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Originally published as Biophys J. BioFAST on October 6, 2006.
doi:10.1529/biophysj.106.087478
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Biophysical Journal 92:278-287 (2007)
© 2007 The Biophysical Society

Microtubule Protofilament Number Is Modulated in a Stepwise Fashion by the Charge Density of an Enveloping Layer

Uri Raviv * {dagger}, Toan Nguyen {ddagger}, Rouzbeh Ghafouri {ddagger}, Daniel J. Needleman * {dagger}, Youli Li * {dagger}, Herbert P. Miller {dagger}, Leslie Wilson {dagger}, Robijn F. Bruinsma {ddagger} and Cyrus R. Safinya * {dagger}

* Materials Department, Physics Department, and {dagger} Molecular, Cellular, and Developmental Biology Department, Biomolecular Science and Engineering Program, University of California, Santa Barbara, California; and {ddagger} Department of Physics and Astronomy, University of California, Los Angeles, California

Correspondence: Address reprint requests to Uri Raviv at his present address, Physical Chemistry Department, The Institute of Chemistry, The Hebrew University of Jerusalem, Givat Ram, 91904, Israel. Tel.: 972-2-6585325; Fax: 972-2-5660425; E-mail: raviv{at}chem.ch.huji.ac.il; or to Cyrus R. Safinya, Materials Research Laboratory, UCSB, Santa Barbara, CA 93106. Tel.: 805-893-8635; Fax: 805-893-7221; E-mail: safinya{at}mrl.ucsb.edu.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX: FORM FACTOR OF...
 ACKNOWLEDGEMENTS
 REFERENCES
 
Microtubules are able to adjust their protofilament (PF) number and, as a consequence, their dynamics and function, to the assembly conditions and presence of cofactors. However, the principle behind such variations is poorly understood. Using synchrotron x-ray scattering and transmission electron microscopy, we studied how charged membranes, which under certain conditions can envelop preassembled MTs, regulate the PF number of those MTs. We show that the mean PF number, <N>, is modulated primarily by the charge density of the membranes. <N> decreases in a stepwise fashion with increasing membrane charge density. <N> does not depend on the membrane-protein stoichiometry or the solution ionic strength. We studied the effect of taxol and found that <N> increases logarithmically with taxol/tubulin stoichiometry. We present a theoretical model, which by balancing the electrostatic and elastic interactions in the system accounts for the trends in our findings and reveals an effective MT bending stiffness of order 10–100 kBT/nm, associated with the observed changes in PF number.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX: FORM FACTOR OF...
 ACKNOWLEDGEMENTS
 REFERENCES
 
Microtubules (MTs) are anionic polymers that self-assemble from tubulin protein subunits into hollow cylinders. Tubulin dimers are arranged head to tail in protofilaments (PFs) that interact laterally and form the MT wall. In eukaryotic cells, a 13-PF arrangement is by far the most common (1Go), though MTs with 11, 12, 14, and 15 PFs have been observed (2Go). For example, it has been found (3Go) that the formation of MT with more than 13 PFs in the ciliate Nyctotherus ovalis Leidy is a highly ordered process. Such MTs are restricted to the nucleoplasm and, moreover, to later stages of nuclear division. They assemble during the anaphase of micronuclear mitosis and during the elongation phase of macronuclear division. About 85% of the MTs that form the large MT bundles assemble in Drosophila wing epidermal cells after the cells have lost their centrosomal MT-organizing centers composed of 15 PFs (4Go,5Go).

When MTs interact with MT-associated proteins or other cofactors they are able to adjust their structure dynamically and self-assemble into bundles and several alternative structures, which are critical components in a broad range of cell functions (6Go–18Go). Although it is well known that MTs are able to adjust their PF number, N, and, as a consequence, their dynamics and function, to assembly conditions such as pH, the presence of cofactors, drugs, and MT-associated proteins (3Go,8Go–10Go,19Go–24Go) or the number of successive disassembly-assembly cycles (2Go), the principle behind those variations is poorly understood. It is also unclear how the PF number is kept at 13 in cells at high fidelity (1Go,2Go).

In earlier articles (16Go,25Go), we studied the interactions between cationic liposomes and MTs. We established the conditions under which the cationic membranes can coat the MTs and form lipid-protein nanotubes (LPN). The LPNs exhibit a rather remarkable architecture, with the cylindrical lipid bilayer sandwiched between a MT and outer tubulin oligomers, forming rings or spirals (Fig. 1). The unique type of self-assembly arises because of a mismatch between the charge densities of the negatively charged MT and the cationic lipid bilayer.


Figure 1
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  		FIGURE 1  (A) A side-view cartoon of the LPN structure showing a microtubule made of tubulin protein subunits (red-blue-yellow-green objects) coated by a lipid bilayer (with yellow tails and green/white headgroups), which in turn is coated by a third layer of tubulin oligomers exposing the side that in MT is facing the lumen. (B) A top-view cartoon of the LPN structure.

