pubs.acs.org/Langmuir © 2009 American Chemical Society
Vimentin Intermediate Filament Formation: In Vitro Measurement and Mathematical Modeling of the Filament Length Distribution during Assembly† )
Stephanie Portet,*,‡,# Norbert Mu¨cke,§,# Robert Kirmse,§,3 Jo¨rg Langowski,§ Michael Beil,^ and Harald Herrmann*, ‡
)
Department of Mathematics, 342 Machray Hall, University of Manitoba, Winnipeg, MB, Canada R3L 2N2, § Division of Biophysics of Macromolecules and Division of Molecular Genetics, German Cancer Research Center (DKFZ), Im Neuenheimer Feld 580, Heidelberg D-69120, Germany, and ^Department of Internal Medicine I, University Hospital Ulm, D-89070 Ulm, Germany. # These authors contributed equally to this work. 3 Current address: Research Group Microenvironment of Tumor Cell Invasion, German Cancer Research Center (DKFZ), Im Neuenheimer Feld 280, Heidelberg D-69120, Germany. Received February 10, 2009. Revised Manuscript Received April 10, 2009 The salt-induced in vitro assembly of cytoplasmic intermediate filament (IF) proteins such as vimentin is characterized by a very rapid lateral association of soluble tetrameric subunits into 60-nm-long full-width “unit-length” filaments (ULFs). We have demonstrated for this prototype IF protein that filament elongation occurs by the longitudinal annealing of ULFs into short IFs. These IFs further longitudinally anneal and thus constitute a progressively elongating filament population that over time yields filaments of several μm in length. Previously, we provided a mathematical model for the kinetics of the assembly process based on the average length distribution of filaments as determined by time-lapse electron and atomic force microscopy. Thereby, we were able to substantiate the concept that end-to-end-annealing of both ULFs and short filaments is obligatory for the formation of long IFs (Kirmse, R.; Portet, S.; Mu¨cke, N. Aebi, U.; Herrmann, H.; Langowski, J. J. Biol. Chem. 2007, 282, 18563-18572). As the next step in understanding the mechanics of IF formation, we have expanded our mathematical model to describe the quantitative aspects of IF assembly by taking into account geometry constraints as well as the diffusion properties of rodlike linear aggregates. Thereby, we have developed a robust model for the time-dependent filament length distribution of IFs under standard conditions.
Introduction Intermediate filaments (IF) constitute, in addition to microtubules (MTs) and microfilaments (MFs), the third principal filament system of the cytoskeleton-structuring metazoan cells. In particular, IFs form, together with several cross-bridging molecules, the cytoskeleton proper. They are highly resistant to disruptive treatments such as the extraction of cells with buffers containing high concentrations of salt and nonionic detergents in † Part of the Molecular and Polymer Gels; Materials with Self-Assembled Fibrillar Networks special issue. *Corresponding authors. E-mail:
[email protected]; h.herrmann@ dkfz-heidelberg.de.
(1) Abumuhor, I. A.; Spencer, P. H.; Cohlberg, J. A. J. Struct. Biol. 1998, 123, 187–198. (2) Ba¨r, H.; Mu¨cke, N.; Kostareva, A.; Sjo¨berg, G.; Aebi, U.; Herrmann, H. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 15099–15104. (3) Bremer, A.; Millonig, R. C.; Su¨tterlin, R.; Engel, A.; Pollard, T. D.; Aebi, U. J. Cell Biol. 1991, 115, 689–703. (4) Bremer, A.; Aebi, U. Curr. Opin. Cell Biol. 1992, 4, 20–26. (5) Cerda, J.; Conrad, M.; Markl, J.; Brand, M.; Herrmann, H. Eur. J. Cell Biol. 1998, 77, 175–187. (6) Coulombe, P. A.; Chan, Y. M.; Albers, K.; Fuchs, E. J. Cell Biol. 1990, 111, 3049–3064. (7) Coulombe, P. A.; Fuchs, E. J. Cell Biol. 1990, 111, 153–169. (8) Dimitrov, A.; Quesnoit, M.; Moutel, S.; Cantaloube, I.; Pous, C.; Perez, F. Science 2008, 322, 1353–1356. (9) Ekani-Nkodo, A.; Kumar, A.; Fygenson, D. Phys. Rev. Lett. 2004, 93, 268301–268305. (10) Esue, O.; Carson, A. A.; Tseng, Y.; Wirtz, D. J. Biol. Chem. 2006, 281, 30393–30399. (11) Ethayaraja, M.; Bandyopadhyaya, R. Langmuir 2007, 23, 6418–6423. (12) Georgakopoulou, S.; Mo¨ller, D.; Sachs, N.; Herrmann, H.; Aebi, U. J. Mol. Biol. 2009, 386, 544–553. (13) Goldman, R. D.; Khuon, S.; Chou, Y. H.; Opal, P.; Steinert, P. M. J. Cell Biol. 1996, 134, 971–983.
