Chapter 6Beam Theory: Architecturefor Cells and CitiesRP: opening picture - cy-toskeleton and bridge6.1 Beams are Everywhere: From Flagella tothe CytoskeletonOne-Dimensional Structural Elements Are the Basis of Much of Macro-molecular and Cellular ArchitectureWhether w e reflect on natural structures such as trees, animal skeletons,cells or the molecules that make up Crick’s “two great polymer languages”, oralternatively, on the manmade structures that range from the bridges that spanour rivers to the airplanes used to traverse oceans, the geometry of structuraldesign has always been a central human obsession. The central point of thischapter is the idea that from the standpoint of model construction for char-acterizing the dynamics of the geometric structures that surround us, there isa great power in dimensional reduction in which such structures are idealizedeither as being pseudo-one- or two-dimensional.The representation of geometric structures as networks of one-dimensionalelements is a perspective of great power and applicability. That part of me-chanics which has grown up around this approximation is known traditionallyas “beam theory”. One of the character traits that is revealed in the greatmodel builders is the ability and imagination to smell out those features of agiven phenomenon that have a true bearing on observation and to recognizethose that are irrelevant. From the standpoint of the present chapter, our aim RP: search for irrelevance isa formal issue, a requirementwill be to see applications of beam theory that at first blush appear inappropri-ate. With a little reflection and imagination, to the contrary, we will find thatbeams surround us wherever we look. The surfer on the beach is reminded ofthis in the form of the swaying palms and the wooden reinforcing stringer thatruns the length of his or her surfboard. The structural mechanics of beams is179180CHAPTER 6. BEAM THEORY: AR CHITECTURE FOR CELLS AND CITIES(A)(F)(E)(G)(B)(C)(D)Figure 6.1: Diverse examples of the way in which structures can be interpretedusing beam theory. (a) Bridge and corresponding beam theory representation,(b) small-scale cantilever used in atomic-force microscope, (c) bacterial cell withflagella, (d) cytoskeleton in a eukaryotic cell, (e) Representation of DNA as anelastic rod. (RP: idea here is something like Fig. 1-11 of ECB where there is areal photo and the cartoon representation with shading showing the beams)RP:put cilia and hair cellequally rewarding whether one studies the flying buttresses of Notre Dame orthe struts used to support the wings of the 777 that carried our tourist to Paris.The biological world has similarly not proceeded in ignorance of the gains to behad through the exploitation of beams. Indeed, as shown in fig. 6.1, there are anumber of different settings within which the notion of a one-dimensional elasticrod serves as a powerful tool. From a technological perspective, figs. 6.1(a) and(b) reveal the use of beams as structural elements in the world of very largevery small structures. More to the point, however, are the advantages to begained when we think of biological structures such as the protrusions on haircells, flagella, filaments in the cytoskeleton, and even molecules such as DNA intheir arrangements in nucleosomes, from the perspective of the elastic theory ofbeams.The argument of the present chapter is first, that it is possible to marshalevidence in fav or of the idea that beam theory merits a place on our list of funda-men t al models and solutions. Indeed, it is exactly in the sense that the analysisof beams presides over the study of structures that we make this argument. Thesecond key thrust of the chapter is to provided a few pointed examples whichserve to illustrate both the traditional applications of the notions of beam the-ory as well as those that take us relatively far from the original intentions ofGalileo, Euler and others.6.2. GEOMETRY AND ENERGETICS OF BEAM DEFORMATION 1816.2 Geometry and Energetics of Beam Deforma-tion6.2.1 Stretch, Bend and Twist, Just Like PolymersWhether we contemplate the dynamics of the flagellum of a swimming bacteriumor the wrapping of DNA around histones to form the nucleosome core particle,the basic picture is one in which the structure of interest has one dimensionthat is much larger than the others. For example, in the case of the bacterialflagellum, we are talking about a structure that is in excess of microns in lengthwith a diameter that is measured in only tens of nanometes. As illustrated infig. 2.9, the same can be said of both tobacco mosaic virus and microtubules (seesection ??) which similarly have a characteristic length of order microns withcross sectional dimensions measured in nanometers. Because of this geometricasymmetry, it is possible when discussing the elasticity of such objects to carryout major simplifications.Beam Deformations Result in Stretching, Bending and TwistingWe begin with a qualitative discussion of the geometric c haracter of thethree key independent modes of deformation to which we may subject a beam.In particular, the deformation of beams can be described in terms of three basicmodes, namely, extension, bending and torsion as shown in fig. 6.2. Extensionaldeformations have already been discussed in section ?? and correspond tosimple elongation from length L to length L + ∆L. For the purposes of thepresent chapter, it is the two other key modes of deformation that are argued tobe of greater significance for thinking about the mechanics of biological systems.In particular, a second key mode of deformation corresponds to the bending ofa beam, with the simplest eventuality being a deformation which results in anarc of a circle. The last key mode of deformation which is a constant feature ofdeformations involving DNA corresponds to twisting the beam about it’s longaxis.A Bent Beam Can Be Analyzed as a Collection of Stretched BeamsThe state of deformation which will animate the majority of our discussion inthis chapter is that of bending. Bending is of interest in many of the examplesalready revealed in fig. 6.1. For example, when determining the free energyassociated with the wrapping of DNA around the histone octamer as shownin fig. 6.1(RP), we will invoke the energetics of beam bending. We begin byexamining the nature of the geometric assumptions that are made about thestate of deformation of a bent beam. Indeed, the first thing we will say aboutthe
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