 
Here, we study in detail, using small angle synchrotron x-ray diffraction (SAXRD) and transmission electron microscopy (TEM), how the mean PF number, <N>, of a preassembled MT is influenced by the tunable properties of an enveloping cationic membrane, which forms the LPNs. We show that the mean PF number, <N>, is modulated primarily by the charge density of the membrane, {sigma}. <N> decreases in a stepwise fashion with increasing {sigma}, toward the value of the uncoated MT, at high {sigma}. <N> does not depend on the membrane-protein stoichiometry or the solution ionic strength. We suggest that the LPN structure demonstrates that <N> and perhaps, as a consequence, MT dynamics, are determined by the attempt of the system to optimize the match between the charge density of the MT wall and that of the layer coating it, which in vivo would primarily consist of MT-associated proteins. Finally, we describe a quantitative physical model to account for our observations, from which we estimate that the effective bending stiffness associated with variation in PF number is of order 10 kBT/nm.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX: FORM FACTOR OF...
 ACKNOWLEDGEMENTS
 REFERENCES
 
Tubulin was purified from bovine brains as described elsewhere(14Go,26Go). Tubulin concentrated to 45 ± 5 µM in PEM buffer (50 mM 1,4-piperazinediethanesulfonic acid, 1 mM MgCl2, 1 mM EGTA, 0.02% (w/v) NaN3, adjusted to pH 6.8 with ~70 mM NaOH), 1 mM guanosine triphosphate (GTP), and 5% glycerol was incubated at 36 ± 1°C for 20 min, as described (14Go,16Go–18Go,26Go,27Go). Unless otherwise indicated, MT depolymerization was suppressed by adding the chemotherapy drug taxol at 1:1 tubulin/taxol molar ratio (20Go,28Go). Liposome solutions were prepared by mixing the cationic lipid, dioleoyl(C18:1) trimethyl ammonium propane (DOTAP) with the homologous neutral lipid, dioleoyl(C18:1) phosphatidylcholine (DOPC) (Avanti Polar Lipids), at a total lipid concentration of 30 mg/ml in Millipore water (18.2 M{Omega} cm), as described (29Go). The mole fraction of cationic lipids is given by

Formula 1(1)
where NCL and NNL are the numbers of cationic and neutral lipids, respectively. The relative cationic lipid/tubulin stoichiometry, RCL/T, is defined as

Formula 2(2)
where NT is the number of tubulin dimers. Lipid solutions were diluted so that equal volumes of preassembled MTs and liposome solutions could be mixed to yield the desired lipid/tubulin stoichiometry. The resulting complexes were characterized by SAXRD and TEM, as described (14Go–18Go). The following results are based on several different experiments, using different tubulin purification preparations and liposome solutions.

Samples were not oriented; thus, SAXRD scans collected on a 2D detector were azimuthally averaged to yield scattering intensity as a function of momentum transfer, q (Fig. 2, C and D). To model the data, as in other MT-related scattering studies (14Go,16Go–20Go,22Go,30Go), a series of power laws that pass through the minima of the scattering intensities was subtracted (Fig. 2 D). The assumption here is that the size distribution is very narrow within each sample.


Figure 2
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  		FIGURE 2  TEM images, SAXRD scans, and analysis of MTs and MTs complexed with DOTAP/DOPC membranes (see Materials and Methods). (A) TEM images of an MT. A whole-mount image is on the left side and a cross section is shown on the right. (B) TEM image of an LPN. The mole fraction of charged lipids, xCL {equiv} NCL/(NCL + NNL) = 0.5, and the cationic lipid/tubulin stoichiometry, RCL/T {equiv} NCL/NT = 120. NCL and NNL are the numbers of cationic and neutral lipids, respectively, and NT is the number of tubulin dimers. A whole mount image is on the left side and a cross section, showing an inner MT with 14 PFs, is on the right. We note that we did not perform a statistical study of such TEM cross section, as our x-ray data is a bulk measurement and inherently includes statistics. The vertical scale bar corresponds to 100 nm. (C) Azimuthally averaged raw SAXRD data (solid symbols) of MTs and LPNs with xCL = 0.4 and RCL/T = 40, as indicated in the figure. Each broken line is a series of power laws that pass through the minima of the scattering intensities. As in other MT-related scattering studies(14Go,16Go,20Go,22Go), this is the assumed background scattering. (D) SAXRD data from C, following background subtraction (open symbols). The blue solid curves are the fitted scattering models. (E) The variation of the radial electron density, {Delta}{rho}(r), relative to water (dotted lines), of MT and LPN walls, as obtained from fitting the data in C to models of isotropic infinitely-long hollow cylinders with nonuniform electron density profile. r is the distance from the center of the cylinders. The fraction of tubulin oligomer coverage at the external LPN wall relative to the internal MT wall, f, obtained from fitting the model to the data, is indicated in the figure. The inner radius, Rin, of the MT wall and that of the internal MT within the LPN complex, obtained from the fitting, are also indicated. (F) A schematic that represents a vertical cut through the LPN wall, corresponding to the top radial electron density profile in E. (G) A cartoon of the LPN. (H) A cartoon of a cross section of the LPN and a magnified slice.

