Article Information

• PDF (983 kb)
• Full Text with Large Figures
• Supporting Material
• Cited by in Scopus (3)

PubMed

• Articles by Fabiana Diotallevi
• Articles by Bela Mulder

Related Articles

• Total Internal Reflection with Fluorescence Correlation Spec...

  Total Internal Reflection with Fluorescence Correlation Spectroscopy: Nonfluorescent Competitors Biophysical Journal, Volume 87, Issue 2, 1 August 2004, Pages 1268-1278 Alena M. Lieto and Nancy L. Thompson Abstract Total internal reflection with fluorescence correlation spectroscopy is a method for measuring the surface association/dissociation rate constants and absolute densities of fluorescent molecules at the interface of a planar substrate and solution. This method can also report the apparent diffusion coefficient and absolute concentration of fluorescent molecules very close to the surface. Theoretical expressions for the fluorescence fluctuation autocorrelation function when both surface association/dissociation kinetics and diffusion through the evanescent wave, in solution, contribute to the fluorescence fluctuations have been published previously. In the work described here, the nature of the autocorrelation function when both surface association/dissociation kinetics and diffusion through the evanescent wave contribute to the fluorescence fluctuations, and when fluorescent and nonfluorescent molecules compete for surface binding sites, is described. The autocorrelation function depends in general on the kinetic association and dissociation rate constants of the fluorescent and nonfluorescent molecules, the surface site density, the concentrations of fluorescent and nonfluorescent molecules in solution, the solution diffusion coefficients of the two chemical species, the depth of the evanescent field, and the size of the observed area on the surface. Both general and approximate expressions are presented. Abstract | Full Text | PDF (224 kb)

• Dynamics of Cellular Focal Adhesions on Deformable Substrate...

  Dynamics of Cellular Focal Adhesions on Deformable Substrates: Consequences for Cell Force Microscopy Biophysical Journal, Volume 95, Issue 2, 15 July 2008, Pages 527-539 Alice Nicolas, Achim Besser and Samuel A. Safran Abstract Cell focal adhesions are micrometer-sized aggregates of proteins that anchor the cell to the extracellular matrix. Within the cell, these adhesions are connected to the contractile, actin cytoskeleton; this allows the adhesions to transmit forces to the surrounding matrix and makes the adhesion assembly sensitive to the rigidity of their environment. In this article, we predict the dynamics of focal adhesions as a function of the rigidity of the substrate. We generalize previous theories and include the fact that the dynamics of proteins that adsorb to adhesions are also driven by their coupling to cell contractility and the deformation of the matrix. We predict that adhesions reach a finite size that is proportional to the elastic compliance of the substrate, on a timescale that also scales with the compliance: focal adhesions quickly reach a relatively small, steady-state size on soft materials. However, their apparent sliding is not sensitive to the rigidity of the substrate. We also suggest some experimental probes of these ideas and discuss the nature of information that can be extracted from cell force microscopy on deformable substrates. Abstract | Full Text | PDF (424 kb)

• Mapping the Energy Landscape of Biomolecules Using Single Mo...

