Cartoons as Models in Molecular Biology · 2015-07-29 · Cartoons as Models in Molecular Biology...

Chapter 2

Cartoons as Models inMolecular Biology

“A picture is worth a thousand words.” — Unknown

RP/JK to Readers: The present draft should be viewed as having the basicstructure of the ultimate final version of this chapter. On the other hand, as itcurrently stands, there is much that is incomplete: i) the figures when in theirfinal form will ALL be drawn in the same style and will resemble fig. 2.25,ii) many sections are incomplete - in particular, one of our main ambitions isto try and complement each section and each cartoon with some quantitativeinsight. As yet these insights are present, at best, in skeletal form. We areinterested in hearing your views on the level of presentation, what is assumedof the reader, whether the concept is clear, your views on the logic - basicallyeverything. Thanks for your time.

2.1 Cartoons and ModelsRP: each cartoon needs anassociated estimate and anassociated set of equations.This chapter and the onethat follows are, after review-ing all of ECB, the two pri-mary sources of much of bi-ological phenomenology. Weneed to do a beautiful jobin this chapter of organiz-ing that phenomenology ac-cording to spatial and tem-poral hierarchies and in thenext chapter on the basis ofparticular models that havecatapulted forward our un-derstanding of biological sys-tems. Also, make sure notto forget that the chapterhas some parallel aims -1) prove that cartoons serveas models,2) introduce bi-ological phenomenology, es-pecially stuff we will try

Biological Cartoons Select Those Features of the Problem Thoughtto Be Essential

We have argued that the fine art of model building ultimately reflects atasteful separation of that which is essential for understanding a given phenom-enon from that which is not. One of the key reflections which struck both ofus while trying to learn something about the beautiful subject that is biologywas the profound and subtle way in which biologists have learned to confrontthe enormous complexity of their problems. In particular, we were struck withthe fact that the visual representations in biology, whether drawn from the im-pressive cartoons that are a mainstay of biological pedagogy or those found onthe pages of the most recent research reports, exhibit precisely those features ofmodel building that have been exploited with success in the physics setting.

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42 CHAPTER 2. CARTOONS AS MODELS IN MOLECULAR BIOLOGY

2-D view of mitochondrion

baffle model crista junction model

intermembranespace

outer membraneinner membrane

crista junctions

matrixcristae

0.1–0.5 mm

1–2 mm

Figure 2.1: Several generations of structural cartoons illustrating the propertiesof mitochondria. JK wonders if we should have history thread like the Pollardand Earnshaw picture on membrane cartoons. show images with the cartoon.

As argued in the previous chapter, a model must contain the essential fea-tures of a system while not containing so much detail as to make it intractable.As an example of the role of cartoons in conveying the essential features of bi-ological structure, fig. 2.1 shows cartoons meant to convey several generationsof structural understanding concerning mitochondria. Though cryo electronmicroscopy has offered some refinements of the original picture, the essentialconceptual elements are present even in earlier cartoons, namely, a) the mi-tochondria are closed, membrane-bound organelles, b) the inner membrane isdecorated with a series of protrusions which segregate different regions of themitochondria and might also serve as the seat of reduced dimensionality diffu-sion.

In this chapter, our intention is to put together a case that biological modelbuilding is practiced without abandon in the form of cartoon making. From itsearliest inception as an experimental science, biology has been built around therole of visual descriptions. Whether in the form of classification of species orthe study of human anatomy or the representation of microscope observations(see fig. 2.2) or even the structure of macromolecules, biology has been un-apologetically visual. Even in this modern era of fluorescent labels for differentbiological molecules and high speed cameras to record them, often the results ofsuch images are framed in the form of cartoons which guide the viewer in theirinterepretation of what from the image is important and what is not.RP: cartoons are in fact

very subtle because they haveto have enough of a re-semblance to reality so thateveryone knows which real-ity is being described - presi-dents always look enough likepresidents to be recognizable.In biology cartoons are afantastic separation of wheatand chaff.

2.2. DIAGRAMMATIC REPRESENTATIONS IN THE PHYSICAL SCIENCES43

Figure 2.2: Sketches of Leeuwenhoek and Hooke of the results of their micro-scope observations.

2.2 Diagrammatic Representations in the Phys-ical Sciences

Physics Also Has a Rich Tradition of Using Cartoons to RepresentPhenomena

Thus far, we have made it sound as though biology is somehow unique in itsexploitation of visual representations in the context of model building. On theother hand, the physical sciences are similarly replete with historic examplesof the role of visual representations. As will be described briefly below, one ofthe most profound chapters in the history of physics concerned the discoveryof the connection between electricity, magnetism and light. The two figuresthat tower over the field of electromagnetism perhaps more than any others areMichael Faraday, a bookbinder’s apprentice with probably less mathematicalequipment at his disposal than a college biology student has today, and JamesClerk Maxwell, one of the most successful theoretical physicists of all time.

2.2.1 Faraday and Lines of Force

Though a discussion of Michael Faraday and the emergence of the concept of theelectromagnetic field in the context of biological cartoons may seem like makinga round-the-world flight to get from San Diego to San Francisco (i.e. a stretch),the relation between the work of Faraday and Maxwell serves as an exampleof precisely the sort of integration of visual and quantitative representationsthat is now taking place in biology and which is the backdrop of the presentbook. In addition, electric fields and potentials are one of the cornerstones ofbiological phenomenology. Indeed, as will be shown in section ??, the notion of


Figure 2.3: Cartoon from the Experimental Researches in Electricity of Faradaywhich helped usher in the “field” concept.

action potentials is central to our understanding of a wide variety of biologicalprocesses. For the purposes of the present discussion, our objective is moremodest with the aim being to recount a historical episode in the physical sciencesin which the role of cartoons could be said to rival the importance they haveassumed in biology today.

The key point of the present discussion is the observation that at the time ofFaraday’s pathbreaking discoveries leading to the elucidation of the concept ofthe electromagnetic field, Faraday’s views of electromagnetic phenomena werelargely visual, in many ways reminiscent of the biological situation presently.Not only was Faraday’s language of description visual, but more palpably, heappealed to cartoons to capture the essence of his model of the electromagneticfield. As shown in fig. 2.3, the idea of “lines of force” that permeate space werevisualized through tricks such as the use of iron filings and were represented inthe pages of Faraday’s papers through cartoons like that shown in the figure.Interestingly, these very same ideas now find themselves occupying a place ofprominence in current biological thinking since biological membranes serve ascapacitive elements in exactly the same way the devices of Faraday did.

2.2.2 Enter Maxwell: The Mathematization of Faraday

The experimental insights of Faraday concerning the relation between electric-ity and magnetism were described verbally and characterized conceptually bycartoons. It is into Faraday’s world of lines of force permeating space thatJames Clerk Maxwell entered, equipped as he was with the mathematical toolsof the theoretical physicist. By his own admission, Maxwell attached enormousimportance to Faraday’s experimental successes: “before I began the study ofelectricity I resolved to read no mathematics on the subject till I had first read

2.3. ONS, SOMES AND EMERGENT PHENOMENA 45

through Faraday’s Experimental Researches in Electricity.” As a result of hisreading of Faraday, Maxwell perceived that the key conceptual elements of themodel of the electromagnetic field were already in place and that what wasrequired was to translate the cartoons and verbal descriptions of Faraday intothe more familiar mathematical language of post Newtonian physics. Indeed,Maxwell himself characterized his successes thus: “ I had translated what Iconsidered to be Faraday’s ideas into a mathematical form”.The central argument of the present chapter is that there are many cartoons

in molecular biology and biochemistry which have served as models of a widevariety of phenomena and that reflect detailed understanding of these systems.On the other hand, to make these models quantitatively predictive they needto be cast in mathematical form, this in keeping with our battle cry that quan-titative data demands quantitative models. As a result, one way of viewingthe chapters that follow is as an attempt to show how the cartoons of molecularbiology and biochemistry have been and can be “translated into a mathematicalform”.

2.2.3 Feynman and the Cartoonization of Field Theory

As an amusing aside, we briefly recount an instance in which the thesis ofthe present section was run in reverse. In particular, in the heady world ofquantum field theory, the mathematical contortions demanded of theoreticalphysicists are legendary. Into that world entered the young Richard Feynmanwho found it convenient to cartoonize various terms in his equations so as topermit bookkeeping to take place visually rather than through the concrete,but longwinded, utilization of explicit mathematical formulae. These cartoonshave become a central part of the pedagogy and daily practice of many branchesof “many-body physics” and are even institutionalized through the presence ofroutines within software such as Mathematica for cartoonizing the equations offield theory.

2.3 Ons, Somes and Emergent PhenomenaRP: can we estimate the typeof wavelengths we are expect-ing?

Another interesting parallel between model building in the physical sciences andthe life sciences is seen in the treatment of collective behavior of assemblies offundamental objects. Indeed, the notion of emergent phenomena has been usedto characterize examples in both settings. For example, when attempting tosimulate the atmosphere, modelers do so without reference to the underlyingmolecular makeup of the atmosphere. Similarly, when contemplating the vibra-tions of a solid, it is convenient to abandon reference to the underlying atomiccoordinates and to refer instead to a new set of degrees of freedom known asphonons. RP: JK intrigued by the size

of ons vs somes - ons have 1024 degrees of freedom, somesdon’t. Somes not as preciseas ons. The whole is greaterthan the sum of the parts.JK argues that our reason fordescribing the hierarchies isnot just that it is organiza-tionally interesting, but dic-tates the way we build mod-els by separating scales andby using subgrid stuff as in-put into models at next scale

As noted above, replacement of some subscale degrees of freedom with ahigher level effective description is referred to as emergence, and one of theintriguing features of emergent phenomena is that the structure or patterns


Figure 2.4: An early example of a Feynman diagram.

exhibited by the system may only be discerned when viewed from the lowerresolution perspective. One palpable example of this is viewed at 10,000 metersin an airplane. What one often notes on a cloudy day is the presence of well-organized structures (known as convective rolls) which amount to a series ofstripes in the cloud formation. Clearly, when viewed at the scale of individualmolecules, each engaged in its own wildly fluctuating trajectory, there is noorder to be seen. On the other hand, when viewed at longer wavelengths, thepresence of such patterns is incontestable.

In the physics world, the presence of such low resolution (or long wave-length) structures has been welcomed with a series of theoretical constructs andan associated nomenclature. For the present purposes, suffice it to say that theexistence of phonons, magnons, excitons, rotons and a boat load of other ons isboth conceptually and mathematically substantive. In particular, the existenceof each on signals the existence of some long wavelength and collective struc-ture involving many microscopic degrees of freedom, whether they be atomicdisplacements or magnetic moments.

The Label Some Refers to Macromolecular Complexes

But what has all this to do with biology? The key thrust of the presentsection is that just as the physicist has a special code language to announcethe presence of some collective (and emergent) property of the system, so toodo biologists. In particular, the use of the label some signifies the collectiveaction of some group of molecules in the name of an overall function. In manycases in the remainder of the book we will find that the use of the appelationsome signifies the presence of a macromolecular assembly built up of multipleindividual molecules. For example, the replisome refers to the collection ofproteins which mediate the making of copies of DNA. This collection can includeproteins for untwisting DNA, for sewing up the Okazaki fragments which result

2.3. ONS, SOMES AND EMERGENT PHENOMENA 47

from the requirement of 5’ to 3’ polymerization of new DNA and others as well.Similarly, the packing of DNA in eucaryotic cells is built up of a macromolecularcomplex (a some) between DNA and a protein octamer known as the histoneoctamer. This entire assembly is known as the nucleosome and will be discussedin greater depth in section ??. Once the DNA message has been translatedinto messenger RNA it is yet another macromolecular complex, namely theribosome, that performs the task of translating the nucleic acid message into acorresponding amino acid sequence. Though the use of somes in the biologicalsetting does not have the same rigorous underpinning as does the notion of an onin the physics setting, we still contend that ribosomes, replisomes, nucleosomes,proteosomes and other assemblages in the biological setting reflect the same sortof collective action.RP: version 2 of the same thingTheoretical modeling in the physical sciences is predicated on the observation

that physical laws operating on a particular spatial or temporal scale are in largepart independent of the goings on at other scales. For example, when attemptingto simulate the atmosphere, modelers do so without reference to its underlyingmolecular makeup. Similarly, when contemplating the vibrations of a solid, itis convenient to abandon reference to the underlying atomic coordinates and torefer instead to a new set of degrees of freedom known as phonons. One couldsay that this relative independence of phenomena at different scales is whatmakes physical laws expressed in the language of mathematics even possible.Model building in the life sciences takes pages from the same play book, as

the notion of emergent phenomena has made its mark here as well. Witness thedifferent representations developed to describe structures such as macromole-cules, and organelles, and processes such as DNA transcription and cell division.We therefore believe that useful theoretical models in biology, that is ones thatcan lead to deeper insight offered by general principles or to new hypothesiswhich can be checked experimentally, should be molded to the scale they seekto describe. Most of the modeling efforts described in this book arise from thisbasic belief. RP: JK intrigued by the size

of ons vs somes - ons have 1024 degrees of freedom, somesdon’t. Somes not as preciseas ons. The whole is greaterthan the sum of the parts

As noted above, replacement of some subscale degrees of freedom with ahigher level effective description is referred to as emergence, and one of theintriguing features of emergent phenomena is that the structure or patternsexhibited by the system may only be discerned when viewed from the lowerresolution perspective. One palpable example of this is viewed at 10,000 metersin an airplane. What one often notes on a cloudy day is the presence of well-organized structures (known as convective rolls) which amount to a series ofstripes in the cloud formation. Clearly, when viewed at the scale of individualmolecules, each engaged in its own wildly fluctuating trajectory, there is noorder to be seen. On the other hand, when viewed at longer wavelengths, thepresence of such patterns is incontestable.In the physics world, the presence of such low resolution (or long wave-

length) structures has been welcomed with a series of theoretical constructs andan associated nomenclature. For the present purposes, suffice it to say that theexistence of phonons, magnons, excitons, rotons and a boat load of other ons is


both conceptually and mathematically substantive. In particular, the existenceof each on signals the existence of some long wavelength and collective struc-ture involving many microscopic degrees of freedom, whether they be atomicdisplacements or magnetic moments.The Label Some Refers to Macromolecular Complexes

But what has all this to do with biology? The key thrust of the presentsection is that just as the physicist has a special code language to announcethe presence of some collective (and emergent) property of the system, so toodo biologists. In particular, the use of the label some signifies the collectiveaction of some group of molecules in the name of an overall function. In manycases in the remainder of the book we will find that the use of the appellationsome signifies the presence of a macromolecular assembly built up of multipleindividual molecules. For example, the replisome refers to the collection ofproteins which mediate the making of copies of DNA. This collection can includeproteins for untwisting DNA, for sewing up the Okazaki fragments which resultfrom the requirement of 5’ to 3’ polymerization of new DNA and others as well.Similarly, the packing of DNA in eucaryotic cells is built up of a macromolecularcomplex (a some) between DNA and a protein octamer known as the histoneoctamer. This entire assembly is known as the nucleosome and will be discussedin greater depth in section ??. Once the DNA message has been translatedinto messenger RNA it is yet another macromolecular complex, namely theribosome, that performs the task of translating the nucleic acid message into acorresponding amino acid sequence. Though the use of somes in the biologicalsetting does not have the same rigorous underpinning as does the notion of an onin the physics setting, we still contend that ribosomes, replisomes, nucleosomes,proteosomes and other assemblages in the biological setting reflect the same sortof collective action.In order to set the stage for biological model building in the next section we

describe the structures and processes that are central to the life of a cell, scaleby scale. In this task we are guided by the idea that often very different modelsare needed to describe phenomena occurring on different scales.