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the cold, which are conditions that lead to the disassembly of MTs and MFs.22 Accordingly, strong denaturants have to be employed to dissociate filaments and to obtain monomeric subunits. Also, in contrast to MTs and MFs, whose subunit proteins tubulin and actin are globular, IFs assemble from fibrous molecules exhibiting an approximately 45-nm-long central R-helical, coiled-coil-forming domain (“rod”). These parallel coiled-coils associate into antiparallel, approximately half-staggered tetrameric complexes already at a relatively high concentration of (14) Goldman, R. D.; Grin, B.; Mendez, M. G.; Kuczmarski, E. R. Curr. Opin. Cell Biol. 2008, 20, 28–34. (15) Herrmann, H.; Eckelt, A.; Brettel, M.; Grund, C.; Franke, W. W. J. Mol. Biol. 1993, 234, 99–113. (16) Herrmann, H.; Munick, M. D.; Brettel, M.; Fouquet, B.; Markl, J. J. Cell Sci. 1996, 109, 569–578. (17) Herrmann, H.; Ha¨ner, M.; Brettel, M.; Mu¨ller, S. A.; Goldie, K. N.; Fedtke, B.; Lustig, A.; Franke, W. W.; Aebi, U. J. Mol. Biol. 1996, 264, 933–953. (18) Herrmann, H.; Aebi, U. Curr. Opin. Struct. Biol. 1998, 8, 177–185. (19) Herrmann, H.; Ha¨ner, M.; Brettel, M.; Ku, N.-O.; Aebi, U. J. Mol. Biol. 1999, 286, 1403–1420. (20) Herrmann, H.; Aebi, U. Curr. Opin. Cell Biol. 2000, 12, 79–90. (21) Herrmann, H.; Aebi, U. Annu. Rev. Biochem. 2004, 73, 749–789. (22) Herrmann, H.; Kreplak, L.; Aebi, U. Methods Cell Biol. 2004, 78, 3–24. (23) Herrmann, H.; Ba¨r, H.; Kreplak, L.; Strelkov, S. V.; Aebi, U. Nat. Rev. Mol. Cell Biol. 2007, 8, 562–573. (24) Hill, T. L. Biophys. J. 1983, 44, 282–288. (25) Hofmann, I.; Herrmann, H.; Franke, W. W. Eur. J. Cell Biol. 1991, 56, 328–341. (26) Ip, W.; Hartzer, M. K.; Pang, Y. Y.; Robson, R. M. J. Mol. Biol. 1985, 183, 365–375. (27) Janmey, P. A.; Euteneuer, U.; Traub, P.; Schliwa, M. J. Cell Biol. 1991, 113, 155–60. (28) Jameson, L.; Caplow, M. J. Biol. Chem. 1980, 255, 2284–2292. (29) Johnson, K. A.; Borisy, G. G. J. Mol. Biol. 1977, 117, 1–31. (30) Kirmse, R.; Portet, S.; Mu¨cke, N.; Aebi, U.; Herrmann, H.; Langowski, J. J. Biol. Chem. 2007, 282, 18563–18572.