  Mapping the Energy Landscape of Biomolecules Using Single Molecule Force Correlation Spectroscopy: Theory and Applications Biophysical Journal, Volume 90, Issue 11, 1 June 2006, Pages 3827-3841 V. Barsegov, D.K. Klimov and D. Thirumalai Abstract We present, to our knowledge, a new theory that takes internal dynamics of proteins into account to describe forced-unfolding and force-quench refolding in single molecule experiments. In the current experimental setup (using either atomic force microscopy or laser optical tweezers) the distribution of unfolding times, (), is measured by applying a constant stretching force from which the apparent -dependent unfolding rate is obtained. To describe the complexity of the underlying energy landscape requires additional probes that can incorporate the dynamics of tension propagation and relaxation of the polypeptide chain upon force quench. We introduce a theory of force correlation spectroscopy to map the parameters of the energy landscape of proteins. In force correlation spectroscopy, the joint distribution (, ) of folding and unfolding times is constructed by repeated application of cycles of stretching at constant separated by release periods during which the force is quenched to <. During the release period, the protein can collapse to a manifold of compact states or refold. We show that (, ) at various and values can be used to resolve the kinetics of unfolding as well as formation of native contacts. We also present methods to extract the parameters of the energy landscape using chain extension as the reaction coordinate and (, ). The theory and a wormlike chain model for the unfolded states allows us to obtain the persistence length and the -dependent relaxation time, giving us an estimate of collapse timescale at the single molecular level, in the coil states of the polypeptide chain. Thus, a more complete description of landscape of protein native interactions can be mapped out if unfolding time data are collected at several values of and . We illustrate the utility of the proposed formalism by analyzing simulations of unfolding-refolding trajectories of a coarse-grained protein (1) with -sheet architecture for several values of , , and =0. The simulations of stretch-relax trajectories are used to map many of the parameters that characterize the energy landscape of 1. Abstract | Full Text | PDF (490 kb)

• …more

Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 92, Issue 8, 2666-2673, 15 April 2007

doi:10.1529/biophysj.106.099473

Biophysical Theory and Modeling

The Cellulose Synthase Complex: A Polymerization Driven Supramolecular Motor

Fabiana Diotallevi and Bela Mulder^,  

FOM Institute for Atomic and Molecular Physics (AMOLF), 1098 SJ Amsterdam, The Netherlands

Address reprint requests to Bela Mulder, Tel.: 31-(0)20-608-1234; Fax: 31-(0)20-668-4106.

A distinctive feature of the cells of higher plants is the cell wall, an extracellular assembly that acts like an external skeleton. Among other things, it allows the cell to support a sizeable internal osmotic pressure, a prerequisite for withstanding the pull of gravity. The cell wall derives its robust mechanical properties from its ingenious construction: it consists of stacks of thin lamellae 1, all deposited parallel to the plasma membrane, that are formed by long parallel almost purely crystalline cellulose microfibrils (CMFs) embedded in a matrix of polysaccharide “packing” material 2,3. Its ubiquitous presence within plant cell walls makes cellulose the most abundant polymeric material in the biosphere. Despite its vital importance both in plant cell function and as a raw material, the primary event of the biosynthesis of cellulose is still only partially understood.

In vascular plants the CMFs are synthesized by a transmembrane protein complex, which we will call the cellulose synthase complex (CSC)^{1 1} We prefer this structurally neutral term to the historical terms “terminal complex (TC)” and “particle rosette”, as they derive from earlier observations of parts of the complex, but are often used in a pars pro toto fashion.

. Although already identified through electron micrograph microscopy for several decades, the definite biochemical proof that these structures indeed are the location of the cellulose synthases, was only provided in 1999 4. Current estimates of the diameter of the CSC on the endoplasmatic of the plasma membrane are in the range of 40–60nm 5, making the CSC one of the largest protein complexes so far observed. Electron micrograph images of freeze fracture preparations of the plasmatic face of plant plasma membranes reveal a characteristic structure of six hexagonally arranged particles with a diameter of ∼8nm forming a ring, or “rosette” 6,7, which has a diameter of ∼25nm (Fig. 1). The current view is that each of the six lobes of the rosette in turn consists of six cellulose synthases that each polymerizes a single glucan chain using UDP-glucose as a substrate 8. These individual chains are then assembled into the CMF, which by implication consists of 6×6=36 chains, consistent with the known crystal structure and the measured cross section of ∼3.5nm 8,9.

Display large version of this figure

Figure 1

Electron micrographs from freeze-fracture preparations of plant cell walls showing a so called terminal-complex, the imprint of a CSC, with an attached CMF in the exoplasmatic face of the plasma membrane (left panel), a so-called particle rosette, the outward facing side of the CSC, within a characteristic depression of the plasma membrane (middle panel, scale bars are 100nm, both images courtesy A. M. C. Emons) and a close-up of the particle rosette with its typical six-sided symmetry (right panel, scale bar is 10nm, image courtesy C. Haigler).