 

   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 APPENDIX: FORM FACTOR OF...
 ACKNOWLEDGEMENTS
 REFERENCES
 
TEM images (Fig. 2 A) and SAXRD measurements (Fig. 2, C–E) performed on pure MT solutions are in agreement with earlier studies (14Go–18Go,20Go,22Go,31Go). The SAXRD profile of MTs is consistent with the form factor of an isotropic hollow cylinder (Fig. 2 D). Based on MT structural data (31Go,32Go), we modeled the MT as three concentric cylindrical shells of a high-electron-density region surrounded by two of low electron density, as shown in Fig. 2 E, keeping the total wall thickness, a1 = 4.9 nm, and mean electron density the same as those of MTs. The thickness and location of the high-electron-density region, within the MT wall, and the inner MT radius, Rin, are fitting parameters in this model (see Appendix for details).

TEM images (Fig. 2 B) reveal that when MTs were mixed with cationic liposomes, unique three-layered LPNs formed. The LPN consists of a MT that is coated by a lipid bilayer (it appears brighter in the images, as the ionic stain avoids the hydrophobic lipid tails), which in turn is coated by tubulin oligomers, made of curved PFs in helical arrangement with different pitches or stacks of rings (Fig. 1). The LPN appears to be the best the system can do to optimize its electrostatic interactions. The formation of tubulin oligomers at the external layer is enabled because the cationic membranes lead to MT depolymerization, resulting in curved PFs. By using a slowly-hydrolyzable GTP analog, GMPCPP, the formation of tubulin oligomers at the external layer of the LPN is prevented (U. Raviv, D. J. Needleman, Y. Li, H. P. Miller, L. Wilson, and C. R. Safinya, unpublished data). It is of interest to note that the kinetochore is believed to recognize and maintain its attachment to the plus-end of spindle MT by a similar three-layered tubular structure induced by MT-associated protein complexes (6Go,7Go). The protein rings that coat the MT allow the attachment of the kinetochore to the spindle MT, whereas the internal MT is able to maintain independently the dynamics required for cell division.

A typical SAXRD scan of the MT-lipid complexes is shown in Fig. 2 C. The broad oscillations are different from that of MTs and correspond to the form factor of the LPNs. To gain quantitative insight into the structure of the complexes, we analyzed the background-subtracted SAXRD data, shown in Fig. 2 D, by fitting to a model. We extended the isotropic concentric cylindrical shells model of MTs to include the second lipid bilayer and the third tubulin layer (Fig. 2, E and F). The radial electron density profile of the inner MT wall and outer tubulin monolayer are taken from the fit to the MT scattering data. The third tubulin layer is assumed to have the mirror image of the inner MT-wall radial electron-density profile, i.e., the PF side directed inward in the MT should be directed outward in the external tubulin layer (8Go) (Fig. 1). See Appendix for details. Apart from providing a good fit to the scattering data, the main supporting evidence for this assumption is the fact that we never found, even in the presence of excess lipids, subsequent external lipid bilayers or lipids inside the MT lumen, showing that both surfaces are similar and have low propensity to interact with cationic liposomes. The electron-density profiles of the lipid bilayer are taken from literature data (33Go,34Go). Using three different lipid solutions with different tail lengths (data not shown), we obtained the expected shifts in the form factor, indicating that we have identified correctly the location of the lipid bilayer. Finally, there are two free parameters in our model: The inner MT radius, Rin, which is allowed to fluctuate within physical reasonable limits and the fraction of tubulin coverage, f, at the external layer, relative to the inner MT wall, which is allowed to float freely between 0 and 1. The scattering model (Fig. 2 D) fits very well to the data.

We are able to control the charge density of the layer that coats the MT and this, based on our observations described below, is a key physical parameter. The membrane charge density, {sigma}, is set by the bilayer thickness, a2 ~ 4 nm, the area per lipid headgroup (29Go), A0 ~ 0.7 nm2, for both lipids, and can be tuned by the mole fraction of cationic lipids, xCL {equiv} NCL/(NCL + NNL), where NCL and NNL are the numbers of cationic and neutral lipids, respectively. When all the lipids are cationic {sigma} = {sigma}cat = 2e/a2A0, where e is the charge of an electron. In general, {sigma} {equiv} xCL{sigma}cat. The relative charged-membrane/tubulin stoichiometry, RCL/T, is given by, RCL/T {equiv} NCL/NT, where NT is the number of tubulin dimers. RCL/T can be tuned to control the overall charge of the complex. RCL/T {approx} 40 corresponds to the mixing isoelectric point.

