Molecular Motors in the Cell

Abstract

Many molecules found in living organisms can bind ATP, and use its energy to perform mechanical actions such as bending, twisting, rotating. In some special proteins such an action can be performed cyclically, as the same molecule can use ATP units at regular intervals, to repeat continuously its mechanical action. If this may not appear at all surprising from a purely chemical perspective, being just one more case of enzymatic chain reaction, it becomes a fascinating subject when seen under an engineering perspective. In fact, such molecules are nothing less than true molecular-scale motors. Dozens of different motor proteins exist in every eukaryotic cell to perform the most diverse functions, and prokaryotic cells also have their share, by employing sophisticated rotating or flapping molecular structures, in their swimming movements.

Appendices

Appendix E: The Cytoskeleton

From a structural point of view, an eukaryote cell is substantially different from the simple picture of a lipid bag filled with water and proteins (and other stuff), because of the presence of a scaffolding structure, the cytoskeleton. The cell cytoskeleton is the dynamically organised ensemble of biological polymers that impart the system the essential component of its mechanical properties. (In Chap. 8, the mechanics of filaments and membrane structures will be separately treated.) The different components of the cytoskeleton are capable of actuating internal forces and responding to the application of external forces, not just opposing a ‘passive’ resistance to the cell deformation. Because of its name, it is tempting to attribute to the cytoskeleton a similar role to that of the skeleton in superior animals; however, the cytoskeleton provides the cell, at the same time, both its internal structure and its force actuating system, thereby resembling at an organ that sums in one all the functions of the bones and the muscles (Fig. 6.20).

Fig. 6.20

Scheme of actin filament structures. G-actin (globular actin) monomers with bound ATP can polymerize, to form F-actin (filamentous actin). They are shown in different colors to highlight the helical winding of the two strings. F-actin may hydrolyze its bound ATP to ADP + Pi and release Pi. However, ADP release from the filament does not occur because the cleft opening is blocked. With the help of many species of actin-binding proteins, F filaments can quickly assemble and disassemble into superstructures, such as actin parallel bundles (above, by binding e.g. $\alpha$-actinin, villin, fimbrin), or actin networks (below, by binding filamin dimers)

Fig. 6.21

Scheme of microtubule structure and growth. $\alpha$- and $\beta$-tubulin monomers combine to form heterodimers; many such heterodimers assemble to form a single microtubule. Thirteen linear protofilaments, made of vertically-stacked tubulin heterodimers, are arranged side-by-side to form a hollow, cylindrical microtubule. The plus end (top) is the faster growing end of the microtubule, and heterodimers are arranged with the $\beta$-tubulin monomers facing the plus end. To the left, a cross-section through the microtubule, with diameter of about 25 nm. At the plus end of the microtubule, some tubulin heterodimers are shown while attaching to the microtubule (polymerisation). These, and the dimers already present at the + end, are bound to guanosine tri-phosphate (GTP, red-green spheres). Over time, GTP loses a phosphate and becomes G-diphosphate (GDP, orange-green spheres)

The cytoskeleton structure and components are quite similar in all eukaryote cells (from uni- to pluricellular organisms), although important differences exists between animal and plant cells. It is constituted of different types of long-chain molecules, or polymers, sometimes called fibres because of their sizeable length compared to the cell scale. These are usually classified into three categories (Fig. 6.21):

• Actin filaments, or F-actin, is formed by monomers of the actin protein (G-actin, of which several variants are known). The long F-actin polymers are twisted in pairs, which then assemble in bundles of variable thickness. Actin is ubiquitous in cells, and is mostly important in muscle cells (myocytes), where it couples with myosin in the sarcomeres. F-actin have a diameter of 6.5–7 nm, a contour length ranging from very small up to a fraction of the cell diameter, and a persistence length (see Chap. 7) $\lambda _p \sim 17 \upmu$m: therefore they are semi-flexible, since their average length is comparable to their persistence length. They have an orientation, due to the asymmetry of the actin monomer (which moreover changes shapes according to whether ATP or ADP is bound), and to their helical self-assembly in which monomers are attached head-to-tail. In particular, this leads to the fact that one of the filament ends (termed +) can polymerise faster than the other end (–). If the length of individual F-actin is generally between 2 and 3 $\upmu$m, however they are usually assembled in tight bundles, whose overall length is rather in the 10–20 $\upmu$m range.