The cell wall is deposited from the inside out, with all the relevant materials delivered through exocytosis of Golgi vesicles. As the CSC is bound to the membrane, the deposition of new CMFs thus takes place in the limited space between the outer surface of the fluid plasma membrane and the earlier deposited rigid cell wall. For this process to work, it had to be assumed that the CSC would have to move in the plane of the membrane 10 leaving behind a CMF in its track. The latter hypothesis has now finally been confirmed by the direct real-time in-vivo observation of fluorescently labeled CSCs 11.

Although the idea that the CSC moves was widely accepted, the question of the origin of this movement has so far received less attention. Obvious candidates for the required force production are motor proteins, molecular chemical energy transducers that are involved in many different biological tasks 12,13. Examples are processive molecular motors such as kinesin, which can transport organelles and vesicles using cytoskeletal elements as tracks, or nonprocessive motors such as myosins that deliver the power strokes for muscle contraction, both using ATP as fuel. Indeed, one of the early theories 14 assumed the CSC to be linked by a motor protein to a cortical microtubule, which then acted as a rail to guide the motion. Another proposal 15 had the cortical microtubules act as force producers themselves, which by setting up a shear flow in the membrane, provide a motive force to the CSC. Later, it was realized that in principle the energy released by the glucose polymerization process could by itself be sufficient to propel the CSC 16. In addition, it was shown that preventing the proper crystallization of the CMF by treatment of cells with the drug Calcofluor led to a thickening of the cell wall, suggesting a dysfunctional dispersion of the CSC along the membrane 17. This observation clearly correlates the movement of the CSC with the polymerization and crystallization processes of the CMFs. To date, however, a detailed mechanistic explanation of how the motion of the CSC is achieved was lacking.

Here we develop an explicit biophysical model of CSC motility. We show that the concept of a Brownian polymerization ratchet, originally proposed by Peskin et al. to explain force production by growing polymeric filaments such as microtubules 13,18, can serve as a basis for describing CSC propulsion. However, we argue that to obtain a full understanding also requires taking into account the geometry of the deposition process, the additional driving force provided by the crystallization of the cellulose and the role of the elastic energy stored in the nascent microfibril as well as in the deformable plasma membrane. To achieve our aim, we develop our approach in three steps: First, we formulate a model that integrates the relevant physical components to obtain a heuristic explanation for the propulsion process. In the next step, we implement this model in a stochastic simulation, providing a proof of principle of the proposed mechanism. In the final step, we simplify the model into a form that allows analytical predictions to be made and show that we can reproduce the experimentally measured value for the CSC speed.

The Monte Carlo scheme used in the stochastic simulation consists of a series of stochastic transitions between different system configurations, all satisfying the imposed constraints. The probability of a given transition depends on the energy difference between the two successive configurations and satisfies the detailed balance condition that ensures the correct sampling of the equilibrium phase space. In the following, even though the simulated systems are discretized for numerical purposes, we will express their Hamiltonians in the continuum limit.

In the simulation, aggregates of six glucan chains are represented by a single effective filament that is modeled as a discrete linear chain of N_f spherical beads of diameter σ that are rigidly connected by bonds of fixed length δ=σ. The length of each chain is thus where the subscript i=(1..6) refers to the ith filament. The angular bending potential between two subsequent bonds in the chain is given by U(θ)=J_f[1–cos(θ)], where θ is the angle and J_f is the bending constant that determines the stiffness of the filament. The first monomer-monomer bond (tip) of each filament is constrained to be oriented along the vertical axis of the laboratory frame and the last monomer-monomer bond (tail) is parallel to the horizontal axis. The tips of the filaments are moreover constrained to be located at the vertices of a regular hexagon, with edge-length R_hex=6σ. No part of any of the filaments is allowed to occupy the space above the rigid wall located at z=0. The individual filaments can be described by inextensible space curves r(s), where s is the arc-length parameter. Because of the inextensibility, so that the local curvature of a filament is given by In the absence of the constraints, the Hamiltonian for the full bundle of 6 filaments is then given by