2.4 Cartoons and the Representation of Biolog-ical Structures: Structure at Many Scales

2.4.1 The Hierarchy of Spatial Scales

Discernible Biological Structures Exist Over a Huge Range of Scales

The spatial scales associated with biological structures runs from the nanome-ter scale that characterizes the individual molecular actors making up the livingworld, all the way to the scale of the patterns of forestation contemplated in ecol-ogy and environmental science. The attempt to convey the essential elementsof structures at these and the scales between is often carried out at the level

2.4. CARTOONS ANDTHE REPRESENTATIONOF BIOLOGICAL STRUCTURES: STRUCTUREATMANY SC

Figure 2.5: Cartoon showing the wide range of important length scales of inter-est in biological systems. RP: things to include: organism (drosophila), organ(drosophila eye) (collection of cells), eucaryotic cell - red blood cell (collectionof organelles), cell (E. coli), virus (lambda), macromolecular assembly (ATPsynthase), single molecule (DNA), water molecule. should we try to show thateach scale is built from stuff in the next level down in the hierarchy. Also, dothe volumes on the figure.

of cartoons. Before embarking on a scale-by-scale description of the hierarchyof spatial scales of interest in biology, we first juxtapose these scales to providea broad perspective of the range of phenomena that elicit geometrical repre-sentation. Classified roughly, we discern the following layers of structure: ionsand small molecules, macromolecules, macromolecular assemblies, organelles,cells, tissues, organisms and communities. We argue that each of these levelsin the structural hierarchy admits of a coarse grained description which can becaptured in cartoon form.

There are several famed books (see Boeke, 1957; Morrison and Morrison,1982 and Kornberg, 1989) which have served to drive home the different struc-tures observed at different scales, in each case building their case by observingstructures using successive powers of ten either to zoom in or out from somereference scale. We imitate the concept used by Kornberg (1989) in fig. 2.5 withthe intention of giving a concrete representation of the meaning of lengths suchas nanometer, micron and millimeter. The strategy of the present section is tofirst examine fig. 2.5 in broad brush stroke form to convince the reader that thisspatial hierarchy is one of the many interesting ways of classifying biologicalphenomenology. Having acknowledged this hierarchy, we will then delve moredeeply into each of the scales shown in the figure on a case-by-case basis. RP: what should we add to

this declarative statement -number of atoms implicatedin thinking about a givenstructure. Range of scalesover which processes that arebiological can be discerned

Broadly speaking, what fig. 2.5 shows is a loose powers of ten representationof different biological structures. We begin with an entire organism, namely,the fruit fly Drosophila melanogaster, with characteristic dimension of several


millimeters. Increasing the level of our spatial resolution, the next structurediscerned is individual organs such as the eye, in this case that of Drosophilaonce again and characterized by dimensions of roughly 100 µm. Continuingour downward descent through yet higher magnification, we now encounter in-dividual eucaryotic cells where the characteristic scale is of order 10µm. Inparticular, because of its importance to our later consideration (see chap. ??)of their equilibrium shapes the figure shows a red blood cell. The next scaleto appear in our spatial descent is that of individual bacterial cells, representedhere by our standard ruler, the bacterium Escherichia coli, with a typical scaleof something like a micron. Yet another factor of ten takes us to the scale of theindividual viruses that attack cells such as E. coli, in this instance bacteriophageT4, with a capsid roughly 100nm across and a tail on the order of 100nm inlength. With another factor of ten increase in spatial resolution, our journeyreveals the presence of individual macromolecular assemblies such as the mole-cular machine (ATP synthase) responsible for ATP synthesis. Single molecules(such as DNA) are the province of yet another factor of ten increase in spatialresolution, and our journey finishes with a final factor of ten descent resultingin the presence of the water molecules which make up the watery environmentfavored by living cells. Again, the point of this exercise has been to presentone of the many ways of organizing biological phenomena, namely, along linesof spatial scale.RP: did this demand more

estimates? One of the key observations the reader is invited to carry away from our firstforay into the biological structure hierarchy is that structures at one scale canoften be thought of as being constructed from building blocks which exist at thenext smaller scale. For example, we note that the eye of Drosophila describedabove can be considered from the perspective of the various cells that make itup. That is, both the numbers and types of cells that are assembled to form thefly’s eye are known and characterized. Similarly, macromolecular assembliessuch as the ATP synthase (or the ribosome, or even viruses) can in turn bethought of in terms of the individual macromolecular building blocks (proteins,RNA, DNA) of which they are comprised. In coming sections we will evaluateeach of the scales introduced above on a case by case basis with the ambition ofdescribing the important biological phenomena that take place at each of thesescales.The Bacterium E. coli Will Serve as Our Standard Ruler

Since our discussion of powers of ten is made with reference to some partic-ular structure, we choose as our standard ruler the bacterial cell, Escherichiacoli. In particular, fig. 2.6 shows a generic view of such a cell and is meant,more than anything, to convey the dimensions of such cells since we will repeat-edly return to them as the basis for describing dimensions of other objects, tocharacterize the volumes of various biological structures and to illustrate themeaning of quantities such as concentrations (see section ??). To put the E.coli scale in perspective, we note that it would take roughly fifty such cells linedup end to end in order to measure out the width of a hair. On the other hand,we would need to divide the cell into roughly five hundred slices of equal width


Figure 2.6: E. coli as our standard ruler. RP: show microscopy image and acartoon with lengths shown.

in order to measure out the diameter of a DNA molecule.

With our standard ruler now in hand, we are prepared to carry out a tele-scopic description of some of the many biological structures that are found atdifferent scales. Fig. 2.5 shows more than five orders of magnitude worth of spa-tial scales, all referenced to our standard ruler of E. coli. The intention of thisfigure is to capture the key classes of structural objects it will be the businessof the remainder of the book to consider: individual molecules, macromolecules(such as DNA, lipids and proteins), macromolecular assemblies (such as theribosome, the proteosome, the nucleosome, etc.), organelles (such as the mito-chondria, the endoplasmic reticulum and the nucleus), cells, tissues and even,organisms. The ambition of the remainder of the section on structure is to fleshout the structural and geometric significance of the range of structures shownin fig. 2.5.

2.4.2 Cells: A Rogue’s Gallery

As noted above, the idea of examining successive powers of ten, starting fromsome reference structure is at once powerful and enlightening. In the booksof Boeke (1957) and Morrison and Morrison (1982), the reference structure ofinterest was a single human being and served as a distinctly human point ofdeparture for examining the cosmos, both large and small. For the purposes ofthe present book, we dethrone the single human as our point of reference with asingle cell on the grounds that it is the cell which is the fundamental unit of theliving world. In keeping with our intention of using E. coli as our standard ruler,we begin with an examination of some of the key features of these bacteria. Asshown in fig. 2.7, E. coli has a shape something like a spherocylinder, alreadyseen to have a length of roughly 1µm and with flagellae which are something


like five times the length of the cell itself (RP: check). The flagellae are utilizedfor powering cellular motility and will arise both in chap. 5 and chap. ?? withreference to their elastic and hydrodynamics properties, respectively.RP: JK wants to have each

cell type referenced with re-gard to type of dynam-ics which arises in thinkingabout this cell. For exam-ple, E. coli motility leads tolow Re dynamics, motilitywithin E. coli is diffusive.But, for a nerve cell, dynam-ics within can’t be diffusive -too big. More general state-ment, he wants each chapterto introduce physical model-ing. E.coli has swimmingand diffusion. REd bloodcells - hemoglobin and bind-ing and also shapes, mus-cle and nerve cells - mole-cular motors and biologicalelectricity, yeast as the ba-sis of central dogma and theorigins of biochemistry - seeKornberg. Also, JK suggeststhat looking through a mi-croscope yields different in-teresting phenomena - differ-ent microscopies and singlemolecule stuff has made thismore exciting time. RP seekmicroscopy images for eachcell. De Rosier pictures ofmotor in E. coli.

Cells Come in a Wide Variety of Shapes and Sizes and With a HugeRange of Functions

Despite the temptation to think of E. coli as a generic representative of thecellular realm, in fact, cells are characterized by a diversity that rivals that whichwe are accustomed to in observing the animal world. An attempt to begin tocome to terms with this diversity is made in fig. 2.7 which is intended to conveyan impression of the range of both sizes and shapes of some of the key cellularactors in the drama of life. For example, fig. 2.7(b) shows a schematic of a yeastcell. As will be emphasized repeatedly, models and model systems form the corephilosophical backdrop for the present book. Just as E. coli serves as the modelprocaryotic system, the yeast cell Saccharomyces cerevisiae serves as the modelsingle-celled eucaryotic organism. As will be shown in chap. 3, yeast has hada rich and glorious history, serving arguably as the model system giving rise tobiochemistry in its modern form. In comparison with many other eucaryotic celltypes, yeast is rather nondescript, with a smooth and regular morphology and acharacteristic size of order 5 µm. In addition to characterizing the yeast cell interms of a length scale, it is also of interest to have an impression of its volume,especially when measured in E. coli volume units, ΩE. coli. In particular, if werecall that ΩE. coli ≈ 0.9µm

3 and think of yeast as a sphere of diameter 5µm,then we have the relation Ωyeast ≈ 600ΩE. coli, that is, roughly 600 E. coli cellswould fit inside of a generic yeast cell.A second class of cells which will assume center stage in chap. 7 are red blood

cells, an example of which is shown in fig. 2.7(c). As will be seen in chap. 7, redblood cells have played an important role in the physical modeling of equilibriumshapes based upon an elastic treatment of the membrane free energy. Red bloodcells come in a variety of different shapes depending upon external conditionssuch as the temperature and the pH and salt conditions of the solution. As seenin the figure, the canonical shape of such cells is a biconcave disc with a diameterof something like 6-8µm and a thickness of roughly 1.5µm. A second modeling

RP: check these numberschallenge that has been posed by the function of red blood cells is related to theirrole as oxygen carriers. In particular, red blood cells are filled with hemoglobinmolecules (RP: quantify). As will be discussed in more detail in chap. ??, thestudy of the binding of oxygen to hemoglobin serves as an important historicalepisode with consequences ranging from biochemistry to statistical mechanicsand is the canonical example of binding cooperativity, itself a modeling themethat will be repeated throughout the text.From a structural perspective, rod cells (fig. 2.7(d)) and nerve cells (fig. 2.7(e))

reveal a great deal more complexity than the examples highlighted above, a factthat will have functional consequences and modeling implications as will be seenin chap. ??. Both of these cell types merit inclusion on our who’s who of celltypes (which is admittedly both severely truncated and idiosyncratic) becauseof the richness of their structures and the relation of these structures to the


Figure 2.7: Representations of several different types of cells. (a) E. coli cell,the canonical bacterial cell, (b) Saccharomyces cerevisiae, yeast cell (c) redblood cell (d) rod cell, - specialized function, (e) eucaryotic nerve cell - outlierstructurally, (f) plant cell. RP; these figures do not show the cellular interior.These choices should be defended - they are not exhaustive, some are chosento illustrate diversity, others because they are key model systems. Red bloodcells because they are simple. Make sure to have the cells have below them alegend saying the volume of the cell, the surface area, the number of occupantmolecules


function of these cells. As will be highlighted in the next chapter, nerve cellshave served as the basis for the entire edifice of biological electricity, actionpotentials and the theory of Hodgkin and Huxley with particular significancebeing assigned to the squid giant axon. We note that one of the most intriguingdynamical questions concerning biological transport is revealed in contemplatingenormous cells such as neurons. In particular, as will be examined more care-fully in chap. ??, the time scale required for diffusion to transport moleculesfrom one extremity of a neuron to the other is of order tdiffusion ≈ x2/D, wherex is the dimension of the cell and D is the diffusion constant. For a protein some5nm in diameter the diffusion constant in water is 100µ2/s; this estimate can beobtained from the Einstein-Stokes equation which gives the diffusion constantof a sphere of radius R moving through a fluid of viscosity η at temperature T ,as D = kBT/6πηR. Therefore the diffusion time for the squid giant axon whichhas a length of the order of 10cm in length is tdiffusion ≈ 108s! For the presentpurposes, the key conclusion to take away from such an estimate is the impos-sibly long time scales associated with diffusion over such distances. Nature’ssolution to this conundrum is to exploit active transport mechanisms in whichATP is consumed in order for motor molecules to carry out directed motion.RP; what about actual mi-

croscopy images? In addition to these cells, fig. 2.7(e) shows a plant cell, again with the ambi-tion of conveying something of an idea of the type of structural diversity acrosscell types. The main objective of both fig. 2.7 and the associated discussion inRP; each one that we have

should have a little descrip-tion and foreshadow whereit will make an appearancelater. Quantitative - vol-ume occupied, linear dimen-sions, energy budget, mem-brane area (external and in-ternal), surface to volume ra-tio, time scales

the present section has been to remind the reader that there is an enormousrange of cell types as revealed both in their sizes and shapes as well as in theirdistinct functions.