• Intermediate filaments. These are the less dynamic components of the cytoskeleton, little known yet but undergoing intense research. In the variant of lamin, they are most important in the structure of the cell nucleus. Intermediate filaments are assembled from a family of related proteins, which share many common features. The definition of ‘intermediate’ comes from their average diameter of 8–10 nm, which is in-between that of F-actin and microtubules. While most abundant in epithelial and neuronal cells, they are however observed in almost all animal cell types.

• Microtubules, the stiffest constituents of the cytoskeleton. Their persistence length is of several mm, largely beyond the cell size, which therefore limits their flexibility. Their diameter is typically $\sim$25 nm, but in some cases can be smaller. The largest contribution to their rigidity comes from the special arrangement of parallel proto-filaments in a hollow tubular structure, since the bending modulus $\kappa _b$ (see greybox on p. 322) grows as the fourth power of the diameter. Compared to the densely twisted F-actin, the gain in stiffness is by a factor of about 80–100. Microtubules are polarised, like actin filaments, but their biochemistry is different. In particular, it is known that a dynamic instability can lead to a very sudden shortening of microtubules, thus originating a large impulsive force (Fig. 6.22).

Fig. 6.22

Optical microscopy images of animal cells (left) and plant cells (right). Different cell elements are stained with fluorescent labels, to visualise the cytoskeleton. In the left image, microtubules are shown in green, actin filaments in red, and the nucleus in blue. Note that microtubules constitute a main scaffold architecture, while actins are in this case concentrated in dense networks just below the cell membrane. In the right image, actin networks are in green, and plastids in red. [Image $\copyright$ left M. Shipman, J. Blyth and L. Cramer, University College London; right E. Blancaflor, The S.R. Noble Foundation, [15]. Repr. w. kind permission from the authors.]

Polymers (see again Chap. 7) can be organised into fibres, bundles or networks, according to the function they are performing. Sometimes, such as for the case of actin, the same polymer fibres can auto-organise into any of the different structures, by dynamically rearranging their configuration, or by breaking and reforming their bonds (Fig. 6.23). Such an outstanding level of organisation is also made possible with the help of a large number of auxiliary proteins:

• Crosslinking proteins. The denomination comes from the physics of polymers (see Chap. 7), where some molecular components can be added to induce cross bridges between long fibres, thereby completely modifying the physical properties of the material (such as adding sulphur to rubber mixtures in the process of vulcanization). This is what happens, in an even more spectacular way, in the cytoskeleton. Most crosslinkers are controlled cyclically by a regulatory network of other proteins, thus allowing a rapid reorganisation of the cytoskeleton.

• Branching proteins, sometimes considered as a special case of the crosslinking proteins, as the name explains these permit the branching out of lateral chains from a central one. They are important for actin filaments.

• Capping and severing proteins, necessary to regulate and eventually arrest the polymerisation rate of the polymer filaments at their extremities.

• Anchor proteins, serve the purpose of transmitting the forces at the interface between cytoskeleton filaments and cell membrane, and participate in cell adhesion. They can attach to integral membrane proteins, or penetrate in the lipid bilayer. Their attachment is reversible, i.e. they can be detached and reconnected at a different place in the cell.

Fig. 6.23

Some possible organisation of actin filaments in the cell. a A migrating cell with a lamellipodium extending in the direction of cell migration, terminal focal adhesions (yellow spots), a meshwork of actin filaments (red stripes next to the lamellipodium membrane), and actin stress fibres (thick, long red stripes). b Isolated, adherent cell with stable focal adhesions and thick stress fibres. c Adherent cell from a dense tissue, with stable focal adhesions, a few thin stress fibres, and a dense meshwork of cortical actin filaments. The grey mass indicates the cell nucleus

The operative keyword of the cytoskeleton structure (as well as for other cell structures) is the concept of remodelling: all the components of the cytoskeleton systems are active structures, which change and evolve during the biological activity of the cell, with a continuous functional reorganisation of the elementary proteins, available as raw material in the cytoplasm.