(1)

The liquid bilayer forming the plasma membrane is modeled as a l×l grid of point particles of size σ that are capable of moving only in the vertical direction. The edges of the membrane are kept fixed at z=0. The Hamiltonian for the membrane is described by the Helfrich functional 19

(2)

In the analytical version of our model, we consider the whole CMF as a single chain whose configurations are constrained to lie in a vertical plane. The chain is modeled as a semiflexible filament with a persistence length ξ_f significantly larger than the typical dimensions of the region where the filament is bent. Again we have the constraint that the tail of the filament is horizontal and the tip vertical, and that no part of the filament can penetrate the wall at z=0. To minimize its elastic energy under the given constraints, the equilibrium shape of the filament will be given by a quarter arc of circle of length with energy E_f(z)=πJ_f/4z_f, where z_f is the distance from the filament tip to the wall. (The geometry of this situation is schematically depicted in Fig. 4.) The force the tip of the filament exerts on the supporting membrane thus is

(3)

The partition function is then given by

(4)

(5)

(6)

(7)

(8)

To study the fluctuations around this equilibrium position, and assuming that these are small with respect to z_f itself, we linearize the force F_f around z_eq yielding

(9)

To determine the velocity of polymerization (see Eq. (14)), we need to evaluate the probability that a gap is opened between the filament tip and the membrane midpoint larger than the monomer size δ. Note that, after linearization of the filament force, both the filament tip and the membrane midpoint can be considered as harmonic oscillators, which are coupled by the requirement that the filament tip is always above the membrane since the two cannot interpenetrate, i.e., that z_m–z_f≥0. Recalling that the probability distribution for a one-dimensional harmonic oscillator in a thermal bath is given by

(10)

(11)

(12)

(13)

The mechanical cycle that we propose is responsible for CSC propulsion is illustrated in a schematic fashion in Fig. 2. We model the CSC as a planar, membrane-bound object. On the side of the object facing away from the cell a regular array of cellulose polymers is extruded. We model these polymers as inextensible semiflexible chains of beads. The configuration of these polymers is constrained by three factors. The first is their attachment to the CSC itself. Here we assume that this attachment not only fixes the location of the polymer tips, but also specifies the orientation of their first bonds to be perpendicular to the plane of the CSC. The latter assumption is consistent with the hypothesis that the chains emerge from narrow channels in the complex. The second constraining factor is the confining influence of the already deposited cell wall, which we model as an impenetrable barrier. The final constraint arises from the fact that the polymers are at their other ends all linked up into a nascent CMF, which on this scale is an effectively rigid linear structure constrained to lie in the plane of the membrane. Because the polymers have a finite resistance to bending, the combination of geometrical constraints imposed on them implies that they are in a nonrelaxed conformation, resulting in forces acting on their attachment points. At the loci where the polymers emerge from the CSC, these forces will typically have both a perpendicular component and an in-plane component. The perpendicular component has two effects: i), it acts as a barrier for the polymerization process, thus influencing the rate of addition of new monomers, and ii), it contributes to a net force that pushes the CSC downward, counteracted by an elastic response of the membrane. The latter effect is consistent with the membrane-indentations that are seen in some freeze-fracture images of the CSC 22 (see also Fig. 1). Note that we disregard the possibility that the CSC “tilts” with respect to the global membrane orientation, as the energetic cost of such short length-scale deformations of the membrane is probably appreciable.

Display large version of this figure

Figure 2

Snapshots at four different time points in a stochastic simulation of the motion of a six filament CSC. Note the rotation of the complex, induced by the helical nature of the crystalline arrangement in the CMF.