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denaturant, such as 6 M urea, during renaturation into buffers of low ionic strength. As determined by analytical ultracentrifugation, tetramers with an s value of 5S to 6S that are stable over extended times are obtained with this procedure.17,33 These tetramers are the original precursors for filament formation. Moreover, because the tetrameric complexes consist of two antiparallel-oriented dimers, the resulting filaments do not exhibit polarity either. IF proteins do not depend on the binding of nucleoside triphosphates for assembly, whereas tubulin and actin are nucleosidases that bind GTP and ATP, respectively.4,37,50 In both cases, the nucleotide plays an essential role in the kinetics of the polymer.28,36 For MTs, recent data indicate that the presence of GTP-tubulin is indeed engaged in the “rescue” of shrinking MTs.8 Different from IFs, both MFs and MTs depend on nucleation reactions for assembly, and assembly proceeds through the reversible addition of monomers in the case of actin and dimers in the case of tubulin.3,29,38 In further contrast, IFs are highly flexible polymers with a persistence length in the submicrometer range, whereas the persistence length of MTs and MFs is orders of magnitudes higher.32 Finally, unlike with MTs and MFs, the administration of mechanical stress to IFs at large deformations does not cause breakage of the filaments but instead IFs become more viscoelastic, a behavior known as “strain stiffening”.27,49 The forces that hold IFs together are very different from those involved in MT and MF formation. This is one of the special features of this “strange” fibrous protein family observed early on.41,46 The principal interactions that mediate the formation of tetrameric complexes, the lateral association of ULFs, the longitudinal annealing of ULFs, and eventually the properties of mature IFs comprise highly negatively charged dimeric rods that interact with neighboring rods via two flexible, unstructured chains, each consisting of the very basic nonR-helical amino-terminal (“head”) domains (12 arginines each without negatively charged amino acids). A strong supportive system of hydrophobic interactions is provided by six aromatic residues within each head domain that mediate subunit-subunit interactions during the maturation phase of IFs as revealed by near-UV circular dichroism spectroscopy experiments.12,21 In addition to these unique molecular and biophysical properties, IF proteins exhibit a distinctly new assembly mechanism. Starting from tetrameric subunits in low ionic strength buffer, the addition of sodium chloride to 50 mM triggers the instantaneous and ordered lateral association of tetramers.33 Thereby full-width “open” filaments of 17 nm diameter and approximately 60 nm length, the so-called “unit-length” filaments (ULFs), are formed within seconds (Figure 1, phase 1). The dimension and the mass of single ULFs have been determined by glycerol-spraying/rotary metal shadowing and scanning transmission electron microscopy (STEM).17,19 It is important to note that, because of the half-staggered overlap of dimers in a tetramer, the center third of an ULF contains 32 molecules per cross-section whereas both ends harbor only 16 chains. Employing time-lapse microscopy, we proposed early on that filament growth occurs by longitudinal (31) Kreplak, L.; Richter, K.; Aebi, U.; Herrmann, H. Methods Cell Biol. 2008, 88, 273–297. (32) Mu¨cke, N.; Kreplak, L.; Kirmse, R.; Wedig, T.; Herrmann, H.; Aebi, U.; Langowski, J. J. Mol. Biol. 2004, 335, 1241–1250. (33) Mu¨cke, N.; Wedig, T.; Bu¨rer, A.; Marekov, L. N.; Steinert, P. M.; Langowski, J.; Aebi, U.; Herrmann, H. J. Mol. Biol. 2004, 340, 97–114. (34) Mu¨ller, M.; Bhattacharya, S. S.; Moore, T.; Prescott, Q.; Wedig, T.; Herrmann, H.; Magin, T. M. Hum. Mol. Genet. 2009, 18, 1052–1057. (35) O’Shaughenessy, B.; Vavylonis, D. Phys. Rev. Lett. 2003, 90, 118301. (36) Pantaloni, D.; Hill, T. L.; Carlier, M. F.; Korn, E. D. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 7207–7211. (37) Pollard, T. D. J. Cell Biol. 1986, 103, 2747–2754. (38) Pollard, T. D. Curr. Opin. Cell Biol. 1990, 2, 33–40.
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Figure 1. Model of the assembly dynamics of intermediate filaments, where n is the length of an ULF and m is the length of a repeating unit.