Fig. 6.24

Microscopy imaging of the localisation of MreB (actin-like) and FtsZ (tubulin-like) proteins in E. coli bacteria. a The left column is a bright-field electron microscopy image of E. coli bacteria. In the right columns, fluorescence microscopy images. Green indicates MreB fluorescence; red indicates anti-FtsZ immunofluorescence; the rightmost image shows the superposition of the two separate images (the scale bar on the left is 1 $\upmu$m). The equipartition of the MreB cytoskeleton into splitting cells is triggered by the membrane association of the FtsZ protein in the cell equatorial plane, and is eventually accomplished by division and segregation of the MreB array. This process ensures that each daughter cell, after splitting at the region indicated by the red FtsZ, inherits one copy of the MreB cytoskeleton. b MreB (purple) has long been thought of as a spiral filament twisting along the cell length, to control cell shape. Likewise, FtsZ proto-filaments (blue) were once thought to wrap around the cell midpoint to organise the division. c Recent work using high-resolution microscopy has revealed that long cytoskeletal filaments are more likely to be short patches of polymers. [Image a from Ref. [16], b, c from Ref. [17], repr. w. permission.]

The cytoskeleton is linked to the cell membrane by specialised proteins, which cluster into focal adhesions, to some extent analog to “feet” allowing the cell to sense and measure the relative stiffness of the surrounding environment (other cell walls, the extracellular matrix, a foreign surface). In response to the measured force, the cytoskeleton can react and rearrange, to adapt the cell shape, to make the cell adherent to (or detach from) the substrate, or follow the commands to activate cell displacement. The cytoskeleton elements also actively participate to the complex process of cell duplication (mitosis), by providing the necessary mechanical forces for chromosome separation and membrane splitting (see Sect. 7.5).

Prokaryote cells seem to lack a properly organised cytoskeleton. This is one of the major differences between nuclear and non-nuclear cells, aside of the different organisation of their respective DNA, the general absence of internal membranes, and the very different sequence of cell reproduction. However, analogues for all major eukaryote cytoskeletal proteins have been found in prokaryotes, and scaffolding structures with similar cytoskeletal function, despite a largely different shape organisation, are being discovered (Fig. 6.24). The functional equivalent of actin contractile rings is thought to be the protein FtsZ, participating in cell division, however from a structural point of view this protein resembles tubulin; by contrast, a protein like ParM resembles actin in its structure, but performs functions resembling those of tubulin in eukaryotes; the structural analog of actin, albeit with a completely different geometric distribution, should be the MreB protein, a polymerisable long filament, running in a coil-like shape about the bacterial membrane; even some analog of the intermediate filaments have been identified, such as the crescentin (CreS) family, which share similarities with both the eukaryotic cytokeratin and lamin-A. Moreover, ‘structural’ proteins with no known eukaryotic homologues have also been discovered. Such polymerisable molecules play essential roles in prokaryote cell division, protection, shape determination, and polarity determination.

Problems

6.1

Swimming bacterium

Consider a bacterium with spherical shape, and radius $R=15\,\upmu$m, swimming in pure water (viscosity $\eta =10$ $^{-3}$ kg/(m s)), with a speed $v_0 = 0.3$ cm/s at time $t=0$. Calculate the stopping distance $t > 0$, by taking a mass $m=\rho V$ for the bacterium.

6.2

Actin polymerisation velocity

Calculate the velocity of polymerisation of a filament of F-actin, formed by monomers of G-actin of average size $\delta =5$ nm, with reaction constants $k_+=7.50$ ($\upmu$M s)$^{-1}$, $k_-=1.25$ s$^{-1}$, in two different solutions of G-actin, with concentration 0.1 and 0.5 $\upmu$M. Comment on the difference between the two cases.