Fig. 3 summarizes a series of SAXRD scans as in Fig. 2C, analyzed as in Fig. 2, D and E. In Fig. 3 A, f is plotted as a function of xCL (or {sigma}) at various RCL/T values. The coverage of the third layer arises primarily from the mismatch between the charge density of the membrane and the MT wall but also due to the mixing entropy of the lipids within the bilayer. There is no difference in the electrostatic energy if the cationic lipid neutralizes the MT or the external tubulin oligomers. When {sigma} is smaller than the charge density of the MT wall, {sigma}MT = 0.2 e/nm3, mixing entropy, which favors random distribution of the charged lipids across the bilayer (35Go), induces coating of tubulin oligomers, yielding f > 0.4 even at low {sigma}. As {sigma} increases, more charged lipids can go to the external monolayer, enable the adsorption of more tubulin oligomers, and account for the monotonic increase in f. Unlike {sigma}, the stoichiometry, RCL/T, has little effect on f.


Figure 3
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  		FIGURE 3  States diagrams of the LPNs as a function of the mole fraction of cationic lipids, xCL, or the membrane charge density, {sigma} (top horizontal axis); {sigma} is calculated from xCL as explained in Results and Discussion. The MT wall charge density, {sigma}MT, as estimated based on the primary structure of tubulin (15Go,16Go,31Go,40Go), is indicated. Each data point is obtained from scattering data and fitting to a model, as demonstrated in Fig. 2. Different symbols correspond to different charged lipid/tubulin ratios, RCL/T, as indicated in Fig. 3 B (inset). For all data points shown, there are enough lipids to cover each MT with a bilayer. Solid symbols, for which RCL/T = 160 · xCL, correspond to a series of data points at which the total number of lipids/tubulin is kept constant and is exactly enough to coat each MT with a bilayer (calculated as in May and Ben-Shaul (35Go)). (A) Fraction of tubulin oligomer coverage at the external layer, f, as a function of xCL (or {sigma}). The solid line indicates the mean values of f(xCL). (B) The inner wall radius, Rin, of the internal MT within the LPN complex and mean PF number, <N>, as a function of xCL (or {sigma}). Rin is obtained from fitting the scattering data to the model, whereas <N> is estimated from Rin (see Results and Discussion). The arrow indicates the <N> value of pure MTs, <N>MT =13.3, as obtained from the fit to the MT form factor, shown in Fig. 2, in good agreement with earlier work (14Go,15Go,20Go,22Go). The three solid lines indicate the mean values of <N> at each step. The broken lines indicate the maximum and the minimum values of <N> at each step.

 
The internal MT size is determined by Rin. <N> was calculated from Rin (Fig. 3 B), assuming (8.10) that the width of a tubulin subunit (31Go), 2a = 5 nm, remains constant at the MT wall center, Rin + a1/2: <N> {equiv} 2{pi}(Rin + a1/2)/2a. If we assume that the width of a tubulin subunit remains constant at Rin (2{pi}Rin/13 {approx} 4 nm) or at Rin+ a1 (2{pi}(Rin + a1)/13 {approx} 6 nm), the values of <N> could change by no more than 1.5%. Rin is obtained directly from fitting the model to the data as described. However, Rin could also vary, by up to 1%, if other assumptions are made to the model, for example, if a1 is allowed to be a function of Rin while keeping the volume of a tubulin subunit constant, and the surface area of tubulin remains constant at Rin or Rin+ a1. Those variations are smaller than the scatter in the data. Finally, Rin is obtained from bulk measurements that benefit from good statistics and are highly reproducible and reliable (Fig. 3 B).

Rin (or <N>) are plotted, in Fig. 3 B, as a function of xCL (or {sigma}) at various RCL/T values. We find that <N> decreases discontinuously with {sigma} and exhibits two steps, within the experimental accessible {sigma} range. At {sigma} < {sigma}MT, Rin = 9.02 ± 0.11 nm and <N> = 14.40 ± 0.15. At {sigma}MT < {sigma} ≤ 2.15{sigma}MT, Rin = 8.48 ± 0.08 nm and <N> = 13.72 ± 0.09, and finally at {sigma} > 2.15{sigma}MT, Rin = 8.13 ± 0.09 nm and <N> = 13.28 ± 0.12, which is similar to values we (14Go,15Go,17Go,18Go) and others (2Go,20Go,22Go) obtained for taxol-stabilized MTs. The nonintegral nature of <N> results from the fact the x-ray data provides the mean PF number. So the variation in the mean PF number is in fact a variation in the distribution of PF numbers. <N> values of 14.4, 13.72, and 13.28 correspond to the high percentage of MTs with 15, 14, and 13 PFs, respectively. As we found for f, the lipid/protein stoichiometry ratio, RCL/T, has little effect on Rin (or <N>), and it is again {sigma} that turns out to be the key parameter. Decreasing <N> with {sigma} appears to be the best the system can do to neutralize itself and compensate for the charge-density mismatch between the MT and the lipid bilayer. As Rin or <N> decrease, the angle between the PFs decreases and they expose a larger fraction of their surface to the lipid layer and thereby are able to neutralize more cationic lipids.

