The AAA-2 ring density could be almost entirely interpreted using this approach and the fit suggests that 3 to 5 subunits are sufficiently closed to bind ATP ( Figure 6—figure supplement 2). We analyzed nucleotide (ADP) binding to wild type ClpB by isothermal calorimetry (ITC) revealing a binding stoichiometry of 7.5 ± 0.1 ADP per hexamer ( Figure 6—figure supplement 3). ITC experiments using ClpB-K212A, which is deficient in nucleotide binding in AAA-1, allowed us to determine a binding stoichiometry of 3.7 ± 0.3 ADP in AAA-2 of the mutant hexamer ( Figure 6—figure supplement 3). The same ADP binding stoichiometry was also found for the repressed and hyperactive variants ( Figure 6—figure supplement 3). The deduced stoichiometries are in good agreement with the distinct AAA-2 conformations observed in the asymmetric EM reconstructions. Similar calculations have been reported for the AAA+ proteins MCM, ClpX and HslU ( Moreau et al., 2007; Yakamavich et al., 2008). The AAA-1 ring is more asymmetric than AAA-2 and it is not easily interpretable by fitting crystallographic dimer models. There is sufficient density to guide rigid body fitting of all AAA-1 domains, but this fitting does not allow deductions of nucleotide occupancy ( Figure 6—figure supplement 2). On February 3, 1959, Buddy Holly, Ritchie Valens, and J.P. Richardson suffered a plane crash which cost them their lives. If not for a coin toss, Valens life would have been spared. Prior to the flight, Valens and Holly’s guitar player (Tommy Allsup) flipped a coin to determine who would get to fly in the plane chartered by Holly. Valens won, and the rest is unfortunate history. In 1971, Don McLean would memorialize this day as “The Day the Music Died” in his hit song “American Pie”. Secretariat, the Triple Crown horse Trus BL, Cheng N, Newcomb WW, Homa FL, Brown JC, Steven AC (2004) Structure and polymorphism of the UL6 portal protein of herpes simplex virus type 1. J Virol 78:12668–12671 Chang JT, Schmid MF, Haase-Pettingell C, Weigele PR, King JA, Chiu W (2010) Visualizing the structural changes of bacteriophage Epsilon15 and its Salmonella host during infection. J Mol Biol 402:731–740 Cuervo A, Carrascosa JL (2011) Viral connectors for DNA encapsulation. Curr Opin Biotechnol 23:529–536

The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the head-to-tail method is employed to determine the vector sum or resultant. A common Physics lab involves a vector walk. Either using centimeter-sized displacements upon a map or meter-sized displacements in a large open area, a student makes several consecutive displacements beginning from a designated starting position. Suppose that you were given a map of your local area and a set of 18 directions to follow. Starting at home base, these 18 displacement vectors could be added together in consecutive fashion to determine the result of adding the set of 18 directions. Perhaps the first vector is measured 5 cm, East. Where this measurement ended, the next measurement would begin. The process would be repeated for all 18 directions. Each time one measurement ended, the next measurement would begin. In essence, you would be using the head-to-tail method of vector addition. Cardarelli L, Lam R, Tuite A, Baker LA, Sadowski PD, Radford DR, Rubinstein JL, Battaile KP, Chirgadze N, Maxwell KL, Davidson AR (2010a) The crystal structure of bacteriophage HK97 gp6: defining a large family of head-tail connector proteins. J Mol Biol 395:754–368 The AAA+ rings have a central opening of ∼30 Å and therefore are not as compact as in one of the previous models ( Lee et al., 2003, 2007, 2010), but not as expanded as in the other ( Wendler et al., 2007). The Arg-fingers are at the interface between subunits, available to catalyse hydrolysis as expected from mutational studies ( Mogk et al., 2003; Yamasaki et al., 2011; Biter et al., 2012).We disagree that an ADP-bound subunit is more similar to an empty state than to an ATP-bound one. HX experiments of ClpB did not reveal differences in protection patterns in ADP and ATPγS, arguing that the conformations are similar [Oguchi, Y, et al, Nat Struct Mol Biol, 2012]. Importantly, both ADP and ATPγS binding led to substantial increase in HX protection compared to nucleotide-free ClpB oligomers, including Walker A motif regions. These data indicate that the overall conformational state of ADP-bound ClpB is more similar to the ATP-bound state than to the empty one. Donate LE, Herranz L, Secilla JP, Carazo JM, Fujisawa H, Carrascosa JL (1988) Bacteriophage T3 connector: three-dimensional structure and comparison with other viral head-tail connecting regions. J Mol Biol 201:91–100 Since negative stain EM of ClpB (BAP) in complex with ClpP gave a clear result different from all previous cryo EM maps of ClpB and Hsp104, we collected cryo EM data on BAP-ClpP as well as on ClpB alone. The same strategy of using only clearly identifiable side views was applied. Complexes were imaged in the presence of ATPγS and independent maps were obtained by de novo angular reconstitution in each case ( Figure 2B,C). Veesler D, Cambillau C (2011) A common evolutionary origin for tailed-bacteriophage functional modules and bacterial machineries. Microbiol Mol Biol Rev 75:423–433