6.3

Chain polymerisation

Let us examine a free-radical addition polymerisation with $k_i = 5.0 \times 10^{-5}$ s$^{-1}$, $u=0.5$, $k_t = 2 \times 10^7$ dm$^3$ mol$^{-1}$ s$^{-1}$, and $k_p = 2640$ dm$^3$ mol$^{-1}$ s$^{-1}$, and with initial concentrations [M] = 2.0 M and $[I] = 8 \times 10^{-3}$ M. Assume the chain termination occurs by combination. Calculate: (a) The steady-state concentration of free radicals. (b) The average kinetic length of the chain. (c) The production rate of the polymer.

6.4

Microtubules association/dissociation constants

In your lab you have only a centrifuge and an UV-absorption spectrometer, and you can play with a solution of tubulin in physiologic medium (pure water with 0.15 M NaCl). With these two instruments, you should design an experiment to measure the association constant of microtubules. (You can do an internet search for additional parameters needed in this case, check for example the websites www.rcsb.org/pdb/ and www.web.expasy.org/protparam/, to obtain some quantity relevant to your problem such as the sequence and the extinction coefficient of the protein tubulin .)

6.5

DNA replication

The mechanisms of DNA replication are very similar in both prokaryotes and eukaryotes, proceeding at the rate of up to 1,000 nucleotides per second in the former, while being slower (50–100 nucl/s) in the latter. Consider the replication of DNA in the E. coli bacterium, which occurs about every 30 min. The replication fork is a structure created by enzyme helicase, which breaks the hydrogen bonds holding the two DNA strands together. The two strands of the DNA open up, and are used as templates by the enzyme polymerase for making the two identical copies. In this action, helicase makes the DNA to turn, with the help of other specialised proteins (the topoisomerases). (a) How much time is needed to make two complete copies of the bacterial DNA? What this tells us about the replication mechanism in prokaryotes? And what about eukaryotes? (b) How fast does the template DNA spins? (c) What is the velocity of a DNA-polymerase-III relative to the template?

6.6

Active and passive diffusion

A membrane with thickness L separates two volumes of fluid $V_A$ and $V_B$, in each of which a constant concentration $c_A$ and $c_B$ of some protein X is maintained. Describe the profile of concentration c(x) inside the membrane. What this has to do with the phenomenon of Brownian motion?

Subsequently, a pressure difference is applied between A and B, which establishes a flux across the membrane at constant velocity. What is the physical coefficient characterising the passage of the protein across the membrane in this condition? What are its physical dimensions? Is Brownian motion playing the same role as before?

6.7

Michaelis-Menten kinetics

The enzyme carbonic anhydrase catalyses the hydration of CO$_2$ in red blood cells to give bicarbonate ion:

$$\begin{aligned} \text {CO}_2 + \text {H}_2\text {O} \rightarrow \text {HCO}_3^- + \text {H}^+ \end{aligned}$$

CO$_2$ is converted to bicarbonate ion, which is transported in the bloodstream and converted back to CO$_2$ in the lungs, a reaction that can be catalyzed by carbonic anhydrase. In an experiment, the following data were obtained for the reaction at pH = 7.1, $T=273.5$ K, and anhydrase enzyme concentration of 2.3 nmol L$^{-1}$:

$$ \begin{array}{ l c c c c } [\mathrm{CO}_2]/(\mathrm{mmol}\,\mathrm{L}^{-1}) &{} 1.25 &{} 2.5 &{} 5.0 &{} 20.0\\ \\ v_P/(\mathrm{mol}\,\mathrm{L}^{-1}s^{-1}) &{} 2.78 \times 10^{-5} &{} 5.02 \times 10^{-5} &{} 8.33 \times 10^{-5} &{} 1.67 \times 10^{-4} \\ \end{array} $$

Determine the catalytic efficiency $e_P$ of the enzyme carbonic anhydrase at 273.5 K.

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(**) The terms of the Creative Commons Attribution 3.0 and 4.0 International License (http://creativecommons.org/licenses/by/3.0/, http://creativecommons.org/licenses/by/4.0/) permit use, duplication, adaptation, distribution and reproduction in any medium or format, as long as appropriate credit is given to the original author(s) and the source, providing a link to the Creative Commons license and indicating if changes were made.