattice by putting spheres of radius Z and volume 27r3xi centered at every lattice point Now space can be divided into cubes of side 2 and volume 8 each containing 1 sphere and 12 quarter spheres see gure 7 Thus the density is 7r3xi m 74 So the fcc lattice packing is denser that the cubic lattice packingi ln fact7 it is the densest sphere packing in three dimensions IVll Y Q5174 h 4 Dansan spun pacle FIGURE 7 0105s section of the face centezed cubic lattice packing 44 Space groups A lattice is the ozigin and its image undez agzoup oftxanslar tions Lattices axe simple pattems They zemain unchanged if you txans ate them in ceztain dizections When you take mole complex euclidean motions and apply them to a pattems instead of Just a point ou get mole complex pattems These axe a good intzoducr 39 to the study of czystallogxaphy 441 2D spaoe grouys Again it is best to stazt with 2D The study of in nitey zepeating 2D pattems of symmetzy has shown that theze axe only 17 possible types T ese pattems axe usually associated with Wallpapez Hexe is a list of 2D space gzoups with pictuzes fzom the papez of Schattschneidez N te t at each pattem is constxucted by applying a few euclidean motions the genezatozs zepeatedly to Just one pazt of the pattezn To constzuct you pattems have fun with Java hall 01 download the Kali pzogzam to you computez it will constxuct a Wallpapez pattem fxom you own small pattem The pattems in 2 dimensions axe called wallyapn groups Note that each gzoup is associated with a lattice symmetzy type fzom the 39st 1 p1 P2 W P57 m Wm my 2299 mm o 22 xx x 22 2 x 222 pazall pazall xect xect zhomb xect xect xect zhomb The p4 p4m p49 p3 p m1 p31m p6 psm 442 442 42 333 333 33 632 632 squats squats squats hexa a exag hexag hexag 142Dspace zou og aphic a dCon a not ion Al o ps g al indicated is the lattice type associated with each gzoup

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