The BAP hexamer has overall outer dimensions of ∼150 × 100 Å, similar to previous structures of ClpB/Hsp104 ( Parsell et al., 1994; Lee et al., 2003; Wendler et al., 2007, 2009; Figure 1C). It encloses a ∼30 Å wide central channel, comparable in size to that in the crystal structure of ClpC ( Wang et al., 2011; Figure 1C,D). In the reconstruction it is possible to identify regions accounting for all the domains, such as L-shaped densities for the AAA+ domains and a rod-like density for the coiled-coil MD. Olia AS, Al-Bassam J, Winn-Stapley DA, Joss L, Casjens SR, Cingolani G (2006) Binding-induced stabilization and assembly of the phage P22 tail accessory factor gp4. J Mol Biol 363:558–576 To see how the method works, consider the following problem: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement. Chattoraj DK, Inman RB (1974) Location of DNA ends in P2, 186, P4 and lambda bacteriophage heads. J Mol Biol 87:11–22The following individuals responsible for the peer review of your submission have agreed to reveal their identity: Andy Martin (Guest Editor); Gabriel Lander (peer reviewer). Zhao L, Kanamaru S, Chaidirek C, Arisaka F (2003) P15 and P3, the tail completion proteins of bacteriophage T4, both form hexameric rings. J Bacteriol 185:1693–1700

Katsura I (1987) Determination of bacteriophage lambda tail length by a protein ruler. Nature 327:73–75 T1 - Head-to-tail cyclization of side chain-protected linear peptides to recapitulate genetically-encoded cyclized peptides Lebedev AA, Krause MH, Isidro AL, Vagin A, Orlova EV, Turner J, Dodson EJ, Tavares P, Antson AA (2007) Structural framework for DNA translocation via the viral portal protein. EMBO J 26:1984–1994 Orlova EV, Gowen B, Dröge A, Stiege A, Weise F, Lurz R, van Heel M, Tavares P (2003) Structure of a viral DNA gatekeeper at 10 Å resolution by cryo-electron microscopy. EMBO J 22:1255–1262Vianelli A, Wang GR, Gingery M, Duda RL, Eiserling FA, Goldberg EB (2000) Bacteriophage T4 self-assembly: localization of gp3 and its role in determining tail length. J Bacteriol 182:680–688 Tang J, Olson N, Jardine PJ, Grimes S, Anderson DL, Baker TS (2008) DNA poised for release in bacteriophage phi29. Structure 16:935–943 The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle. In fact, it is said that Sigmund Freud himself realized this phenomenon and used it to his advantage. When faced with someone who was struggling to make a decision, Freud would sometimes suggest that they flip a coin. Allegedly, Freud would then say “Look into your own reactions. Ask yourself: Am I pleased? Am I disappointed?” This method of flipping is sometimes called the Freudian Coin Toss. By analyzing your reaction, you may realize that you actually knew the choice that you wanted to make all along!

At birth, an infant’s head is typically larger and rounder than the rest of their body. As they grow, their head will continue to grow faster than other parts of their body. This phenomenon occurs due to neurological changes that occur in infancy that help facilitate certain skills. For example, infants may be able to use their upper limbs before their lower limbs due to cephalocaudal development. Gaussier H, Yang Q, Catalano CE (2006) Building a virus from scratch: assembly of an infectious virus using purified components in a rigorously defined biochemical assay system. J Mol Biol 357:1154–1166 Proximodistal is a term used in anatomy to describe the direction of growth or development from the center of the body or a local structure outwards towards its distal ends. This term is often used to refer to the growth of limbs, for example, where the shoulder joint is located more proximally than the fingertip. It can also be used to describe how certain structures develop and grow throughout the body, such as organs or muscles. In general, it describes an outward-growing pattern from a central point. Understanding Proximodistal Development Lengyel JA, Goldstein RN, Marsh M, Calendar R (1974) Structure of the bacteriophage P2 tail. Virology 62:161–174 Leiman PG, Chipman PR, Kostyuchenko VA, Mesyanzhinov VV, Rossmann MG (2004) Three-dimensional rearrangement of proteins in the tail of bacteriophage T4 on infection of its host. Cell 118:419–429

Guasch A, Pous J, Ibarra B, Gomis-Rüth FX, Valpuesta JM, Sousa N, Carrascosa JL, Coll M (2002) Detailed architecture of a DNA translocating machine: the high-resolution structure of the bacteriophages phi29 connector particle. J Mol Biol 315:663–676 The best example of proximodistal and cephalocaudal pattern is seen in the development of a foetus during pregnancy. As the foetus grows, it follows a certain pattern of development, beginning with the head and working its way down to the toes. The spinal cord develops first, followed by internal organs, arms, legs and other extremities such as fingers and toes. This is an example of both proximodistal (starting from the center and working outward) and cephalocaudal (starting from the head and working downward) patterns of development. The Cephalocaudal and Proximodistal Principles Sborgi L, Verma A, Muñoz V, de Alba E (2011) Revisiting the NMR structure of the ultrafast downhill folding protein gpW from bacteriophage λ. PLoS One 6:e26409 Maxwell KL, Davidson AR, Murialdo H, Gold M (2000) Thermodynamic and functional characterization of protein W from bacteriophage lambda. The three C-terminal residues are critical for activity. J Biol Chem 275:18879–18886