By mixing DOTAP with the neutral lipid dioleoyl(C18:1) phosphatidylethanolamine (DOPE), which has a smaller headgroup than that of DOPC, we obtained negative membrane spontaneous curvatures (36Go). The cationic lipid dilauryl(C12:0) trimethyl ammonium propane (DLTAP) and the homologous neutral lipid dilauryl(C12:0) phosphatidylcholine (DLPC) (Avanti Polar Lipids) have shorter hydrophobic tailgroups (~1.2 nm) compared to DOTAP/DOPC (~1.4 nm). As the bending rigidity of a fluid membrane (37Go,38Go), {kappa}, is given by {kappa} {propto} (a2)3, where a2 is the membrane thickness, DLTAP/DLPC membranes have ~60% lower {kappa} compared to DOTAP/DOPC membranes. Fig. 4, A and C, shows that when MTs are complexed with DLTAP/DLPC or DOTAP/DOPE membranes, the behavior is to a great extent similar to that obtained with DOTAP/DOPC membranes, the main difference being when {sigma} {approx} {sigma}MT, where the boundaries between the steps may have shifted a bit. This indicates that although the energy barrier for the formation of the LPN is a function of {kappa} (16Go), once the LPN has formed, the charge density is the key parameter in determining <N>.


Figure 4
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  		FIGURE 4  The effect of salt, taxol, and membrane spontaneous curvature and rigidity on the mean PF number, <N>, and tubulin oligomer coverage, f. The cationic lipid/tubulin stoichiometry RCL/T = 160 · xCL for all data points. (A) <N> (or Rin) as a function of xCL (or {sigma}). The solid and broken lines are taken from Fig. 3 B, indicating the mean values of <N> and the upper and lower limits of <N> at each step, respectively, for MTs complexed with DOTAP/DOPC membranes. Solid diamonds indicate the <N> values of MTs complexed with DLTAP/DLPC membranes and solid circles indicate the values for MTs complexed with DOTAP/DOPE membranes. Open symbols indicate the effect of added salt when MTs are complexed with DOTAP/DOPC membranes. Stars indicate the addition of 50 mM KCl (leading to Debye length of {kappa}–1 = 0.9 nm, when the buffer is taken into account) at several membrane charge densities. Triangles indicate the addition of different salt concentrations when xCL = 0.5. The inset shows the variation of <N> with {kappa}–1, when xCL = 0.5. (B) The variation of f with {kappa}–1 (i.e., salt) for xCL = 0.5, for MTs complexed with DOTAP/DOPC membranes. The broken line indicates the mean value of f for the complexes in the buffer solution with no added salt. The solid line is a guide for the eye. (C) f as a function of xCL. The solid line is taken from Fig. 3 A. Other symbols are as in Fig. 4 A. (D) <N> (or Rin) as a function of the molar ratio, t, between taxol and tubulin for xCL = 0.5 with DOTAP/DOPC membranes. The solid line is a fit to a logarithmic expression: <N> = ln(142218 + 883725 x t). The inset is a TEM cross section (bottom) corresponding to t = 0 (arrow), showing the inner MT with 12 PFs and a TEM side-view image (top) showing a short LPN. (Scale bar, 50 nm.) We note that we did not perform a statistical study of such TEM cross sections, as our x-ray data is a bulk measurement and inherently includes statistics. (E) <N> (or Rin) as a function of xCL (or {sigma}) for t = 0, corresponding to no added taxol. Open squares correspond to tubulin, which was directly mixed with DOTAP/DOPC membranes (with no added GTP). Solid squares correspond to DOTAP/DOPC membranes that were mixed with MTs polymerized with GTP at 36 ± 1°C but not taxol-stabilized.

 
Similarly, for xCL = 0.5, the addition of salt has, within the scatter, no effect on Rin (or <N>) (Fig. 4 A). This is attributed to the fact that at the interface between the internal MT and the lipid bilayer, the complex is highly charged and the ion concentration is a few molar and therefore not sensitive to small variation in the solution ionic strength outside the complex, which is at a much lower concentration. However, the addition of salt significantly increases f (Fig. 4, B and C), because it screens the electrostatic repulsion between the negatively charged tubulin oligomers, which are exposed to the salt solution. This is the way to achieve full tubulin coverage at the external layer (without added salt, f < 0.8, see Fig. 3 A). Above some critical salt concentration, which increases with xCL, the complexes do not form (indicated by f = 0 in Fig. 4, B and C), because at high salt concentration the propensity of the solution to accept more counterions is reduced and thus counterion release, which is the driving force for the complex formation (36Go), is not favorable.