RP: JK asks: Why are cellsso large? Why are cellsso small? Discussion inLehninger - pg.21. Can weuse energy density or powerdensity for cells and macro-molecules and add up forcells to get total power. Arecells big? Are cells small?- number of ribosomes in E.coli is big, but the fluctu-ations are big too. diffu-sion time tells us which cellsare big (neurons) and whichare small (E. coli) - fluxhas to scale linearly with Rif we keep drho by dt fixed- the mass and energy re-quirements for a given unitvolume are probably fixed,but this means that as cellgets bigger then flux willhave to increase. For cellscompute the dimensionlessSeifer/Boal parameter

From the perspective of providing quantitative estimates one way to come toterms with these different cell types is to ask are such cells big or are they smallwhen viewed through the prism of the physical processes they undergo? Think-ing about the packing of various cellular parts provides one perspective. Forexample ribosomes, which are responsible for protein synthesis, can be thoughtof as roughly spherical macromolecular complexes roughly 20nm in diameter.There are approximately 15,000 ribosomes in E. coli which is a sphero-cylinder2µm in height and 0.8µm in diameter; this amounts to a volume of roughly 1µm3

per cell. If we take a uniform distribution of ribosomes throughout the cellularinterior then the mean separation between ribosomes is 40nm, comparable totheir size, once again attesting to the crowded nature of the cellular environ-ment. Indeed, the tight packing of ribosomes is evident in fig. ??. Furthermorewe now understand that the cell can not be much smaller as this would lead toinsufficient volume to accommodate for all the ribosomes necessary for proteinproduction.

RP: left hanging on big vs

2.4.3 The Cellular Interior: Organelles and Other Super-structures

The Cellular Interior Has a Variety of Conserved Internal StructuresCalled Organelles


As we descend from the scale of the cell itself, a host of new structures knownas organelles come into relief as shown in fig. 2.8. In particular, these organellesserve as the specialized apparatus of cell function, serving in capacities rangingfrom energy generation (mitochondria and chloroplasts) to protein synthesisand modification (endoplasmic reticulum and Golgi apparatus). Our intentionin the remainder of this section is to provide a quick guided tour of some of thekey organelles, acknowledging from the outset the superficial nature of our tourand presenting a corresponding invitation to the reader to delve more deeplyinto these structures in traditional books such as Alberts et al..The most generic single observation that one might make about organelles

is that they are compartmentalized structures which are usually separated fromthe remainder of the cell by bilayer membranes. Nowhere is this more evidentthat in the case of the nucleus, itself one of the most striking features of eu-caryotic cells. The nucleus is a membrane-bound region, within which is foundthe genetic material (in highly compacted form, a point we take up again insection 2.4.4). A characteristic dimension to bear in mind when thinking of thenucleus is 10µm, which translates into a volume of ≈ 5 × 1014A3 if we thinkof the nucleus as a sphere. One theme that will arise repeatedly (see sections?? and ??) is that of DNA packing in different organisms. One interestingcontext in which to address this question is that of eucaryotic DNA packing,where as mentioned above, the genetic material (all ≈ 3 × 109 base pairs of it inthe case of human cells which translates into roughly a meter long molecule) iscontained in a nucleus with linear dimensions which are a factor of 10−5 timessmaller than the DNA molecule itself. RP: RP: nuclear size, nu-

clear import and exportAs shown in fig. 2.8, the nucleus is structurally tied to the endoplasmic retic-ulum in a continuous fashion. That is, the membranes which bound the nucleusform exvaginations to form a second membrane bound region which is the seatof much important biosynthesis and is known as the ER lumen. One of themost compelling features of the endoplasmic reticulum for our present purposesis the enormity of its surface area. In particular, if we consider both the smooth RP: see the table on pg. 661

of MBOC4 for relative areasand rough endoplasmic reticulum, it can be argued that between 40-50 % ofthe lipid bilayer surface area in cells is tied up in these organelles. As hintedat above, from a functional perspective, the ER garners much attention in itsrole as the seat of lipid biosynthesis in addition to its role as the source of muchprotein biosynthesis as well. Indeed, the rough ER membrane has its character-istic rough morphology precisely because of the presence of membrane anchoredribosomes on these membranes which carry out protein synthesis. Another of RP: make estimate of num-

ber of ribsomes on membraneand areal density. Use to es-timate total number of mem-brane bound ribosomes

the key modeling challenges we will face in thinking about the functional char-acteristics of both the nucleus and the endoplasmic reticulum is tied to the factthat in both of these organelles there is a steady traffic of molecules being im-ported and exported across their membranes, demanding a consideration of theprocess of translocation, a subject we will broach again in chap. ??.Another beautiful and fascinating class of organelles are the mitochondria,

already featured in fig. 2.1, of interest for a variety of reasons, most especially intheir role as factories for ATP. One of the intriguing quantitative questions whicharises upon reflection on the organelles described thus far concerns the rich,


pleated structures adopted by the membranes making up these organelles. Inparticular, by what mechanisms are such membranes constructed? In chap. ??,RP: see pg. 449 of ECB for

exhaustive list and pg. 18.Quantitative - volumes, sur-face to volume ratio, molecu-lar content. Make list of theMODELING challenges eachof these organelles provides.Pleated membrane motif tiestogether organelles in a waythey usually aren’t. Nucleusis going to take us to translo-cation - translocation is ageneric theme - transport be-tween.

we will undertake an examination of the free energy associated with the highlyconvoluted membrane structures exhibited by both the endoplasmic reticulumand mitochondria. One possible insight to emerge from that discussion is thehypothesis that the ER and mitochondria are structural partners in the sensethat both organelles are examples of what happens in double membrane systemswhen there is a huge mismatch in the number of lipid molecules between thetwo layers.RP: Golgi apparatusRP: lysosomesRP: peroxisomes -see pg. 6 of Pollard and Earnshaw and pg. 449 of EBC

for a list of organellesRP: endosomesOne of the techniques that has revolutionized our understanding of structures

like those described above is the use of cryo-electron tomography. This tech-nique is one of the centerpieces of structural biology and is built around unitingelectron microscopy with sample preparation techniques which have abandonedchemical fixation techniques in favor of rapid freezing of the sample. The use oftomographic methods has made it possible to go beyond the planar sections seenin conventional electron microscopy images. The basic idea of the technique isindicated schematically in fig. 2.9, and is built around the idea of rotating thesample over a wide range of orientations and then to build up a correspondingthree-dimensional reconstruction on the basis of the entirety of these images.These techniques have already revolutionized our understanding of particularorganelles and are now being used to image entire cells. In conjunction with theimage analysis and pattern recognition software which goes hand in hand withthese techniques, profound insights have been garnered not only into the struc-tures of cells, organelles and macromolecular assemblies such as the ribosome,but also into the function of these structures.Macromolecular Assemblies Are Arranged in Superstructures

One of other features to emerge from the improved resolution and three-dimensionality of cryo-electron microscopy images is the clear presence of or-dered arrangements of macromolecular complexes such as ribosomes. For ex-ample, it has long been known that the rough endoplasmic reticulum is charac-terized by arrays of ribosomes (RP: make an estimate of how many and theirdensity and area of interface). However, recent work using cryo-electron tomog-raphy reveals that even free ribosomes are not arranged in a random fashion,and instead exist in superstructural arrangements. As shown in fig. 2.10, layersof ribosomes are juxtaposed between arrays of actin filaments, hinting at thepossibility that a variety of loosely ordered structures remain to be discerned.The Cellular Interior Is Highly Crowded With Mean Spacings Be-tween Molecules That Are Comparable to Molecular Dimensions

One recurring theme that will rear its head in many different modeling con-


Figure 2.8: Cartoons of various organelles within the cell. RP: like the ideaof having a single real image of a cell and then having cartoons of variousorganelles. Examples are 1-8, 1-11, 1-13 and 1-15 of ECB. RP: this figure willbe a cartoon of a single eucaryotic cell in the center and then with surroundingblow ups of various organelles. Nucleus and ER together. Mitochondria. Golgi.Use Animal Cell of pg.18 of ECB


cryo-electronmicroscopy(A) image reconstruction(B)

Figure 2.9: Cartoon illustrating the idea of tomography used for revolutionizingour knowledge of organelles such as mitochondria and the Golgi apparatus. RP:make sure to reference Baumeister/Steven paper that has similar concept

Figure 2.10: Image and associated cartoon illustrating the organization of as-semblies into specific arrangements. RP: Baumeister/Frangankis arrangementof ribosomes, arrangements of ribosomes in rough ER, ATP synthase in mito-chondria


Figure 2.11: Cartoon illustrating the tight packing of molecules within the cell.

texts is reflected in fig. 2.10 and has been driven home by the compelling car-toons of Goodsell, namely, the interior of the cell is highly crowded. The cartoonrepresentation of such crowding is illustrated in fig. 2.11 which has a remark-able faithfulness to the type of experimental pictures shown in fig. 2.10, thoughit came before such experiments. In coming chapters, we will see that thiscrowding effect will force us to call in question our simplest models of chemicalpotentials, the properties of water and the nature of diffusion. We have alreadymade an estimate of the typical spacing of ribosomes in bacterial cells and willsimilarly examine the mean spacing of hemoglobin in red blood cells in section??, leading to the generic conclusion that the mean spacing of proteins and theirassemblies is comparable to the dimensions of these macromolecules themselves.The cell is a very crowded place!

2.4.4 Macromolecular Assemblies: The Whole is Greaterthan the Sum of the Parts

Macromolecules Come Together to Form Assemblies (Somes)

One of the main structural levels that will capture our interest throughoutthe remainder of the book is that of macromolecular assemblies. At this point,the reader should think of yet another factor of ten magnification relative tothe previous section, and with this increase of magnification comes the realiza-tion that the cell is populated by a number of important assemblies such as areshown in fig. 2.12. For example, the genetic material in the eucaryotic nucleusis organized into chromatin fibers which themselves are built up of protein-DNAassemblies known as nucleosomes. The replication complex that copies DNA be-fore cell division is similarly a collection of a host of molecules which has beencalled the replisome. When the genetic message is exported to the cytoplasmfor translation into proteins, it is a yet another macromolecular assembly that


Nucleosome

Nuclear Pore Complex

Replisome

Ribosome Proteosome

ATPosome

Figure 2.12: Cartoons of variety of key macromolecular complexes (ribosome,replisome, proteosome, nucleosome, ATPsome, divisome - pg. 561 of Moat)

lays in wait, namely, the ribosome. The production of ATP, the energy currencyof the cell, in mitochondria is similarly mediated by a macromolecular complexknown as ATP synthase. When proteins are old and tired and have been tar-geted for degradation, they are sent to another macromolecular assembly knownas the proteosome.

Ultimately, it is not the task of the present work to be encyclopedic and asa result we are unable to give a case-by-case analysis of the many importantmacromolecular assemblies that populate cells. On the other hand, our missionis different in the present section and can be summarized succinctly throughthe observation that many of the most important processes in cellular life aremediated not by one or two molecules, but rather by complex multicomponentassemblies (loosely designated as somes) and which will require a diligent mod-eling effort, just as has the analysis of individual macromolecules. We notethat for the purposes of model building, there has been a strong reliance on theuse of cartoons which attempt to remove those structural details which are notcentral and keeping those features that figure into the verbal or mathematicaldescription of the process of interest.RP: “chromosomes”, “nu-

cleosomes” “signalosomes”- signaling complexes “ribo-somes” - protein factories“replisome” - “primosome”nuclear pore complex -opportunity to foreshadowtranslocation Go with thetheme of complexes and howthey are emergent and leadto emergent properties incell. NEed to highlight ATPpower in later chapters,self-assembly, right at theedge between the living and

li i ld E

RP: add here dynamics of assemblies - database of macromolecular motions

Helical Motifs Are Seen Repeatedly in Molecular Assemblies

A second class of macromolecular assemblies, characterized not by functionbut rather by structure is the wide variety of macromolecular complexes which


(A)

microtubule

(B)

bacterialflagellum

(C)

tobaccomosaic virus

25 nm 20 nm 20 nm

Figure 2.13: Cartoons of macromolecular complexes with helical geometries. (a)Microtubule, (b) bacterial flagellum, (c) Tobacco mosaic virus.

adopt a helical geometry. Several representative examples are shown in fig. 2.13.In particular, in fig. 2.13(a), we show the geometric structure of microtubules.As will be described in more detail later, these structures are built up of indi-vidual protein units called tubulin. RP: more details. A second example shownin fig. 2.13(b) is the bacterial flagellum of RP, already discussed in section ??.Here too, the same basic structural idea is repeated with the helical geometrybuilt up from individual protein units, in this case flagellin. The third examplegiven in the figure is that of a filamentous virus, with tobacco mosaic virus(TMV) chosen as one of the most well studied of viruses. RP: tempted to use actin

as one of the examples be-cause it will come up repeat-edly throughout book, becausein some cells it is the mostabundant protein, illustratesboth alpha helices and betasheets

The concluding statement of the present section amounts to two observa-tions. First, we note that many of the most important processes of cellular lifeare mediated by the presence of macromolecular complexes. The second part ofthis observation is the recognition that it is by virtue of the collective nature ofthese complexes that they have the function they do.