The result of all the in-plane force components due to the individual polymers is the net force that drives the linear motion of the CSC. Energy is injected into this system by the polymerization as well as the crystallization process, as both will tend to increase the stress in the polymers. The energy is dissipated by the work the CSC as a whole performs against the frictional forces it experiences. We stress the fact that thermal fluctuations, which are a dominant effect at the relevant molecular scale, play a crucial role in the whole process. It is these fluctuations that allow the system to cross (and also to recross) the energetic barriers associated with the mechanical constraints imposed on the polymerization process. In fact, it is the rectification of these fluctuations that allow the system to convert chemical energy into directed motion.

We now implement the conceptual model presented above in terms of a stochastic simulation. For simplicity, we consider a CSC producing six effective filaments (EFs), each representing six cellulose chains. This simplification is consistent with the mechanism proposed by Cousins et al. 23 in which the ∼36 cellulose strands that emerge from the CSC are first assembled into six glucan chain aggregates, which subsequently coalesce crystallizing into a CMF. The EFs are modeled as bead chains with a bending potential governing the relative orientation of pairs of neighboring bonds. The beads on different chains have a short-range attractive interaction allowing them to crystallize into a compact arrangement. The already extant cell wall is taken to be a rigid wall, into which the beads are not allowed to enter. The EFs emerge from six hexagonally arranged channels representing the CSC, with their first bond constrained to be perpendicular to the plane of the CSC. This whole construct interacts with a fluid membrane modeled as a dynamically reconfigurable squared network of beads and springs. The tips of the EFs cannot penetrate the membrane, thus coupling the EFs energetically to the membrane. Starting from an initial condition in which the end of one of the chains is clamped, the simulation now proceeds as follows. An attempt is made to move one of the particles in the system (either a chain or a membrane bead). The proposed movement is accepted with a probability proportional to the Boltzmann weight of the associated change in energy of the whole system (including the energies associated with various constraints). This procedure is then repeated for several sweeps over all the particles in the system. This standard Metropolis Monte Carlo scheme allows the system to equilibrate its state to the current lengths of the individuals EFs. After this equilibration step, the gap between the tips of the EFs and the membrane is monitored for all the EFs. If this gap is larger than the size of a chain bead, a new bead is added at the tip, preserving the perpendicular orientation of the first bond. The system is then allowed to equilibrate again, and the whole cycle is repeated. The justification for this procedure is found in the large separation in timescale between the molecular relaxation mechanisms and the rate at which the polymerization process progresses, which allows one to treat the system in a quasi-stationary manner. Note that this algorithm is therefore a microscopic implementation of a Brownian ratchet 18, in the case that the fluctuations are fast compared to the polymerization rate. The full details of the simulation were discussed in the Materials and Methods section.

The results of our simulation show that after initial effects have died down, a stationary regime is reached in which the CSC moves with a statistically stationary velocity in a direction dictated by the essentially straight CMF produced. This shows that indeed, the polymerization and crystallization processes, both exothermic, are coupled by thermal fluctuations to the membrane and the partially flexible chains as energy transducers, are sufficient to obtain the directed motion of the synthase.

Fig. 3 shows four snapshots of our simulation at successive times (a short movie is available in the Supplementary Material ). We note that although the trajectory of the CSC is approximately linear, the complex itself undergoes a rotation during its motion. This is caused by the helical nature of the crystalline structure of our pseudo-CMF, which is a natural consequence of the maximization of the binding energy between the spherical monomers. Intriguingly, cellulose microfibrils have also been observed experimentally to “twist” 24,25, an effect generally attributed to the chirality of the planar glucan chains, which spontaneously “twist” to relieve the strain built around the oxygen bridge that connects the successive glucan units together. Clear evidence of this phenomenon is provided by the twisted cellulose ribbon produced by Acetobacter, the so-called vinegar bacterium that lives at water-air interfaces and propels itself by forces derived from cellulose polymerization. Interestingly, moving Acetobacter cells undergo a continuous rotation about their longitudinal axis: again, this is believed to be caused by the relaxation of the torques generated by the crystallization during the biogenesis of the CMFs 26. Also clearly visible in the side views of Fig. 3 is the marked indentation of the membrane at the locus of CSC, over an area several times the area of the complex itself. This indentation is a consequence of the forces generated by the bent EFs, and has been observed experimentally (see Fig. 1 and Emons 22), and as such provides direct evidence in favor of our model.