The fact that, within the scatter, Rin (or <N>) is stationary, whereas f changes dramatically, in the presence of salt (Fig. 4, A and B) shows unambiguously that the coverage of tubulin oligomers has little effect on <N> and it is {sigma} that predominantly controls the MT PF number. This may well be due to the diameter of the tubulin rings or spirals at the external layer, which happens to be similar to the diameter of free tubulin rings in solution (30Go), implying that the rings do not exert large tension on the internal MT. The membrane bending rigidity, {kappa}, sets an energy barrier for the formation of the LPN (16Go). However, once the LPN is formed, it seems that the bending rigidity and spontaneous curvature of the membrane do not play a role, within our experimental conditions. We may conclude that the elastic properties of the layer that coats the MT in the LPN have, to a certain degree, little effect on <N> compared to the membrane charge density.

We thus used the LPN system to examine the effect of the chemotherapy drug taxol, which is known to stabilize MTs (15Go,19Go,20Go,28Go). Without taxol, similar LPNs are obtained and Rin (or <N>) again decreases with {sigma} (Fig. 4 E), though perhaps more data are needed to determine the exact form of this decrease. However, SAXRD analysis and TEM images (Fig. 4 D) show that the LPNs are shorter than with taxol, indicating that taxol mainly stabilized the straight curvature of the tubulin subunits along the PFs, thereby leading to longer polymers. This is complementary to our earlier osmotic stress measurements (15Go), which showed that taxol does not change the lateral interactions between PFs.

The second difference is that in the absence of taxol, <N> is smaller than in the presence of taxol (Fig. 4, A and E). Perhaps the reason for this is that in the absence of taxol, it is somewhat easier for the complex to adjust its size, and by going to a smaller size, the matching between the charge densities of the MT and the lipid bilayer improves. Interestingly, we found that <N> increases logarithmically with the molar ratio, t, between Taxol and tubulin (Fig. 4 D), for xCL = 0.5. This suggests that the stabilization of the MT PFs increases logarithmically with t, implying that taxol stabilizes the straight PF conformation in a global fashion and clearly beyond its local attachment to specific tubulin subunits, which would yield a linear dependence on t. This is consistent with the manner in which taxol suppresses MT dynamics (28Go); a small amount of taxol significantly suppresses MT dynamics.

To understand how MT PF number is regulated by the charge density of an enveloping layer, we provide a simple physical description of the energy associated with the coassembly of an MT with an oppositely charged lipid bilayer. The basic assumption of the model is that, even though an MT is highly resistant against deformations that require changes in PF length, the binding between two adjacent PFs in an MT is quite weak (15Go). As a result, even weak noncovalent interactions between an MT and the environment—such as the mechanical torque exerted on a MT by the adhering lipid bilayer or electrostatic interactions—can alter MT PF number.

Assume that a MT consists of m negatively charged PFs of length l in the form of a circular bundle. The total PF length L = ml is proportional to the number of tubulin monomers, which will be assumed fixed in the following. For a given cross section of the MT, draw a line from each PF to the center of the MT so that the angle between adjacent lines equals 2{pi}/m. Let {varphi}* be the preferred value of this angle in the absence of electrostatic interaction between PFs. The lateral bending energy cost of a MT, associated with change in PF number and hence deviations of 2{pi}/m from {varphi}* is, to the lowest order,

Formula 3(3)
where k is an effective bending stiffness per unit length associated with variation in PF number.

The MT is surrounded by a cationic lipid bilayer with a thickness denoted by a2. Let s be the arc-distance along the center line of this bilayer, again along a cross section (Fig. 5), and let {rho}(s) be the local curvature radius of the center line. The Helfrich bending energy cost of the lipid bilayer is then

Formula 4(4)
with {kappa} ~ 10 kBT the membrane bending modulus (16Go). We will assume that lipid material is freely exchangeable with a reservoir, so that the MT is fully covered. Equation 3 again does not include the electrostatic self-energy of the lipid material.


Figure 5
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  		FIGURE 5  Cartoon demonstrating the geometry associated with a membrane that coats the MT PFs.