RP: Pinchosomes - dy-namin, FtsZ - try to lookat as helical assemblies in acircle

2.4.5 From PDB Files to Ribbon Diagrams

The living world is often argued to be built up of four great families of molecules,namely, nucleic acids, proteins, sugars and lipids. Indeed, each of these broadclasses will take centerstage at various times in the story to follow. On theother hand, sugars and lipids are often treated as the poor stepsisters of whatCrick has referred to as the “Two Great Polymer Languages”. In particular, asshown in fig. 2.14, both nucleic acids and proteins are built up from a limitedalphabet of units. In particular, the nucleic acids are built up from an alphabet


N

G T

AC

MRY

I

V F P

E

C

K

D

S

N

GWHQT

L

A

nucleotides amino acids

α helix β strand

codon

gene

protein

ALPHABET

WORDS

SENTENCES

NUCLEIC ACIDS PROTEINS

G

GC

G

C

CA

AT

T E

E M

T

Y

KS

IE

L

R

Y

Figure 2.14: Illustration of Crick’s “two great polymer languages”. RP: thiscartoon should show the molecules but illustrate the language aspect as well.RP: Note that the codon picture as is is screwed up - should just show threebase pairs.

of four letters (A - Adenine, G - Guanine, T - Thymine and C - Cytosine,for DNA). The protein language is constructed from an alphabet of 20 distinctamino acids. The words in the DNA language are built up from collectionsof three letters and are referred to as codons, with each such word having acorresponding letter in the protein language. One way to think of the wordsin the protein language is as collections of amino acids which give rise to thefundamental units of protein secondary structure - namely, alpha helices andbeta strands, about which more later. Finally, in the DNA language, sentencesare formed by collections of word (i.e. collections of codons) and correspond togenes, where we note that a given gene codes for a corresponding protein, forexample. The ambition of the present subsection is to examine with atomistic(or nearly) resolution, the various molecules that implement Crick’s two greatpolymer languages, as well as a number of other important molecular actors inthe living realm. In addition, we argue that the types of cartoons that havearisen in order to describe molecular structure reflect those features of thesestructures that are robust and provide a rich and compelling example of thepower and subtlety of biological cartoons.JK: interested in under-

standing what ribbon dia-grams buy you. That is, canyou learn something fromthese structures.

Macromolecular Structure Is Characterized Fundamentally By AtomicCoordinates

We live in the era of structural biology. This era was kicked off fifty yearsago nearly to the day from the time of this writing with the determinationof the atomic level structure of DNA. This discovery had been preceded bythe brilliant insights of Pauling and Corey into the nature of the secondarystructures of proteins and on the heels of these model building advances came the


determination of explicit structures from X-ray crystallography for myoglobinand hemoglobin. In the time since, the conjunction of X-ray crystallography,nuclear magnetic resonance and cryo electron microscopy have ushered in an erawhen the determination of new structures is literally a daily experience. Indeed,as will be asked of the reader in the problems at the end of the chapter, a visitto websites such as the Protein Data Bank or VIPER reveals just how manymolecular and macromolecular structures are now known. RP: make sure also to go

to Database of Macromole-cular Motions and to reallypush hard for dynamics inthis chapter.

Though the word structure can mean different things to different people(indeed, that is one of the primary messages of the present chapter), at the levelof structural biology, the determination of structure ultimately refers to a listof atomic coordinates for the various atoms making up the structure of interest.As an example, fig. 2.15 shows detailed atomic portraits of both nucleic acidsand proteins. In such descriptions, the structural characterization of the systemamounts to a set of coordinates

ri = xii+ yij+ zik, (2.1)

where, having chosen some origin of coordinates, the coordinates of the ith atomin the structure are given by (xi, yi, zi). The reader is invited to get a visceralfeel for the type of data we are describing here by visiting the Protein Data Bankand searching for their favorite protein. Having done that, we ask the reader todownload the coordinates for this protein and to examine the (x, y, z) lists thatconstitute the “structure” of the protein of interest. Though we certainly agreethat the determination of such structures has been profoundly enlightening, wesimultaneously register two complaints: i) the description of macromoleculesby a set of atomic-level coordinates provides a mistaken impression that suchmolecules are static rather than engaged in the endless thermal dance that istheir true destiny and ii) it is desirable to find a way to extract only thosefeatures of the molecule of interest that are really central to their function andto cast aside the remaining degrees of freedom as uninteresting. The latter pointhas already become a part of the equipment of molecular biology in the form ofthe ribbon diagrams we discuss presently.Macromolecules Are Often Described Using “Ribbon” Diagrams

Our description thus far has been founded upon the all-atom mentality en-couraged by the successes of structural biology. On the other hand, it is clearthat when possible, it would be desirable to construct representations which donot require reference to each and every atom. Indeed, we will argue that oneof the central challenges of modern biological model building is to find a wayto strike a balance between atomic-level specificity and the desire to reduce thenumber of parameters need to describe a given system.The most naive implementation of structural coarse graining like that de-

scribed above is an age-old part of chemistry and concerns the identification ofparticular chemical groups. The key point associated with adopting such coarse-grained structural descriptions is that we mentally impose a pseudo-rigidity withthese groups which is implicitly indifferent to the small jiggling motions that


Figure 2.15: Atomic-level structural representation of nucleic acids and proteins.(a) atomic structure of DNA in the B-form, (b) atomic structure of lysozyme.

are surely always associated with such groups. Further, we identify such groupswith a roughly context-independent chemical behavior. Fig. 2.16 shows a varietyof chemical groups that are of interest in biochemistry and molecular biology.Note that different people will make different choices for which groups will meritplacement on their chemical group hall of fame. We defend our choices on thegrounds that they will make an appearance later when discussing particularbiological processes. However, we also wish to argue that the mindset adoptedhere, namely, of portable Tinker-Toy or Lego-like units that can be swapped inand out of various contexts will make appearances at other scales as well. Inparticular, it will be seen in section ?? that thinking about gene regulationcan be built around the idea of portable units that serve to activate and repressparticular genes. All told, though such ideas are almost certainly an oversim-plification, they serve as a compelling point of departure for trying to tame thenearly overwhelming chemical complexity of many systems.

Though chemical groups of all types are surely of interest in biochemistry andmolecular biology, it is perhaps those associated with nucleic acids and proteinsthat garner the most interest. In fig. 2.17, we show the way in which individualgroups can be seen as the building blocks of DNA structures such as that shownin fig. 2.15(a). In particular, we note that the backbone of the double helixis built up of sugars (represented as pentagons) and phosphates. Similarly,the nitrogenous bases which mediate the pairing between the complementarystrands of the backbone are represented diagrammatically via hexagons andpentagons, with hydrogen bonds depicted as shown in the figure. Our mainpoint in presenting these complementary views of DNA is to note that dependingupon the type of question being asked, different structural levels of descriptionwill be appropriate. Indeed, in section ??, we will show that for the purposes


Figure 2.16: Cartoon representation of some of the most important chemicalgroups that are used in biochemistry. JK: must have a rationale for the choiceswe make. This version taken from Lehninger. RP: make sure to include thenumber of atoms and the volume and surface area (effective)

of estimating the energetics of DNA packing in bacteriophage, DNA may bethought of as an elastic rod. A second coarse-grained view of DNA will beoffered in section 6.2.2, where we will examine the spatial extent of DNA insolution from lysed bacteriophage and bacteria. This analysis will be completelyindifferent to the chemical particulars of DNA and will only exploit the ideathat these molecules are mechanically rigid over some length scale. By way ofcontrast, in section ??, we will see that neither the featureless elastic rod nor therandom walk description will suffice to characterize experiments on nucleosomeformation.Structural coarse-graining has assumed even greater importance in the con-

text of proteins and their assemblies than in the context of nucleic acids. Infig. 2.18, we reveal the protein language alphabet in the form of the various sidechains of all of the different amino acids. The physical properties of these side RP: make a table with their

volumes and other proper-ties.

chains will make an appearance in different contexts in the following chapters.For example, in chap. ??, we will construct estimates of the free energy costof embedding the hydrophobic amino acids in water. In chap. ??, we will takeup the question of what charge state to assign to side chains such as lysine andarginine which typically present positive charges when in solution at normal pH.A compelling and universal way of characterizing protein structures with-

out making reference to the entirety of atoms making up a given structure isrevealed through the existence of ribbon diagrams. Such diagrams character-ize the key elements of protein secondary structure such as alpha helices andbeta sheets through a well-defined visual mnemonic, without the necessity ofcontemplating the coordinates of each and every atom. Fig. 2.19 gives a few


Figure 2.17: Representation of nucleotides using schematic representation ofsugars, phosphates and bases. RP: figs. 6-4,6-5, 6-6 of ECB are models forwhat we might do.

Figure 2.18: Cartoon representation of all the amino acid side chains. (RP: thisis from Branden and Tooze - ours will be more three dimensional and will havethe volumes and areas of each residue as part of the figure.)


representative examples of the way in which protein structure can be capturedwith ribbon diagrams. Fig. 2.19(a) reminds the reader of the ribbon diagramrepresentation of an alpha helix, while fig. 2.19(b) makes a similar reminderwith respect to beta strands. Fig. 2.19(c) shows one of the poster children ofprotein structure prediction, lysozyme, which will makes it appearance again insection ??. Hemoglobin has a central role in the history of both biochemistry(with regard to the notion of cooperativity to be elaborated in section ??) andstructural biology (as one of the first proteins to have its atomic-level struc-ture determined) and is shown in fig. 2.19(d) in ribbon representation. Anotherbeautiful protein which has assumed gigantic proportions in a wide range ofprofoundly clever experiments is the green fluorescent protein (GFP) shown infig. 2.19(e). We round out this rogue’s gallery of proteins in fig. 2.19(f) with themechanosensitive ion channel. RP: use to give some num-

bers on DNA, proteins, lipidsand sugars - simple empir-ical formulas for their sizesvs number of monomers - doboth mass and volume andcharge

Matching Structures From Electron Microscopy and X-Ray Crystal-lography Is a Coarse Graining Operation

RP: here talk about the dock-ing of xray into cryo pictures- beautiful synergy and showscoarse graining. Grant to tryto do at whole cell levelRP: outstanding challenge -time development of such co-ordinates. Biology lives inthe time evolution of thesecoordinates. Ligand/receptorbinding, motor motion, etc.(kinetic verbs)

The question of “coarse-graining” is not only within the purview of modelbuilders. We note that structural biologists find themselves asking exactly thesame types of questions described above in light of the existence of solved struc-tures at different levels of resolution. Indeed, one of the most beautiful recentdevelopments in structural biology has been the emergence of techniques forexploiting both low- and high-resolution structural data simultaneously. Forexample, the atomic coordinates for a given macromolecular assembly might beknown in one configuration, while a low-resolution structure from cryo-electronmicroscopy might be known for a second configuration. Using docking methods,the atomic coordinates of the second configuration can be guessed by using theatomic coordinates known for the first configuration and docking them into thesecond. An example of this strategy is shown in fig. 2.20.Static Structural Snapshots Can Be Misleading: Think Dynamics

There is no doubt that the achievements of structural biology rank amongthe most compelling and impressive from the entire history of science. Never-theless, there is also something fundamentally misleading about the collectionsof coordinates to be found on the various structural databanks. In particular,the living world is dynamic, and worse yet, far from equilibrium. As a result,the structural snapshots embodied in atomic coordinates for a given structurefail to convey the fact that each and every atom is engaged in a constant thermaldance. Hence, at the very least, the coordinates are really of the form

ri(t) = xi(t)i+ yi(t)j+ zi(t)k, (2.2)

where the t reminds us that the coordinates depend upon time and what ismeasured in experiments might be best represented as hri(t)itime, where thebrackets hitime signify an average over time. One of the central theses of thisentire book is that the next great challenge for experimentalists and model-ers alike will be to restore time to its just place in thinking about biologicalstructures.


Figure 2.19: Variety of different representations of protein structure. (a) alphahelix, (b) beta strand, (c) lysozyme, (d) hemoglobin, (e) green fluorescent pro-tein (GFP), (f) mechanosensitive ion channel (MscL). RP: Make sure they areall drawn to scale and that volumes and areas are given also.


Figure 2.20: Docking of X-ray crystal structure atomic coordinates into a cryoelectron microscopy image. RP: this figure from Rossmann but is really only toshow the kind of figure we want - this is not the real one.

Biological Structures From Molecules To Cells Are Represented byInsightful Cartoons

Our discussion of structure given thus far has centered on descending in scalefrom individual cells all the way down to the individual molecules that populatethem. In our journey of successive magnifications we have argued that cartoonrepresentations of these various structures are much more subtle than casual useof the word “cartoon” suggests. Indeed, our thesis is that the construction ofsuch cartoons reflects model building either implicitly or explicitly in the sensethat such cartoons account for as much information as is needed to convey theconnection between structure, dynamics and function. We now undertake atelescoping up in scale, starting at the level of our standard ruler, namely asingle E. coli cell and working up all the way to organisms.

2.4.6 Telescoping Up in Scale

Cells Form Assemblies Through Cell-Cell LinkagesRP: cell dimers, immunolog-ical synapse, synapses, adhe-sion complexes. Note thatcell contact types and distrib-utions are part of the overallstory of development sincethey dictate the nature of tis-sues.