Display large version of this figure

Figure 3

Schematic representation of the mechanical cycle in our model of CSC propulsion, illustrated for the case of a single CMF. (From top to bottom) Step 1: the filament and the membrane are in thermal equilibrium. Step 2: due to fluctuations of the filament and/or the membrane, a gap of sufficient size is opened allowing a new monomer to be added to the filament. Step 3: the increase in length of the filament causes an accumulation of elastic energy in the system, which generates a force on the tip of the filament. Step 4: the accumulated energy is released in a unidirectional motion of the tip of the filament, which is attached to the CSC. Step 5: the filament has effectively advanced by one monomer unit δ and equilibrium is restored, allowing the cycle to repeat itself.

Although the simulation presented above is able to illustrate the mechanism we propose, its inherent complexity nevertheless impedes a fully quantitative analysis of how the different factors work together to produce the outcome: directed motion of the synthase at a given average speed. We therefore undertake to strip the model to its bare essentials, focusing on a single growing polymer constrained to a two-dimensional planar geometry interacting with a three-dimensional elastic membrane. In this simplified setting, whose projection on a vertical plane is already illustrated in Fig. 2, the model can be solved exactly, allowing the polymerization velocity to be determined. The details of the full calculation were presented in the Materials and Methods section.

In the following, we neglect the thickness of the polymer and the membrane. The membrane is fixed at its edges to a rigid frame of size Ω=l². The equilibrium position of the membrane is taken to coincide with the hard top wall that represents the already extant cell wall. We neglect the spatial extent of the CSC, which is now simply represented by the constraint on the verticality of the first bond of the polymer. We first investigate the equilibrium configuration of the polymer. The active part of the filament can be considered as an elastic rod clamped at one end horizontally to a rigid part, representing the crystallized CMF, and vertically at the tip by the CSC. Such a rod adopts a quarter arc of circle configuration, whose length is πz_f/2, where z_f is the vertical distance of the filament tip to the wall. In the following, we will consider the polymer so stiff as to always maintain the definite shape of an arc of circle: thermal fluctuations have the only effect to modify its radius of curvature z_f. The force the filament exerts on the membrane at its tip is given by where J_f is the bending modulus of the filament. In response to this force, the membrane will deform, generating a counterforce on the tip of the polymer. This counterforce is given by F_m(z_m)=−2A γ z_m, where A=2π/[ log(1+γΩ/J_mπ²)], γ the membrane surface tension, J_i the membrane bending modulus, and z_m the vertical displacement from the equilibrium configuration of the membrane. Balancing these two opposing forces yields the equilibrium position Note that the force exerted by the membrane is already linear in the displacement z_m. We now also linearize the force exerted by the polymer around the equilibrium position, anticipating the fact that we will be concerned only with small fluctuations around it. This procedure maps our model conceptually onto a system of two linear springs acting in opposite directions, the polymer downward and the membrane upward, with the constraint that the tip of neither spring may pass the other, reflecting the fact that the polymer and the membrane cannot interpenetrate (Fig. 4). The dynamics of the growing polymer is governed by the balance between the rate of addition of new monomers and the rate of removal of monomers (assuming a reversible polymerization reaction). In the case that the timescale of the thermal fluctuations is fast compared to the polymerization kinetics, and under the common assumption that the effect of the applied force only influences the on-rate and not the off-rate, the polymerization speed is given by 18

(14)

(15)

Display large version of this figure

Figure 4

(a) Schematic cross-sectional drawing of the geometry used in the analytical model. (b) The effective model of two coupled springs the analytical model maps onto.