 
The main contribution to the gain in electrostatic energy of the system comes from the free energy gain due to the counterion release(39Go) that produced the association of the two macroions of opposite charge (the MT and the cationic lipid layer). The interface between the MT and the lipid bilayer is a cylinder of radius R {approx} ma1 and surface area A {approx} La1 (a1 is the size of a PF monomer). The net surface charge density, {sigma}cyl, of the cylinder at the interface between the MT and the lipid layer depends on the mole fraction, xCL, of cationic lipid in the membrane as

Formula 5(5)

Here, {sigma}cat and {sigma}MT are, respectively, the charge densities per unit volume of a completely cationic lipid bilayer and of the MT wall, c is the fraction of total MT wall charge per unit area that is at the interface between the MT wall and the lipid bilayer, and the factor 0.5 reflects the symmetry of the lipid bilayer, i.e., only half of the membrane charge is located at the interface between the membrane and the MT and the other half is at the external lipid bilayer. The mole fraction at the isoelectric point is xiso = ca1{sigma}MT/0.5a2{sigma}cat {approx} 0.7c under the conditions of our experiments, described in Materials and Methods.

The entropic free-energy gain due to counterion release will be included as an adhesion energy per unit area {gamma} between the lipid bilayer and the oppositely charged PF (39Go). This counterion-release adhesion energy is of the order of the thermal energy times the number of charges per unit area in the contact region between the two macroions, i.e., Formula 5. The constant {gamma}0 is included to allow for any residual van der Waals attraction between lipid and tubulin material, and p is the fraction of counterions that are released.

An important point of the model is that when we evaluate the adhesion energy, we should not treat the MT as circular, but must account for the surface structure of the MT provided by the individual PFs. As the lipid bilayer wraps around the profile of the MT, sections that adhere to a PF will alternate with sections, between PFs, that do not adhere, since the bending stiffness of the bilayer prevents it from perfect local adjustment to the MT surface profile.

Let {theta} be the arc distance of the contact line between the lipid bilayer and one PF (in cross section). If we approximate a PF cross section as circular, with radius a, then the adhesive contact area per PF equals la{theta}, so the total contact area per MT is La{theta}. The adhesion energy is then

Formula 6(6)

The first term, with {lambda} equal to {gamma} times a microscopic length, is the adhesion energy per unit length between a locally flat lipid bilayer and a PF.

We now can minimize the bending energy (Eq. 4) of the bilayer sections between the PFs if we know the cross-sectional shape of a PF. For PFs with a circular cross section with radius a, this is a straightforward calculation with the following result:

Formula 7(7)
where all terms are dimensionless. The first term is the adhesive line energy, the second term is the sum of the adhesive surface energy and the bending energy of the adhering lipid bilayer. The third term is the bending energy of the connecting nonadhering sections, and the fourth term is the sum of the MT bending and electrostatic energies. The effective MT bending stiffness per unit length, k, and the preferred angle between protofilaments, {varphi}*, are in principle functions of {sigma}cyl, although in our experiment, the screening condition is strong. The Debye-Huckel electrostatic screening radius is typically ~1 nm (see Fig. 4 A) and thus much smaller than the thickness of the lipid bilayer and the PF diameter. In this case, the dependence of k and {varphi}* on {sigma}cyl is very weak. In the discussion below, to the lowest order, we ignore this dependence and regard k and {varphi}* as constants.

This result can be viewed as a variational expression that must be minimized with respect to the adhesion angle {theta}. The outcome of this minimization depends on the key dimensionless parameter

Formula 8(8)
When {Gamma} = 1/2, a continuous transition takes place from an adhesive to a nonadhesive state.

When {Gamma} < 1/2 (corresponding to membranes with low charge density, xCL < 1/4p), a "weak-adhesion" regime, the bending energy of the lipid bilayer exceeds the adhesion energy, and Eq. 3 is minimized by {theta} = 0. The lipid bilayer is either a perfect cylinder, only touching each of the PFs in turn, or it does not adhere at all to the MT (i.e., the lipid vesicles stick to the MT, forming a "beads on a rod" structure (16Go)). In this case, the total energy reduces to:

Formula 9(9)

The first term, the contact-line energy, is the only negative contribution. For adhesion, the total energy must be negative, so Formula 9 must exceedFormula 9. Minimization of the energy with respect to m in that case gives, for the optimal number m* of PFs,

Formula 10(10)

The fact that the optimal value of 2{pi}/m is <{varphi}* (corresponding to a PF number greater than that of uncoated MT) is due to the lipid bending energy, which can be reduced by increasing the radius of the MT. This result is in accordance with our findings (Fig. 3 B).

In the regime where {Gamma} > 1/2, the adhesion energy of the bilayer exceeds the bending energy. The bilayer now partially follows the outer contour of the MT. The total energy, which is minimized when the arc length of the adhesive sections is Formula 10, equals

Formula 11(11)

The third term is the lipid bending energy which now favors smaller m values, since that allows for extra contact area between the bilayer and a PF. Minimization with respect to m now gives, for the optimal number m* of PF's,

Formula 12(12)

This result also predicts a decrease in the PF number with increasing membrane charge density.

By comparing Eqs. 10 and 12 with our results we find that k should be of order 10–100 kBT/nm to account for the variation we observe in MT PF number.