Thus far, our discussion has centered on the way in which molecules organizeto form structures ranging from molecular motors to macromolecular assembliessuch as the ribosome to organelles and ultimately to cells themselves. It is alsoof interest to examine the organization of cells to form higher level structures.The physics of such organization is driven in part by the formation of physicallinkages between different cells. For concreteness, we consider several distinct

RP: estimates for numberof contacts, estimate the to-tal energy contained in ad-hesion, estimate the forceof deadhesion, self-assemblyand development - epithelialsheets and contacts, C ele-gans development

classes of linkages between cells, starting with the linkages between epithelialcells to form epithelial sheets and then turning to the formation of synapses


between neurons. Our choice of these particular examples is based largely onthe fact that each of these examples will reveal itself later in some other context- for epithelia, in the context of cell adhesion and neurons in the context ofbiological electricity. In addition, these two examples reveal contacts with widelydifferent functions since in the epithelial setting contacts between epithelial cellsmediate the formation of structures with a particular mechanical role whilein the neuronal context, the formation of these cellular partnerships mediatesignaling.One Class of Multicellular Structures Is the Epithelial Sheets

RP: opportunity for effec-tive elasticity (which is ac-tively driven internally byadherens junctions) of ep-ithelial sheets. They canform interesting shapes asshown in fig. 19-25 ofECB. Estimates on integrinstrengths by looking at whatleads to blisters. Shouldthe model be one of per-colation - integrins are al-ways being broken but onlong hike, there are so manybroken that one gets macro-scopic detachment? Need toesimate the number of cellsthat are connected in some ofthese sheets

One of the most important types of multicellular structure is that formed assheets of epithelial cells as shown in fig. 2.21. One way to think of such epithelialsheets is as the multicell analog of the various lipid bilayer membranes thatenclose cells and their various organelles. In particular, such epithelial sheetshave as one of their functions (as do lipid bilayers) the segregation of differentmedia which can have highly different ionic concentrations, pH, macromolecularconcentrations and so on. One interesting question we will examine in detailin section ?? is the extent to which it is possible to write down an effectivemembrane theory for epithelial sheets themselves.Epithelial sheets form part of the structural backdrop in organs ranging

from the skin to the bladder. Functionally, such cells have roles ranging fromserving as a barrier to transport of molecules, providing an interface at whichmolecules can be absorbed into cells and as the seat of certain molecular se-cretions. The morphology of epithelial sheets is quite diverse as illustrated infig. 2.21 where it is seen that the morphology of the individual cells are dis-tinct in different epithelial sheets and further, the collective structures are quitediverse. In particular, note that the different structures can be broadly classi-fied into those structures which are a monolayer sheet (simple epithelium) andthose which are a multilayer (stratified epithelium). Within these two broadclasses of structures, it is possible to further refine the structural distinctionsby attending to the structures of the individual cells themselves. In particular,the cells making up a given epithelial sheet can be flat, pancake-like cells, de-noted as squamous epithelium. If the cells making up the epithelial sheet haveno preferred orientation, they are referred to as cuboidal, while those whichare elongated perpendicular to the extracellular support matrix are known ascolumnar epithelia. A reasonable criticism of a purely structural classificationRP: make estimates of aver-

age number of contacts (6 forhexagonal net) and averagenumber of such cells in vari-ous contexts.

like that given above is the observation that within a given structural type thereare still a range of different possible functions.The linkages between different epithelial cells come with a wide variety of

structures and functions. For a detailed accounting of this topic, the reader isurged to consult chap. 19 of Alberts et al. (1998) with particular reference totheir fig. 19-21. Though our current discussion is but a weak echo of theirs,fig. 2.22 shows in schematic form, once again in rogue’s gallery format, some ofthe key epithelial linkages. As usual, we note that the cartoons used to capturetight junctions, adherens junctions, desmosome junctions, gap junctions andhemidesmosome junctions aim to strike a compromise between simplicity (and


Figure 2.21: Cartoon illustrating the various ways in which epithelial cells canbe organized. (this figures from pg. 53 of Andrew Rogers, Cells and Tissues)


Figure 2.22: Cartoon illustrating the way in which epithelial cells assemblethrough binary interactions. RP: fig. 19-21 of ECB. Might make sure to includein figure the dimensions of linking proteins, the density of contacts, and theenergy scale for destroying the contacts (also forces). Also, prefer to do thecartoon in 3D taking advantage of the structural information that has beengained since the original figure was made.

generality) and realism. In particular, we note that as a result of a number ofdifferent structural studies, the relevant molecular actors and their associatedstructures are becoming known with increasing confidence. One outcome ofsuch studies is that the types of linkages under discussion here are more com-plex both structurally and chemically than is revealed in the various cartoons.As an aside, we also note that though our discussion of epithelial sheets cer-tainly constitutes a foray into “telescoping up in scale”, our discussion of thevarious cell-cell adhesion junctions, when viewed from the structural hierarchywhich forms the logical backdrop of the present section, would be more ap-propriately placed with our earlier discussion of macromolecular assemblies insection 2.4.4. Broadly speaking, the different junction types seen in fig. 2.22 canbe classified functionally according to whether or not they serve to i) create adiffusive barrier between different regions (tight junctions), ii) form mechanicallinkages providing structural integrity to collections of cells (desmosome junc-tions) and iii) mediate communication between adjacent cells (gap junctions).

RP: give some numbers ondensity of contacts, dime-sions of proteins that medi-ate contacts and also, wheredo we put adherens junc-tions?

Nerve Cells Form Complex, Multicellular Complexes

A second class of cell-cell contacts of enormous importance are the connec-tions which take place between nerve cells (neurons) and other cells (includingother neurons). We remind the reader that, roughly speaking, neurons are con-structed such that they have many input terminals known as dendrites and asingle output terminal, the axon. The class of contacts of present interest con-

2.5. CARTOONS ANDTHE REPRESENTATIONOF BIOLOGICAL PROCESSES: STOPWATCHES ATMANY S

Figure 2.23: Schematic illustration of the nature of a synapse. (from Science,Cohen-Cory, 298, 770 (2002)); RP: this figure also needs a larger scale descrip-tion of complexes formed by nerve cells

cern the way in which the output of one cell can be connected to the input ofthe next. Such a contact is represented schematically in fig. 2.23 and is knownas a synapse.Tissues Are Collections of Cells and Extracellular Matrix

We have seen that cells can interact to form complexes. One of the richestexamples of collections of cells is tissues.RP: organs

Cells Differentiate During Development Leading to Entire Organisms

The highest level in the structural hierarchy to be entertained in the presentdiscussion is that associated with individual organisms. The emergence of or-ganisms is largely the province of developmental biology.Higher Level Structures Exist as Colonies of Organisms

2.5 Cartoons and the Representation of Biolog-ical Processes: Stopwatches at Many Scales

RP: structural biology hasunfinished business in theform of dynamics. We con-tinue to have spatial scaleissues even when discussingprocesses. Reference Feyn-man QED book for stop-watches to do phase

2.5.1 The Hierarchy of Temporal Scales

As shown in section ??, discernible biological structures come in a broad varietyof sizes. Indeed, the goal of the entirety of the previous section was to highlightthe existence of these structures and the various visual representations that have


Figure 2.24: Cartoons of various tissues

Figure 2.25: Formation of patterns in bacterial colony.


been set forth to greet them. One of the allied thrusts of our discussion of theadvances in structural biology was to strike a tone of caution on the groundsthat structure in the absence of dynamics is, at best, misleading. The goal ofthe present section is to turn to the question of the variety of time scales thatexist in biological processes. Though our use of the word “process” should beintuitively obvious, the key aim in using this word is that it should conjure inthe reader’s mind the notion of time. For our purposes, a process is a timeevolution of a given system through a series of connected events. Our presentthesis is that the acknowledgement of temporal evolution has to be the highestpriority in biological model building.Biological Processes Are Characterized By a Huge Diversity of TimeScales

As already hinted at above, one of the defining features of living systemsand the property that could be argued to make them the most interesting isthe fact that they are dynamic. Our use of the term dynamic refers to the factthat in discussing living systems, we are automatically invited to consider time.In fig. 2.5, we showed the way in which consideration of successive measuringsticks revealed new levels of structural organization which have an existencewhich can be thought of independently of other scales. This same reasoning canbe brought to bear on biological processes. Our intention is to consider the typesof processes that are observed when we consider successive increases or decreasesin our temporal resolution. A range of different processes, and their associatedtime scales, is shown in fig. 2.26. Broadly speaking, what fig. 2.26 shows is aloose powers of ten representation of different biological processes. We begin(fig. 2.26(a)) with the development of an entire organism, namely, the fruit flyDrosophila melanogaster, with a characteristic time scale of roughly ten days.If we are to increase our temporal resolution by a factor of ten, what comesinto relief is the process whereby individual cells in the development processbegin to differentiate giving rise to the fruit fly’s body. Yet another factor often increase in our temporal evolution brings us to the time scales associatedwith the cell cycle, a process during which cells undergo division and as shownin fig. 2.26(c). Increasing the rate of our stopwatch by another factor of tennow allows us to capture (fig. 2.26(d)) the time scales over which an individualbacterium such as E. coli exercises its random excursion during movement. Ifour stopwatch now runs a factor of ten faster we are now operating at the scaleof deciseconds, a scale which characterizes the rate of amino acid incorporationduring protein synthesis as shown in fig. 2.26(e). The rate of incorporation ofnucleotides during DNA transcription happens roughly ten times as fast as doesthe rate of amino acid incorporation as indicated schematically in fig. 2.26(f). Agreat number of biological processes are dictated by the passage of ions across ionchannels, with a characteristic time scale of milliseconds as shown in fig. 2.26(g).A factor of thousand speed up of our stopwatch brings us to the world of enzymekinetics at the microsecond time scale (fig. 2.26(h)). We note that though wefind the conceptual plan of fig. 2.26 to be a compelling way to acknowledge thediversity of biological time scales, it is important to avoid being overzealous


in precisely attributing a time scale to certain classes of processes since suchprocesses themselves occur over a range of different scales in different organisms.Our plan in the remainder of the entirety of this section is to hang substantive

meat on fig. 2.26 and our brief discussion of it. In particular, we intend to takea stopwatch-by-stopwatch view of each of the time scales we described and toexamine the types of processes observed at each of these distinct scales. We alsopoint out that processes at different time scales are related in a similar fashionas structures at different length scales. Namely, fast processes in the cell oftencombine in an interesting way to produce a cellular process on a slower timescale. An example is provided by the amino-acid addition rate of the ribosomewhich sets a scale for the cell division since all the proteins of the daughter cellneed to be produced.RP: do estimates, put in the

actual time scales and saywhat they are, elaborate oneach of them. RP: note thatthe spatial and temporal areto some extent coupled - fastand small go together - is itroughly true that the prod-uct is roughly constant. Thatis, large rates go with smallstructures and the product ofrate and length is roughlyconstant.

2.5.2 The Cell Cycle and the Standard Clock

The E. coli Cell Cycle Will Serve as Our Standard Stopwatch

Recall that in fig. ?? we used the size of an E. coli cell as our standardmeasuring stick. Similarly, we presently invoke the time scale of the cell cycle asour standard stopwatch since, ultimately, the cell itself is the defining biologicalentity. The goal of fig. 2.26 was to illustrate the variety of different processesthat occur in cell biology and the time scales over which they are operative. Inkeeping with our discussion of structural hierarchies, we once again used thedodge of invoking various multiples of ten to illustrate some of the dominanttime scales associated with processes to be considered throughout this book.The standard stopwatch for the purposes of the present discussion will be

the time scale associated with the bacterial cell cycle, and in particular, thatof E. coli. As shown in fig. 2.27, the bacterial cell cycle will be defined as thetime between the “birth” of a given cell resulting from division of a parentalcell to the time of its own subsequent division. This cell cycle is characterizedstructurally by the formation of a septum and a concomitant segregation ofthe bacterial chromosome, followed by the division of the original cell into twodaughters. Note that the cross sectional area of a dividing cell such as E.coli remains roughly constant and hence the cell growth (and volume) duringdivision can be read off from the cell length. The time scale associated withthe binary fission process of interest here is of order an hour (to within a factorof two), though division can take place as quickly as 20 minutes under optimalgrowth conditions. We note that the subject of DNA packaging in cells willbe of interest in several different contexts such as sections ?? and ??. Oneof the modeling challenges indicated by the present discussion is the dynamicalquestion of what physical and chemical processes attend the segregation of thebacterial chromosome and the formation of the septum.The bacterial cell cycle offers plenty of opportunities for estimates which

raise interesting physical questions. For example, given that the volume of theE. coli cell doubles in roughly 1000 seconds we can estimate the rate of additionof water molecules to the cell. The volume of an E. coli cell, determined earlier


egg

egg

larva

pupaadult fly

days 2 5 9

(A) development of Drosophila

(B) early development of Drosophila

hours

(C) bacterial cell division

minutes

(D) cell movements

seconds

PK 2.19i

(E) protein synthesis

PK 2.19ii

0 0 0 0 0 0 0 1 0 0 0 5 0 0 1 0deciseconds

(F) DNA transcription

0 0 0 0 centiseconds 0 0 0 1 0 0 0 3

(G) gating of ion channels

(H) enzyme catalysis

0 0 0 0 0 0 0 1 0 0 0 2milliseconds

DNAtemplategrowing mRNA

transcript

RNA polymerase

closed ionchannel

++

+

ions

lipidbilayer

open ionchannel

+

+

+

closed ionchannel

+

+

+

lipidbilayer

substrate

enzyme

0 0 x 10–60 1 x 10–6

0 5 x 10–61 0 x 10–6

Figure 2.26: Cartoon which considers factors of ten in time scale. Use our clockicon - logarithmic clock? RP: ATPsynthase rotation rate, viral life cycle, selfassembly, DNA replication, translation, diffusion,


in this chapter, is roughly 1µm3, and with roughly 70% of it occupied by watergives an estimated 2 × 1010 water molecules per cell. If the volume of the celldoubles in 1000 seconds, this implies an estimate that some 20 million watermolecules must enter the cell every second during rapid growth. Similarly, thecell life cycle provides an estimate for the rate of DNA production by DNApolymerases. The number of polymerases responsible for DNA replication inE. coli (DNA polymerase III) is of order 10. Given that the complete E. coligenome is about 5× 106 base pairs (bp) in size, the rate of DNA replication perpolymerase must be about 500 bp/s in order to keep up with the cell cycle itself.Biochemical studies have found rates in the 250-1000 bp/s range, in agreementwith our crude estimate. Yet another cellular process of great importance isprotein synthesis by ribosomes. Since a typical cell is 20% by volume protein,and the density of proteins is roughly that of water, we estimate the total weightof proteins in E. coli to be of the order of 2 × 10−17kg or 1010Da. Given that asingle amino acid has a mass of roughly 100Da, we arrive at an estimate of 108

amino acids per E. coli cell which are produced over 1000 seconds of the cellcycle. Since the number of ribosomes is roughly 2000 this gives a polymerizationrate of 50 amino acids per second for each ribosome, while the measured valueis 25 amino acids incorporated per second.RP: estimates needed here -

something on dynamics andmotion of polymers. Also,estimate the number of mole-cules being synthesized persecond (average) in orderto double the mass in 20minutes - think about fluxesacross surface that are im-plied - don’t really get thephysical aspects yet!

