Chapter 6Random Walks and theStructure ofMacromolecules“The journey of a thousand miles begins with a single step.” - Chinese proverb(RP: check)6.1 What is a Structure: PDB or RG?We have already belabored the idea that the study of structure is often a pre-requisite to being able to tackle the more interesting question of the functionaldynamics of a particular macromolecule or macromolecular assembly. Indeed,this notion of the relation between structure and function has been elevated tothe status of the true central dogma of molecular biology, namely, “sequencedetermines structure determines function” (Petsko and Ringe, 2004), whereinour mission is to uncover the relation between sequence and consequence. Wehave also already noted that the idea of structure is hierarchical and subtle, withthe relevant detail that is needed to uncover function often living at totally dis-parate spatial scales. For example, in thinking about phosphorylation-inducedconformational changes, an atom-b y-atom description is required, whereas inthinking about cell division, a much coarser description of DNA is likely moreuseful. The key message of the present chapter is that there is much to begained in some circumstances by abandoning the deterministic, pdb mentalitydescribedinearlierchaptersforastatistical description in which we attemptonly to characterize certain av erage properties of the structure. Indeed, we willargue that this type of thinking permits immediate and potent contact with arange of experiments.To elaborate, we continue with an aside o n the concept of structure itselfwith the aim of probing the subtlety that attends this idea. The use of the word221222CHAPTER 6. RANDOM WALKS AND THE STR U CTURE OF MACR OMOLECULESstructure is in the end an attempt to lay order on top of the geometric dispositionof huge collections of atoms. When discussing the structure of a bridge such asthe Golden Gate Bridge we certainly make no mention of the underlying atomiccoordinates. Rather, in the search for a reduced description, we make referenceto the dimensions of overall objects which are treated as though they are of aperfect rigidity and hence once the dimensions, the center of mass coordinatesand the orientation are specified,wehavesaidallthereistosay. Attheoppositeextreme, when describing a crystal such as that of the crystals that make up themicroprocessors on which these pages are being written, once again, we bypassan atom-by-atom reckoning with the idea of exploiting the perfect monotony ofthe underlying crystal lattice. There are other cases yet! When con templatingthe chemically complex and atomistically enormous molecules that make up theliving world, the notion of structure reveals other facets. In particular, we findthat a statistical description of the structure becomes entirely appropriate. Thatis, rather than attempting to state the precise location of each and every atomthat makes up the macromolecule, we settle for a description of the structurewhich specifies its compactness.6.1.1 Deterministic vs Statistical Descriptions of Struc-turePDB Files Reflect a Deterministic Description of MacromolecularStructureThe notion of structure is complex and ambiguous. In the context of crystals,we can think of structure at the level of the monotonous regular packing of theatoms into the unit cells of which the crystal is built. This thinking applies evento crystals of nucleic acids, proteins or complexes such as ribosomes, viruses andRNA polymerase. Indeed, it is precisely this regularity that makes it possibleto deposit huge pdb files containing atomic-coordinates on databases such asthe Protein Databank and VIPER. In this world view, a structure is the set(r1, r2, ···rN), where riis the vector postion ri=(xi,yi,zi)oftheithatomin this N-atom molecule. However, the structural descriptions that emergefrom x-ray crystallography provide a deceptively static picture which can onlybe viewed as a starting point for thinking about the functional dynamics ofmacromolecules and their complexes in the crowded innards of a cell.Statistical Descriptions of Structure Emphasize Average Size andShape Rather Than Atomic CoordinatesAs noted above, in the context of polymeric systems, the notion of structureis subtle and brings us immediately to the question of the relative importance ofuniversality and specificity in macromolecules. In particular, there are certainthings that we might wish to say about the structure of polymeric systems thatare indifferent to the precise chemical details of these systems. For example,when a DNA molecule is ejected from a bacteriophage into a bacterial cell,all that we may really care to say about the disposition of that molecule is6.2. MACROMOLECULES AS RANDOM WALKS 223how much space it takes up and where within the cell it does so. Similarly,in describing the geometric character of a bacterial genome, it may suffice toprovide a description of structure only at the level of characterizing a blob of agiv en size and shape. Indeed, these considerations bring us immediately to theexamination of statistical measures of structure. As hinted at in the title to thissection, one such statistical measure of structure is provided by the radius ofgyration, RG, which, roughly speaking, gives a measure of the size of a polymerblob. It is the business of the remainder of the chapter to hang quantitativemeat on this idea and to show the calculable consequences of adopting such adescription of structure.6.2 Macromolecules as Random WalksRecall that one of our primary missions in this book is to illustrate the ways inwhich good models rear their heads under widely different circumstances. Thepresent chapter introduces one such model, namely, the random walk,whichis so important that it will see double duty. In the present setting, we arguethat the random walk and its cousins are just the models we need to providea statistical characterization of the structure of macromolecules. On the otherhand, in chap. ?? the random walk will once again rear it’s head, though in thatcontext with relevance to the diffusion of various molecules within cells.Random Walk Models of Macromolecules View Them as Rigid Seg-ments Connected by HingesAs already seen in chap. 5, one way to characterize the geometric dispo-sition of a macromolecule such as DNA is through the deterministic functionr(s). This function tells us the position (r)ofthatpartofthepolymerwhichis a distance s along its cont our. An
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