If we vary xCL, then mainly the adhesive energy is affected. With decreasing mole fraction, the optimal number of PFs steadily increases until we reach the critical point where adhesion between the lipid bilayer and the MT is lost. Note that due to the van der Waals attraction, it is necessary to use more rigid membranes to study this "wrapping transition" (16Go). It should be noted that the physics of this wrapping transition—with its competition between adhesion energy and bending energy—is essentially similar to the well-known Marky-Manning transition of DNA/nucleosome complexation.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
APPENDIX: FORM FACTOR OF...
 ACKNOWLEDGEMENTS
 REFERENCES
 
We have shown that the electrostatic interactions between a MT and a charged layer coating it influence the MT PF number in LPNs. We find that the mean PF number decreases in a stepwise fashion with the lipid-bilayer charge density. The physical model we presented to account for our results suggests that the energy associated with the PF number change is of order 10 kBT. This model system may provide insight into one of the mechanisms through which MT size is regulated in cells. The fact that the range of charge densities that lead to each mean value of PF number is relatively broad allows variations in the composition of the MT enveloping layer while maintaining the same PF number.


   APPENDIX: FORM FACTOR OF CONCENTRIC HOLLOW CYLINDERS AND ITS IMPLICATION TO MICROTUBULE AND LIPID-PROTEIN NANOTUBES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX: FORM FACTOR OF...
ACKNOWLEDGEMENTS
 REFERENCES
 
We start by considering the form factor of a single hollow cylinder of core radius Rc and shell radius Rs with a total height 2H. We assume that the inside and outside of the tube have the same electron density and that the inside of the tube has a uniform electron density that differs by Formula 12 from the outside of the tube. The scattering amplitude F is proportional to the Fourier transform of the electron density of the hollow cylinder:

Formula 12
where the integration is over the volume V of the hollow cylinder.

In cylindrical coordinates, we obtain

Formula 12
where J0 and J1 are the zero and first Bessel functions of the first kind.

The intensity I is given byFormula 12, but since our solutions are isotropic we need to perform a powder average in the reciprocal q space:

Formula 12
By setting Formula 12 we get: Formula 12 and Formula 12, so finally the intensity is given by

Formula 12
where A and B are constants.

In the more general case, we have a series of n concentric homogenous hollow cylinders with an overall radial electron density profile given by the set of parameters (Formula 12, {rho}k, Hk). Formula 12 is the difference between the electron density of the surrounding (the solvent in our case) and the kth homogenous hollow cylinder with a core radius Formula 12 and a shell radius Formula 12. 2Hk is the height of the kth hollow cylinder (Hn+1 = 0) and k =1,2,...,n+1. The scattering intensity of such randomly oriented n concentric cylinders is

Formula 12
For n infinitely long concentric hollow cylinders, we get

Formula 12

In our case, we reduced the number of parameters by having Formula 12, {rho}k be a function of a subset of parameters, ai, out of which a much smaller subset of parameters was free to float.

For the case of pure microtubule solutions we have the set of parameters shown in Table 1 For the microtubule-lipid complexes the set of parameters is given in Table 2. The values of Formula 12 and Formula 12 are calculated (based on the parameters of Tables 1 and 2) as described in Table 3.


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  		TABLE 1  The values of ai in the case of pure MT

 

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  		TABLE 2  The values of ai in the case of the LPN

 

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  		TABLE 3  Calculation of Rk and {rho}k

 

   ACKNOWLEDGEMENTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX: FORM FACTOR OF...
 ACKNOWLEDGEMENTS
REFERENCES
 
We thank K. Ewert, S. Richardson, and K. Linberg for experimental help; D. McLaren and P. Allen for help with cartoons; and M. A. Jordan, A. Gopinathan, A. Zilman, N. Gov, T. Deming, and P. Pincus for discussions.

This work was supported by National Institutes of Health grant GM-59288 (to U.R., D.J.N., Y.L., and C.R.S.), National Science Foundation grants DMR-0503347 and CTS-0404444, Dept. of Energy grant DE-FG02-06ER46314 (to U.R., D.J.N., Y.L., and C.R.S.), and National Institutes of Health grant NS13560 (to H.P.M. and L.W.). The University of California, Santa Barbara, Material Research Laboratory received support from National Science Foundation grant DMR-0080034. The Stanford Synchrotron Radiation Laboratory, where some of this work was done, is supported by the U.S. Dept. of Energy. U.R. received fellowship support from the International Human Frontier Science Program Organization and the European Molecular Biology Organization.


   FOOTNOTES
 
Daniel J. Needleman's present address is Harvard Medical School, Harvard University, Boston, MA 02115.

Submitted on May 9, 2006; accepted for publication September 5, 2006.


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 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 APPENDIX: FORM FACTOR OF...
 ACKNOWLEDGEMENTS
 REFERENCES
 
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