We also note that one of the other classes of processes to be discussed belowis that of genetic networks, and we ask the reader to anticipate the regulatorycomplexity involved in orchestrating the cell cycle, indeed, as noted by Neid-hardt et al. (1990), “the regulatory system that operates the cell cycle standsat the highest level of organization of any global control system, because theformation of new cells is the ultimate function of all the structure, metabolism,and physiological processes of the cell.”The Eucaryotic Cell Cycle Consists of Four Phases Involving Molec-ular Synthesis and Organization

The bacterial cell cycle introduced above is but one example of the dazzlinglycomplex processes of cell division which take place more generally. In fig. 2.28,we show the conceptual organization associated with our understanding of theeucaryotic cell cycle. As shown in Table 17-1 of Alberts et al. (1998), there isnot one universal time scale for the eucaryotic cell cycle, with some embryo cellswith cell cycles that rival those found in bacterial cells, single cell eucaryoteslike yeast with cell cycles of several hours and other specialized cells such asthose in the liver with cell cycles in the thousands of hours. The eucaryoticRP/JK to readers: this all

needs to be written, esti-mates made, physical ques-tions posed. Like for thebacterial case, work out themass/time budget and sur-face fluxes! As it currentlystands, this is a rehash ofAlberts and needs interestingestimates like those suggestedhere.

cell cycle is usually described in terms of four stages denoted as G1, S, G2 andM, with the M phase including the most recognized features, namely, nucleardivision (mitosis) and explicit cell division (cytokinesis), the two G phases (gap)as periods of growth and the S phase (synthesis) during which the nuclear DNAis replicated. Together, the phases other than the M phase constitute interphase.During interphase, the mass content of the cell increases as does its size.We have already introduced a typical cell cycle as setting the time scale for

our standard stopwatch throughout the remainder of the book, though we willalso several other time scales as standards as well such as the rate of rotation of


Figure 2.27: Cartoon showing the key elements in the cell cycle. RP: make thecircular cartoon also show physical pictures of the process. This picture comesfrom the great book of Neidhardt et al. and is B. subtilus and not E. coli. Weneed to do a better job on the formation of septum and the timing and showthe DNA also. Chap. 17 of Moat on Microbial Physiology also has some goodstuff.

Figure 2.28: Cartoon showing the key elements in the eucayotic cell cycle. RP:make sure that a) we have structural pictures for each part of cycle and b) havestopwatches. During G phases maybe we can write dn/dt of various moleculetypes. During S phase, make estimate (will be too short) of length of S byfiguring out how long it takes to replicate DNA.


ATP synthase and the vibrational period of molecular vibrations. Just as ourjourney through the hierarchy of time scales started with our standard rule andthen involved increasing levels of magnification followed by telescoping out inlength scale, our current discussion of temporal scales will follow the same basicmotif. In particular, we begin with the E. coli cell cycle as our fundamentalclock and consider temporal magnifications to ever shorter time scales. Notethat as we proceed to smaller time scales, many of the individual processes tobe considered are themselves key parts of the overall process of the cell cycle.Once we have considered ever smaller time scales we will consider a telescopingup in time scales to temporal processes that take longer than the cell cycle.

2.5.3 Self-Assembly: From the Cytoskeleton to VirusesJK: the idea about self-assembly is that it involveselements both of diffusiveand active - like puttingtogether molecules to makecomplexes gives function,putting together differenttypes of processes give usfunction. Jane’ says we haveto justify the examples.

Many Structures Are Formed Through the Process of Self-Assembly

RP: this is subtle. Gelbartclaims it is not self-assemblyif energy is input, but I amnot sure I buy that. Pack-ing is still part of the self-assembly process.

As already noted above, cellular life is characterized by a wide range ofprocesses which take place at a wide range of time scales. As our first example ofprocesses that take place on time scales smaller than that of the cell cycle itself,we consider processes whereby individual molecular units come together to formmacromolecular assemblies and commonly referred to as self-assembly. In thediscussion below we give two distinct examples of the self-assembly, with eachchoice made in part because these are examples where both experimentationand physical model building have enjoyed some level of success. Obviously, thislist of examples is in no sense exhaustive and the reader is invited to dream uphis or her own.The Cytoskeleton is Always Under ConstructionThe advent of video microscopy in conjunction with fluorescent molecules

that allow for optically labelling other molecules has made it possible to watchthe temporal evolution of cells and their components in real time. One classof problems of great richness and diversity are those related to cell motility.Fig. 2.29(a) shows a series of time lapse images of a fibroblast migrating on asurface. From the perspective of the current discussion, the key point we wish totake away from this figure is the mechanisms that mediate this motion. In par-ticular, we call attention to the fact that actin filaments are polymerizing at theleading edge of the cell and pushing the membrane forward with the cell mov-ing at a mean rate of RP µm/sec. Interestingly, this same fundamental process,namely, orchestrated cytoskeletal assembly can be hijacked by bacteria such asListeria monocytogenes and used for the basis of their own motility as shownin fig. 2.29(b). The basic idea is that the bacterium hijacks the cytoskeletalself-assembly process to form a comet tail of sorts which pushes the bacteriumalong at rates of order RP. Both of the processes of motility described abovewill be considered in greater depth in sections ?? and ??.It is of interest to the current discussion to make a cursory examination of the

time scales associated with the process of polymerization (self-assembly) of actinfilaments. Fig. 2.30 provides a simple schematic view of the actin polymerizationprocess. Since we will consider this process in detail in section ??, we content


Figure 2.29: Series of images showing the motion of a fibroblast on a surface.The accompanying cartoon shows the dynamical evolution of actin filamentsduring this process. RP: should we use Listeria instead or use both. Make sureto use stopwatches. Listeria picture from Mitchison at Harvard.


Figure 2.30: Cartoon which gives a schematic view of the polymerization ofactin. JK notes this can be an example cartoon of binding/unbinding.

ourselves for the moment with using the observed rate of cell migration in both ofthe examples considered in fig. 2.29 to estimate the mean rate of polymerizationof actin filaments. Given that the mean rate of motion of the Listeria bacteriumRP; try to use the observed

rate of cellular migration tothe rate of actin polymeriza-tion. Make sure we fore-shadow the later discussionof polymerization ratchet.Estimate how much actin isin cell. Also, use the mea-sured critical concentrationto estimate both kon andkoff. How? By using theknown spacing of moleculesto estimate kon as diffusionto capture and then use ccritto obtain koff by definition.

is in the range 0.02 - 0.2 µm/sec., and that the size of the G-actin monomersthat are being incorporated in the growing filaments is of order 4 nm, if weassume that there is a perfect linear relation between the growth of individualfilaments and the macroscopic motion of the cell (obviously, this is a grossoversimplification, but nevertheless provides a useful bound), then we estimatethat the mean incorporation rate is of order 25 monomers/sec. Concretely,

RP: all numbers must be re-done - did from memory

what we have done is to extract from observations on the motion of Listeriathe mean addition rate of actin monomers. The microscopic processes thatattend polymerization are more subtle than hinted at here since the monomerscan also detach from the actin polymer, there are hydrolysis reactions thattake place within this polymer and the rate at which monomers are addedto the growing polymer depends upon the concentration of actin monomersavailable for polymerization. We note that our estimated rate of addition of25 monomers/second includes those effects and will require further reflection,to be undertaken in section ??, to extract the actual kinetic rate coefficientsthemselves.

As will be seen repeatedly throughout the remainder of the text, polymer-ization is one of the most fundamental of all processes that occurs in the cellularrealm, though it occurs under a huge range of different guises. In table RP wegive a collection of examples of important biological polymerization reactionswith their corresponding rates in monomers/second.RP: this table should include

filamentous phage, actin andmicrotubules, RNA, DNA,protein synthesis in the ribo-some, polysaccharides, amy-loid fibers, collagen, etc.

Viral Life Cycles Include a Series of Self-Assembly Processes

We have already described the cell cycle as a master process characterized


by an enormous number of subprocesses. A more manageable example of anentire “life cycle” is that of viruses, which not only illustrate the intricate timingof biological processes, but further exemplify the process of self-assembly. Anexample of the viral life cycle of a particular class of viruses that attack bacterialcells (known as a bacteriophage) is shown in fig. 2.31 which shows the keycomponents of one of the stars in the bacteriophage line up, T4. The key stagesin this life cycle are captured in kinetic verbs such as infect, which refers to theentry of the viral DNA into the host cell, transcribe and translate, which referto the hijacking of the cellular machinery so as to produce viral building blocks(both nucleic acid and proteins), assemble, which refers to the coming togetherof these building blocks to form the viral capsid, package, which is the part ofthe life cycle when the viral genome is enclosed within the capsid and lyse, whichrefers to the dissolution of the host cell and the emergence of a new generationof phage to go out and repeat their evil deeds elsewhere. As illustrated by the RP: JK points out that every

one of these sections needs toallude to the modeling ques-tions that can be posed andwhere this will be addressedlater. In this case, every ki-netic verb poses a problemfor a model builder

cartoon in fig. 2.31 and in particular, by our use of the stopwatch motifs, the timebetween the arrival of the virus at the bacterial surface and the destruction ofthat very same membrane during the lysis phase when the newly formed virusesare released seems very short at 30 minutes. Indeed, one of our charters in thechapters that follow will be to come to terms with the 30 minute characteristictime scale of the viral life cycle and the various processes that make it up.

RP: Fig. 9-15 of ECB to beused explicitly and attentiondrawn to every single kineticverb. RP: do ananimal virusalso to show diversity in viralstrategies.

As a second example of the nature of viral self-assembly (and because of theirbeautiful structures) we examine a second class of viruses with very different lifestyles and strategies, namely, animal viruses and in particular, that of HIV. Asshown in fig. 2.32, already the infection process itself is much richer and morecomplex than it is in the context of bacterial viruses. In particular, we note thepresence of a membrane coat on the virus which allows the entire virus to beswallowed up by the cell which is under attack. Once the virus has entered theembattled cell, it exploits linear cytoskeletal filaments like those described aboveto actively transport the viral particle to the nucleus. Once the genetic materialhas been delivered to the nucleus, a process of self-assembly (capsid formation,DNA replication, etc.) with similarity to that described for bacteriophage takesplace resulting in the creation of infectious virions. RP: make sure to do some

useful estimates about num-ber of protein subunits anddiffusion times, rate of repli-cation, etc.

RP: figure from pg. 41 ofLevine, Viruses. JK - aquestion we pose in think-ing about this is is diffusionenough? Jane’ prefers thatthis virus section shows boththe ATP using and not ATPusing self-assembly processesfor viruses.

Recall that our current mission is a series of magnifications in temporal scale,starting from the cell cycle. The discussion of the present section has centeredon a class of self-assembly processes with a characteristic time scale of minutes(RP: check all this - hard to be too precise). We now continue our descent intime scale first, through a consideration of the diffusive processes whereby thevarious molecules implicated in self-assembly find each other and then followedby the metabolic and genetic pathways that lead to the creation of the variousmolecular units described here in the context of self-assembly.

2.5.4 Diffusive Processes: The Random Walk in Biology

RP: this should be seen astaking place on the secondstime scale. This can includemotility, signaling, translo-cation

One of the most fascinating aspects of biological dynamics results from the factthat the cellular realm is a world in which deterministic forces are on nearlyan equal footing with the thermal forces which begin to make their presence


2 mm

50 nm

minutes

bacteriophage bacterium

host cellDNA

viral attachment

DNA injection

macromolecular synthesis andself-assembly

DNA packaging

assembly completion and lysis

mRNA

host cell RNApolymerase

viral DNAreplication

empty capsid

capsids tail sheath

x100

Figure 2.31: Cartoon showing stages in the life cycle of a bacteriophage. Notethat the head and the tail follow distinct assembly pathways and join only afterthey have separately assembled. RP: this is T4 and comes from Rossmann in2003 Phage Assembly meeting. Make sure to add stopwatches.