The model proposed for the mechanism of the CSC propulsion in this article achieves three goals. First of all, on a conceptual level it provides an explicit and physically consistent heuristic for understanding CSC motion. Secondly, our stochastic simulations, albeit simplified with respect to reality, provide a proof of principle of this mechanism. Finally, the analytical approach, which abstracts the model to its bare essentials, allows the observable result of the mechanism, i.e., the velocity of motion of the CSC, to be quantified in terms of the underlying physico-chemical parameters. Our fundamental assumption that the microscopic fluctuations occur on a timescale fast compared to that of the motion of the CSC justifies the use of the coarse graining that underlies the analytical approach, which replaces the many individual microscopic degrees of freedom, with a small number of effective ones. We can therefore use the results of the analytical approach in an attempt to estimate the velocity of the CSC. In this attempt we are of course limited by the availability of quantitative estimates of the relevant parameters.

We first consider the bending modulus of the effective filament, arguably the least well-determined parameter. Using crystallographic data, we can provide an upper and a lower bound to the value of J_f, as the Young’s modulus Y of cellulose is known to vary between 5 GPa for the amorphous state and 150 GPa for the crystalline state 27. We can, however, extract a more appropriative estimate from the depth of the observed indentation of the plasma membrane at the locus of the CSC, which provides an estimate for the equilibrium value of the filament tip to wall distance z_eq. Through the use of relation Eq. (8), we can then determine For typical values of the membrane surface tension γ=5×10⁻⁵ N/m, the membrane bending modulus J_m=2×10⁻²⁰ Nm, an indentation depth of z_eq=100nm, and size of the relevant membrane patch l ≈ 300nm, we find J_f=2.5×10⁻²⁵ Pa m⁴. Taking the radius of the effective filament to be r=1/2 diameter of the CMF=2nm, we obtain an estimate of Y=4J_f/(πr⁴)=20 Gpa, which falls squarely between the bounds mentioned above. We have to keep in mind that, even if the structure of the CMF is almost perfectly crystalline, at the moment of the extrusion from the CSC, the glucan chain aggregates are in a noncrystallized state. Thus the effective Young’s modulus of the aggregate is much lower that the one of a cellulose crystal.

For the polymerization rate, we use the value of free polymerization of cellulose achieved by the bacterium Acetobacter, 26,28. Strictly speaking, these experiments determine the net rate K_on–K_off, but we assume that the off rate for these almost irreversible chemical bonding processes is negligible. The final parameter necessary is the size of the monomer, which is equal to the size of a glucose subunit, making With these ingredients, we can now estimate the speed of the CSC to be This number compares favorably to the measured average speed observed by the Somerville group 11. We can also compare our results to the estimate of Hirai et al. 29, who observed calcofluor-stained CMFs growing from membrane fragments isolated from tobacco BY-2 protoplasts. Their estimated elongation rate of is higher than that observed in vivo, presumably due to the absence of the spatial constraint of an existing cell wall, which lowers the counterforce experienced by the polymerization process. Nevertheless, given the inevitable effects of friction with surrounding aqueous medium, this value is still lower than that which we would estimate for totally unobstructed deposition, in which case P(Z>δ)=1 and we obtain .

We contend that the biophysical model presented here provides a solid basis for understanding the propulsion of an individual CSC. Moreover, it gives an estimate for the polymerization velocity of the CMF that is consistent with the observed speed of the CSC within the uncertainty imposed by the approximations used. This opens the way for considering the much more challenging problem of understanding the full dynamics of cell wall deposition, which involves the simultaneous and apparently highly coordinated deposition of many CMFs. Indeed, the question of the origin of the highly regular CMF textures found in secondary cell walls is still actively debated. Although the textbook explanation involves the guidance of CSCs by ordered arrays of cortical microtubules 30, there is also a body of evidence against this idea, and a few models have been suggested that rely more strongly on the physical interactions between the CSCs and the CMFs they produce (see, e.g., Emons and Mulder 31 and Baskin 32). Clearly, all these approaches will benefit from a fuller understanding of the motive processes of the CSC in interaction with its direct environment.