Figure 2.32: Cartoon showing stages in the life cycle of the HIV virus. RP: thisfigure comes from B. Sodeik (JCB, 159, 393 (2002) - infection, Lingappa et al.,JCB, 136, 567 (1997) - assembly and Ganser et al., Science 283, 80 (1999) forstructure. This is all tentative and is meant just to remind us that we want toconstruct a figure if enough is known to do so, otherwise, use Herpes.


known for particles in solution which are of micron size and smaller. Evidencefor the fundamental role played by random forces was already in hand in the1820s when Robert Brown made his systematic investigations of the motionsthat now bear his name.Many Biological Processes Are Diffusive

As noted in fig. 2.26, one of the representative classes of process of interesthere is that of diffusion. What we have in mind here is the fact that as a resultof the random thermal jiggling that is characteristic of particles with sizes likethose found in cells, a particle will undergo a net random movement. WhetherRP: numbers and estimateswe think of the motion of proteins that bind to DNA (one-dimensional diffusionalong the DNA molecule) or the G-actin monomers that are added to the endof an actin filament during polymerization (three-dimensional diffusion) or theassembly of viral capsids, the basic picture is that of molecules being batteredabout and every now and then ending up in the same place at the same time.As will be shown in more detail later, the characteristic time it takes a diffusiveparticle to move a distance x is given by

tdiffusion =x2

D, (2.3)

where D is the diffusion constant for the particle of interest in the solutionunder consideration. As illustrated in the figure, the time scale associated witha typical protein to diffuse a distance of our standard ruler (i.e. across an E.coli) is roughly 0.01s where we have used x = 1µm and D = 100µm2/s to arriveat this estimate.One of the schemes we will return to repeatedly for trying to determine

the rates of biological processes is to bound the rate of a given process on theassumption that the rate of that process is dictated entirely by diffusion. Inreality, many of the processes of interest involve several steps including thati) the relevant molecular participants find each other (both in terms of spatialposition and orientation) and ii) once they do find each other, that they passover some barrier which further reduces the rate. Nevertheless, it will still beof great interest to estimate the rates implied by the diffusive process wherebythe molecular participants find each other. Examples of interest include: thereaction H++OH− → H2O, the binding of oxygen to hemoglobin, the bindingof lac repressor to DNA and the assembly of individual monomeric units to formviral capsids. As will be shown in section ??, the outcome of the most naivediffusive arguments (and in the case where the two diffusing particles are treatedas being of the same size) is that the rate constant for molecular association isgiven by

kdiffusive =8kBT

3η, (2.4)

where η is the viscosity of the medium. If we use the numbers for water at roomtemperature to deterimine the viscosity, this results in the value kdiffusive ≈7× 106m3/ mole · sec, a value which is helpful for estimating rates of point-like


Figure 2.33: Film clip of E. coli diffusive trajectories. (RP: self similarlityinteresting or not). This is from Berg J. Bacterio and Phys. Today articles.

particles, but overestimates the rates associated with protein-ligand interactions,a shortcoming that can (and will in section ??) be amended by acknowledgingthat a ligand can only interact correctly with its receptor in certain preciseorientations.

Part of the hugely powerful mathematical apparatus that has grown up inresponse to diffusive motions is the theory of the random walk, so importantas a “hydrogen atom” to merit a dual entry in our modeling hall of fame.One of the powerful applications of the theory of the random walk is for thecharacterization of the dynamical trajectories of particles that exercise motionsin which the direction during one step is completely indifferent to the directionsof all steps that have gone before. An example of a trajectory of this type isrealized during cellular motility of cells like E. coli as shown in fig. 2.33. RP: make some little toy es-

timates to foreshadow laterchapter

Some Diffusive Processes Are Biased or Ratcheted

RP; here explain the way inwhich diffusion process canbe carried out in presence ofsecond mechanism.

Interestingly, there are some diffusive processes which take place under theinfluence of a ratcheting action. One idea of particular conceptual importanceand which will be discussed in detail in section ?? is that of the translocationratchet. Such ideas are forced on us by virtue of the fact that there are manybiological instances in which a biopolymer is required to pass from one side of amembrane to another through some form of pore. For example, nuclear exportis mediated by complexes such as the nuclear pore complex. The mitochondriaand endoplasmic reticulum are themselves both also constant participants inthe translocation drama, given that various proteins are passed to and fromthese membrane-bound organelles repeatedly. If we recall the significance of theHershey-Chase experiment in explicating part of the life cycle of bacteriophage,we note that there too the genetic material is passed across the membrane of thebacterium, leaving the empty capsid behind. The present argument is reflected


in the statement that the diffusive motion of a biopolymer traversing such amembrane can have its random walk across the membrane sped up as a resultof the binding of molecules (chaperonins) to the polymer at a progression ofsites once they have crossed the membrane. In this case, the translocation timetakes the form

ttranslocate ≈ nd2

D, (2.5)

where d is the mean spacing between binding sites, and n is defined throughreference to the fact that the total length of the polymer is L = nd. Thisexpression can be argued for simply by noting that d2/D is the time for a singlebinding site to diffuse over a distance d. Assuming that this exposes the bindingsite to a chaperonin molecule, which then binds and prevents any backwardmotion of the polymer past this position, the total time for translocation is thetime for all n binding sites to complete this process. In the absence of thisratchet-like mechanism the diffusion time for the polymer would be

tdiffusion ≈n2d2

D, (2.6)

or n = L/d times longer. In the case of a bacteriophage injecting its DNA intoa bacterial cell the binding of RNA polymerases can serve to produce a ratcheteffect. For DNA some 10µm in length, typical of a genome of a bacterial virus,and with polymerase binding sites located roughly one per micron, the speedup due to polymerase binding will be n = L/d = 10. This will in fact be anupper bound given the fact that all binding is reversible and once in a while apolymerase will detach from the DNA spoiling the ratcheting action.RP: poorly explained, need

numbers, need to give hon-est and concrete example andforeshadow where this willturn up later.

2.5.5 Metabolic and Genetic Networks: Time Scales ForMolecular Construction

Many Important Chemical Reactions Are Linked in Networks

One man’s reactant is another man’s product. What this means concretelyfor the current discussion is that there are a huge number of chemical reactionsin cellular life that are linked in chains such that the output (the product) of onereaction is the input (the reactant) to some other reaction. Indeed, to garnera sense of the diversity of biochemical networks, the reader is encouraged toexamine Michal (1999). One might say that the “hydrogen atom” of chemicalnetworks is that provided by the glycolysis pathway and indicated schematicallyin fig. 2.34. One of our other reasons for raising metabolic and genetic networksin the present chapter is to introduce yet another class of cartoons, namely ofnetworks, which are highly instructive. Interestingly, just as we argued that setsof atomic coordinates for a given macromolecular structure can be misleadingsince they fail to suggest the dynamic aspects of molecular structure, so too domost network diagrams fail to endow the description of these networks with aproper sense of the passage of time.RP: glycolysis and metabolic

networks - include stop-watches and also have graphsof concentrations of all thereactants and products as afunction of time, all on samepicture - the key point is tofind a way to improve thestandard diagrams of net-works to show more than justconnectivity. Also, should


Metabolic Networks Are Reactions AssociatedWithMolecular Trans-formation and Energy Generation

Metabolic networks such as the glycolysis pathway introduced above have acentral role in cellular life. In particular, such processes are the seat of energygenerating molecular transformations whereby molecules taken on board by anorganism are broken down and exploited for energetic and synthetic purposes.Since one of the key themes of the present chapter is the role of cartoons incapturing essential biological processes, we call the reader’s attention to our ownenthusiasm for the set of cartoons (and associated text) describing the glycolysispathway to be found in Goodsell (1998). The glycolysis pathway describes theseries of enzyme-mediated and coupled reactions whereby a glucose molecule, ahexagonal sugar, is broken down to yield two molecules of pyruvate, a smallermolecule containing three carbons. A favorable by-product of this series ofreactions is the production of ATP. In fig. 2.34 we schematize this pathwayillustrating each of its molecular steps with stopwatches illustrating the averagelifetime of each reaction and the structural cartoons illustrating the relative sizesof enzyme and substrate. Note that from a modeling perspective, the entiretyof the glycolysis pathway represents a series of coupled chemical reactions whichcan be considered using the types of models described in chap. ??. We remind RP: make sure to try and

estimate how much of thisgoes on in our body eachday. Also, make sure that inthe chemical kinetics chap-ter, we come back to glycoly-sis - UCSD guy does calcula-tions. the models are big andugly - is that the best we cando?

the reader that though we have fixated on the specific example of the glycolysismetabolic pathway, there is a seemingly endless variety of such pathways (lipidsynthesis, protein synthesis, respiration, formation of polysaccharides, proteindigestion, etc.) which can be thought of from the standpoint of exactly thekinds of pathways illustrated in fig. 2.34, again as illustrated in detail in Michal(1999).Genetic Networks Are Collections of Genes Whose Expression Is In-terrelated

RP: is this where I should dothe bacterial flagellum case.- a second treatment, nowfrom perspective of Alon ex-periment

A second class of networks which are of far reaching significance are known asgenetic networks. The conceptual picture which should be imagined is indicatedschematically in fig. 2.35 where we show the coupled genes associated with asynthetic genetic network known as the repressilator. The idea embodied insuch diagrams can be understood both at the abstract level and at the levelof the individual molecular actors which participate in the various reactions.At the abstract level, we note that a given gene can be under the control ofother genes in a variety of different ways. Perhaps the simplest eventuality isdirect negative control in which a specific protein binds to the promoter regionon DNA of a particular gene and physically blocks binding of RNA polymeraseand subsequent transcription. This protein is itself the result of some othergene which can in turn be subject to control by yet other proteins (or perhapsthe output of the gene that it controls). The most celebrated example of thistype of genetic network (and which can be thought of as the “hydrogen atom”of such networks) is the lac operon to be considered in more detail in chap. ??,where we examine model systems in biology. RP: try to estimate or re-

port some numbers on rela-tive rates of gene expressionas a function of presence ofrepressor.

The Formation of the Bacterial Flagellum Is Intricately Organized in


Figure 2.34: Representation of glycolysis pathway. RP current idea, horizontalfilm strips for each of the ten steps and a vertical stopwatch flow showing theelapsed time of the entire process. On horizontal film strips show the enzymeand substrate and have the stop watch showing elapsed time for the subprocess.This picture is lifted from the web just to indicate idea. RP: use stopwatchesin a way that depends upon concentration - perhaps multiple stopwatches forSAME reaction since the elapsed time depends on concentration, even if therate constant does not.

Figure 2.35: Representation of genetic network for the repressilator. RP: thisfigure lifted from Powerpoint presentation of Khaled Rhaman. RP: put in theactual structures and try to add stopwatches


Figure 2.36: Cartoon illustrating the bacterial flagellum and the labels for thevarious gene products.

Space and TimeRP: rationale is a) the con-stant appearance of flagel-lum in later chapters, and,the existence of a profoundexperiment that examinestiming. This section has todo with the structural assem-bly. RP: must do the UriAlon timing for the flagellumprocess. Check that what Iwrote is correct.

As we have already begun to belabor, the role of time in biological systemsmust be constantly attended to, both from the point of view of experimentaldesign and modeling. One of the most beautiful recent experiments which haspermitted us to peer directly into the dynamics of genetic networks concerns theprocess of assembly of the bacterial flagellum. The basic idea of the experimentis to induce the growth of flagellae in starved E. coli cells and to use a reportergene, namely, a gene leading to the expression of green fluorescent protein, toreport on when each of the different genes associated with the flagellar pathwayare being expressed. In particular, fig. 2.36 shows the various gene products(FlgK, MotB, etc.) that are involved in the formation of the bacterial flagellum.Essentially, each of these products corresponds to one of the protein buildingblocks associated with flagellar construction.The concept of the experiment being described here was to use the green

fluorescent protein to report the presence of a given gene product. By attachingthe reporter to different genes for different cells, it was thereby possible toobserving which genes were being expressed when as shown in fig. 2.37. Inparticular, fig. 2.37 shows the intensity of GFP associated with different genesas a function of time and resulted in the ordering of the genes. RP: are their control genes

we can explain in the contextof network.

JK: would like to have anestimate for every exampleof the time scale for theprocess of interest. One ofthe tools for the flagellumproblem (and self-assembly)is diffusion to capture. JKchallenges the logic sincethe Alon experiment doesn’tmeasure the real self assem-bly, it measures a step in

The main objective of the present subsection was to illustrate the types ofprocesses that are associated with the various metabolic and genetic pathwaysso central to life. We have seen that such processes are characterized by time


Figure 2.37: Illustration of the timing of gene expression during the process offormation of the bacterial flagellum. The figure on the left shows which genesare being expressed as a function of time and the figure on the right gives aschematic interpretation of what the experiment teaches us. RP: show the geneproduct on the graph associated with each of the curves. This comes from Kaliret al., Science, 292, 2080, 2001. The big message here is put time in biologyfront and center.

scales which range from RP to RP.

2.5.6 The Processes of the Central Dogma

The Central Dogma Describes the Processes Whereby the GeneticInformation Is Expressed Chemically

We have already had occasion to mention Crick’s quip concerning the “twogreat polymer languages”, visualized in fig. ??. The central dogma refers to theset of mechanisms where the molecules of these two great languages are synthe-sized. The fundamental processes of replication, transcription and translationand their linkages are shown in fig. 2.38. The basic message of this “dogma” inits least sophisticated form is that DNA leads to RNA which leads to proteins.

DNA Is Copied in a Process Known as Replication

One of the primary processes shown in fig. 2.38 is replication, the processwhereby a copy of the genetic material itself is made. DNA replication must takeplace when a cell divides, for example. Also, recall that in our discussion of theviral life cycle, the machinery of the host cell is hijacked to do the virus’ bidding,and part of that hijacking includes making a copy of the viral genome. As shownin fig. 2.38, the process of DNA replication is mediated by a macromolecularcomplex (the replisome) which has a variety of intricate parts such as the enzymeDNA polymerase which incorporates new nucleotides onto the nascent DNAmolecule.RP: estimates and numerical

facts


REPLICATION

TRANSCRIPTION

TRANSLATION

DNA

RNA

RNA

DNA polymerase

RNA polymerase

helicase

primase

DNA

DNA

DNA template

RNA message

ribosomegrowing polypeptide chain

protein

Figure 2.38: Cartoon showing various processes associated with the centraldogma. RP: this cartoon shows the dogma and the structural players.


The DNA Message Is Carried By a Messenger Molecule Following aProcess Known as Transcription

The DNA molecule serves as a template in two different capacities. Asdescribed in the previous paragraphs, a given DNAmolecule serves as a templatefor its own replication. However, in its second capacity as the carrier of thegenetic material, a DNA molecule must also dictate the synthesis of proteins(the expression of its genes). The first stage in this process of gene expressionis the export of the genetic material from the nucleus (in the eucaryotic setting)in the form of a messenger molecule (mRNA) which serves as the template forprotein synthesis. In fig. 2.38, this process links the DNA → RNA vertices andis mediated by a molecular machine known as RNA polymerase.RP: Seth Darst structure,

database of macromolecularmotions, estimates and nu-merical facts

New Proteins Are Produced During the Translation Process

Once the messenger molecule (mRNA) has been synthesized and exportedto the cytoplasm, the translation process can begin in earnest, (RNA → Pro-tein). As already described in section ??, translation is mediated by one ofthe most fascinating macromolecular assemblies, namely, the ribosome. Theribosome is the apparatus that speaks both of the two great polymer languagesand in particular, forms a string of amino acids (a polypeptide chain) whichare dictated by the codons (collections of three letters) on the mRNA molecule.The structure of the ribosome is indicated in cartoon form in fig. 2.38. As themessenger molecule is threaded through the ribosome, special RNA molecules(tRNA) which are charged with the various amino acids and each of which recog-nizes (and binds to) a particular mRNA sequence (RP: check the precision ofthe wording), are placed in conjunction with the relevant mRNA sequence andtheir amino acid cargo is taken by the ribosome and added to the polypeptidechain of ever increasing length.This subsection has had as its ambition the idea of elaborating on several of

the processes shown in fig. ??, and known collectively as the processes of thecentral dogma with the added aim of illustrating the sense in which biologicalcartoons capture those features of a given structure or process that are centralto understanding the phenomenon of interest. We note that in terms of themolecular participants in the central dogma, many of the key structures havebeen determined using both cryo-electron microscopy and x-ray crystallogra-phy. As a result, the cartoons that describe these molecules (and assemblies)have become more precise, but ultimately, they remain a caricature of the fullatomic-level reality of these structures and we argue that in many instancesthese coarse structural descriptions are actually more enlightening than theirall-atom counterparts.

2.5.7 Active Transport: Molecular Motors and Pumps

We have already noted that biological motility is based in large measure ondiffusion. On the other hand, there are a host of processes that cannot wait thetime required for diffusion. In particular, recall that our estimate for the time


required to diffuse a distance x, is tdiffusion ≈ x2/D. If we are to substitute adiffusion constant relevant to a typical protein (D ≈ 10−10m2/s) and demandthe diffusion time for this protein to traverse an axon (x ≈ 10cm), this time is awhopping 108 seconds or roughly three years. For comparison we can estimatethe transport time for kinesin moving on a microtubule over the same distance.As the typical speed of kinesin in a living cell is 1µ/s, the time for it to transporta protein over a distance of 10cm is 105 seconds or some 28 hours.

Like the Wells Fargo stage coach, molecular transport must go on. Collec-tively, we have in mind the processes we will refer to as active transport with theidea that directed motion is effected through the consumption of some energysource (usually ATP). Broadly construed, the subject of active transport allowsus to classify a wide variety of molecules as molecular motors. We have alreadyseen the existence of such motors in a number of different contexts, with bothDNA polymerase and RNA polymerase introduced in the previous section bothsatisfying the definition of active transport.

Much Cellular Transport Is Effected by Molecular Motors

Concretely, the class of motors of interest in the present discussion are thosethat mediate transport of molecules from one place in the cell to another. Often,such transport takes place as vesicular traffic, with the cargo enclosed in a vesiclewhich is in turn attached to some molecular motor. These molecular motorstravel in a directed fashion on the cytoskeletal network which traverses thecell. For example, traffic on microtubules runs in both directions (recall thatmicrotubules have a polarity) as a result of two translational motors, kinesinand dynein. Molecular motor mediated transport on actin filaments is shown in RP: why should reader recall

this - introduce earliercartoon form in fig. 2.39. Note that this cartoon is meant to indicate a roughidea of the relative proportions of the motors and the microtubules and to conveythe overall structural features, such as two heads, of the motors themselves.

Membrane Bound Proteins Transport Molecules From One Side of aMembrane to the Other

The second class of active transport of interest to the current discussion hasto do with the transport of ionic and molecular species across membranes. Notethat the molecular motors described above remedy the conundrum associatedwith the inadequacy of diffusive transport in some circumstances. This samedifficulty arises in the context of ion transport, for example, in those instanceswhere the cell requires to move ions up a concentration gradient. One of the mostnoteworthy examples of this necessity arises in the context of the ATP synthase.Recall that these rotary chemical factories are driven by a proton gradient,an imbalance in the concentration of proteins across the mitochondrial innermembrane which is established through the action of the type of membrane-bound active-transport proteins described here. In fig. 2.26(RP) we showed theprocess of ion transport across a membrane with a characteristic time of RP.


Figure 2.39: Motion of myosin on an actin filament (RP: like 16-16,16-17 ofECB). RP: general question, should all of these cartoons in the process sectionhave stopwatches on them? I think so. More likely to use figure from PaulSelvin recent paper on Myosin V - that really gives sense of fact that this isdynamic.

Figure 2.40: Cartoon representing the active process whereby ions are trans-ported up a concentration gradient.

2.6. THE MATHEMATIZATION OF BIOLOGICAL CARTOONS 97

2.5.8 Chemical Kinetics and Enzyme Turnover

RP: this bit is on the rates of reactions, final of the scales we cared about. Atthe fastest rates considered in fig. 2.26 we invoked enzyme kinetics as a class ofprocesses which can take place over very fast time scales. Before delving moredeeply into this class of processes, we note that the division of the second half ofthis chapter is a bit artificial and that our discussion of metabolic pathways andthe central dogma, for example, have already brought us head on into discussionsof enzymes and their kinetics.

2.5.9 Telescoping Back Up in Time: Developmental Biol-ogy

2.6 The Mathematization of Biological CartoonsRP: what I want to do hereis to try and show howthe examples given earlierin the chapter can be math-ematicized. For example,galaxy structures and corre-lation functions, rate equa-tions for processes, coor-dinates and kinematics forgeometry of structure.

We opened the present chapter with observations on episodes from the history ofphysics, with special reference to the discoveries of Faraday concerning the elec-tromagnetic field, discoveries that were couched in words rather than equationsand which were characterized visually by cartoons. As part of that discussion,we appealed to the intriguing comments of Maxwell who saw his challenge asthe need to translate Faraday’s insights into a “mathematical form”. One of thetheses of the present chapter, and for that matter, of the entire book, is the ideathat as a result of the increasing extent to which biological data is quantitativeit is now not only opportune but even necessary to engage in translating someof the precise biological cartoons into a mathematical form. We have alreadyforeshadowed some of the modeling challenges presented in thinking about thevarious phenomena described in this chapter and round out the chapter with abrief discussion of some of the most important large-scale quantitative modelingchallenges to be faced in coming chapters.

2.6.1 Geometry and Structures

The moment we invoke the word “structure” we have automatically endowed ourproblem with a geometric character. One of the themes that will be considered incoming chapters is how best to characterize the geometry of biological systems.Several powerful questions that can be posed at the outset and which we willanswer in several distinct ways in the chapters that follow are: i) should wecharacterize structure deterministically or statistically, ii) should we characterizestructure discretely or continuously?Structure Can Be Described Both Deterministically and Statistically

There is a fundamental tension in describing macromolecular structure thathas already been alluded to repeatedly throughout this chapter. In particular,despite the existence of databanks that are filled with atomic coordinates fordifferent molecules, in the agitated environment of the cell these molecules arebattered about more than the ships seen in Victory at Sea. Consequently, there


are circumstances in which the “structure” is changing continuously and hencestatistical measures of that structure are more useful than their deterministicanalogs. In chap. 5 we consider the mechanics of polymers such as DNA from thedeterministic perspective in which the molecule is assigned a precise position,described by the space curve r(s), where s is the arclength along the polymer.It will be seen that this description provides a variety of powerful insights intoproblems such as DNA packing in viruses and eucaryotes and the buckling ofcytoskeletal filaments. On the other hand, in chap. 6 we will see the advantagesgarnered by thinking of a polymer such as DNA as bustling about through avariety of different conformations, with the result that our treatment of structurewill center on statistical quantities such as the average end-to-end distance andthe radius of gyration.

Structure Can Be Described Both Discretely and Continuously

A second set of related questions concerning the mathematization of cartoonscenter on the possibility of discrete and continuous representations of structure.At the simplest level, atomic-structures as captured in a pdb file are the mostcompelling example of a discrete representation of structure. There are a finiteset of coordinates, namely, (x1, y1, z1, x2, y2, z2, · · · xN , yN , zN), where xi is thex-coordinate of the ith atom and so on. The object of the present discussionis to remark that this is by no means the only way to think about structure.A second possibility which is still discrete is to assign a molecule some set ofcollective coordinates. Here again, the literature is replete with examples ofsuch thinking with one example being models in which the coordinates of onlythe Cα atoms in the protein backbone are considered.

Indeed, as mentioned above, an equally valid though admittedly lower-resolution description of the structure of a long DNA molecule is to think ofit as the space curve r(s). This same type of thinking can (and will) be used indescribing membranes. The full atomic-level description is characterized by thecoordinates of each and every atom in each and every lipid molecule. On theother hand, we can invoke the tools of differential geometry wherein a surfaceis (in simplest terms) thought of as a function which gives the height of thesurface at every point (x, y) on the plane, h(x, y). These two eventualities areshown side by side in fig. 2.41.RP: Here I do stuff like the

alpha helix, etc... RP: theidea of putting some stuffin here on what has beenlearned about the statisticsand connection to genome isvery appealing RP: want todo a bit here on mathematicsof parametrization of curvesand surfaces and why thisis relevant to mathematizingcartoons Collective Coordi-nates Structural Motifs

2.6.2 Evolution Equations and Processes: Beyond KineticVerbs

The Mathematics of Entanglement.

RP: this may not belong herebut I want to talk about en-tangled networks. Polymers,dislocations, etc..

Deterministic Evolution.

RP: rate equations and self-

Stochastic Evolution.

2.7. FROM CARTOONS TO ANIMATIONS AND MOVIES 99

xx

yy

h(x,y)z

ri

(A) (B)

Figure 2.41: Cartoon showing both atomistic and continuum description ofmembrane.

Figure 2.42: Cartoon representation of self-assembly via polymerization of mi-crotubules. (RP: model on fig. 16-12 of ECB - we would like to have the pictureshow the pieces of the mathematics such as kon, koff and the concentration


2.7 From Cartoons to Animations and Movies

RP: transcendence of sta-tic structures by dynamics.Animations and movies asthe basis of such anima-tions. Try to really be cre-ative here and give good ex-amples, such as cell motil-ity, Sheetz experiments. Be-gin with Harold Edgerton,talk about time lapse pho-tography for weather, plantsblooming, cellular dynamicsand then modern video mi-croscopy. Time lapse. Theserepresent enormous exper-imental advances that arejust starting to change howwe REALLY think. Tryto key in on this kind ofdata. Emphasize dynamicsand processes.

RP: Data sets from high per-formance computing. Ques-tions on higher dimensionalthings and how to make car-toons. Banchoff RP; A quietrevolution that has accom-panied the use of comput-ers is the advent of computergraphics and an increasingreliance on cartoons both forpedagogy, but even for thecommunication of the resultsof research.

2.8 A Parting Shot

RP: key ideas of chapter - role of cartoons in representing complex phenomena ina way that boils off fat and leaves us the essential elements, introduce biologicalphenomenology and especially those parts of biological phenomenology thatwe intend to answer to with model building, to try and provide some of thequantitative backdrop - estimates aimed at getting a sense of the scales of thingsin space and time. Use quote from The Little Prince where he talks about adultsliking numbers - in this case, we contest the spirit of St. Exupery, this numberssense can be every bit as playful as drawing a mouton.

2.9 Further Reading

Schweber -both article in Rev. Mod. Phys. and the book.

Boeke K., Cosmic View The Universe in 40 Jumps, The John Day Com-pany, New York: New York, 1957. Boeke starts with a picture of a child holdinga cat and proceeds to view her situation both by decreasing (26 orders of mag-nitude) and increasing (13 orders of magnitude) the resolution of the view.Mattuck R. D.,AGuide to Feynman Diagrams in the Many-Body Prob-lem, McGraw-Hill, Inc., New York, New York, 1976. Mattuck’s wonderful bookhas a number of different virtues, one of which is the debunking of the mysticalaura associated with Feynman diagrams. For the present purposes, it is ratherin its role as a representation of Feynman diagrams that we appeal to the book.

Ptashne M., A Genetic Switch, Blackwell Science and Cell Press, Cambridge:Massachusetts, 1992. Ptashne’s fantastic book (and Kornberg and Goodsell,mentioned below) are really the places that first got me thinking about theamount of science captured in cartoons. I cannot recommend Ptashne’s bookhighly enough.

Goodsell D., The Machinery of Life, Springer-Verlag, New York: New York,1998. Following on the work of Minton and Zimmermann (RP: check), Goodsellhad the very clever idea of trying to represent the cell as it really is, crowded andto reveal the connection between structure and function, explicitly and visually.His books make for fascinating reading.

Kornberg A., For the Love of Enzymes, Harvard University Press, Cam-bridge: Massachusetts, 1991. Kornberg’s beautiful book, though featured herebecause of its constant appeal to cartoons, is hugely fascinating. It is a greatpleasure to see his passion for science, with this book in particular reading likeone long ode to enzymes. Part of the relevance to the current chapter is hisFigure 2-2 where he gives a biological view of the structures that are seen as

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