annealing of ULFs via interdigitation of their ends and further end-to-end annealing of short IFs (i.e., multiples of ULFs) with ULFs (Figure 1, phase 2a,b). After all ULFs are used up, IF growth proceeds exclusively by the longitudinal annealing of individual IFs with one another. Thereby groups of filaments with an unknown length distribution are obtained at successive time points of assembly (Figure 1, phase 2c). During filament elongation, major rearrangements within the “open”, 17-nm-wide filaments mediate a radial compaction process that eventually yields smooth, flexible, and approximately 10-nm-wide mature filaments (Figure 1, phase 3).12 This three-phase model has gained additional strong support from the discovery that disease mutants of the human muscle-specific IF molecule desmin decay from the normal IF assembly pathway at distinct stages compatible with the above-described assembly phases.2,23 In further support, it has been demonstrated by quantifying the time-dependent length of growing filaments and subsequent mathematical modeling, surveying eight different potential subunit association scenarios, that the above three-phase model is the only one that appropriately describes the dynamics of the filament assembly process.30 In particular, it was demonstrated that tetramer addition to filament ends is not a significant factor for the filament elongation process. In general, association reactions involving subfilamentous structures, such as tetramers and octamers, occur in the very early stages of the in vitro assembly experiments within the first seconds but do not play a significant role in the elongation of filaments. Finally, the model demands that end-to-end addition of ULFs and filaments powers filament elongation. In our previous model of the vimentin assembly kinetics, we have focused on the various possible reaction constants driving assembly from tetrameric complexes to mature filaments.30 As a central term derived from the measured data, we employed the mean length of filaments in that study. For the present work, we have developed a model to describe the dynamics of the length distribution of IFs based on the assumption that the ULF is the (39) Prahlad, V.; Yoon, M.; Moir, R. D.; Vale, R. D.; Goldman, R. D. J. Cell Biol. 1998, 143, 159–170. (40) Quinlan, R. A.; Brenner, M.; Goldman, J. E.; Messing, A. Exp. Cell Res. 2007, 313, 2077–2087. (41) Renner, W.; Franke, W. W.; Schmid, E.; Geisler, N.; Weber, K.; Mandelkow, E. J. Mol. Biol. 1981, 149, 285–306. (42) Riseman, J.; Kirkwood, J. G. J. Chem. Phys. 1950, 18, 512–516. (43) Schaffeld, M.; Herrmann, H.; Schultess, J.; Markl, J. Eur. J. Cell Biol. 2001, 80, 692–702. (44) Schopferer, M., Ba¨r, H., Hochstein, B., Sharma, S., Mu¨cke, N., Herrmann, H., Willenbacher, N. J. Mol. Biol. 2009, 388, 133-143.
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minimal IF. Therefore, the model is designed in a general framework using the ULF as the unit of measurement. Using an ordinary differential equation formalism, this model describes the rate of change of the concentrations of filamentous populations (i.e., filaments composed of i ULFs). The modeling approach follows a proposal that Terrell L. Hill made for the association of long linear aggregates in solution.24 A central idea in this treatise is that geometry constraints as well as diffusion properties of rodlike linear aggregates are essential parameters for the calculation of association rates. In summary, in our current model, only the elongation process but not the formation of ULFs is considered because of the exhaustive, instantaneous formation of ULFs after the initiation of assembly and because filament elongation as such proceeds by the longitudinal annealing of ULFs and filaments with one another.30 Filament disassembly is not considered because it is not observed under the experimental conditions employed. Moreover, this model can now be applied to the study of any IF proteins that assemble in the vimentin mode and that can form mixed filaments with vimentin such as desmin, peripherin, R-internexin, and neurofilament triplet proteins.21,52
Figure 2. In vitro vimentin filaments assembled after 3000 s with a protein concentration of 0.02 mg/mL.
Experimental Procedures Protein preparation, assembly, and both electron and atomic force microscopy were performed as published previously.17,30,32 In brief, the proteins were freshly reconstituted into buffer of low ionic strength for every experiment. For electron microscopy, assembly was started by the addition of potassium chloride to 100 mM and stopped by adding an equal volume of assembly buffer containing 0.2% glutaraldehyde. The samples were allowed to absorb to glowdischarged carbon-coated copper grids for 2 min, followed by standard staining, washing, and drying steps.31 A representative electron microscopy image used for the determination of filament length is shown in Figure 2. In the case of atomic force microscopy, assembly was stopped by diluting the sample 1:80 and adding it onto freshly cleaved mica (incubation time 2 min). The backbones of imaged filaments were traced manually, and the individual filament lengths were calculated from the coordinates. The data presented in Figure 3 were recalculated from our previous publication.30 As we have shown in the latter study that the mean length of filaments measured by both methods does not differ significantly, data sets from electron and atomic force microscopy have been combined for statistical reasons. For each time point, the length of 650 to 1140 filaments was measured. For Figure 5, we determined the length of more than 1000 filaments obtained by electron microscopy of newly assembled filament preparations (protein concentrations 0.1 and 0.02 mg/mL). Whereas the whole length of one ULF monomer (n) is around 60 nm, the length (m) of one ULF finally adopted within the IF is assumed to be 43 nm.48 Vimentin assembly proceeds via ULF annealing at both nonpolar ends, whereby two annealing ULF ends interdigitate by approximately 20 nm. Hence, two ULFs of 60 nm length each will form, after annealing, an IF of ∼100 nm length. To convert lengths from nanometers (L) to number of ULFs (i), the following expression is used: [(L - n)/m] + 1 = i. (45) Sciortino, F.; De Michele, C.; Douglas, J. F. J. Phys.: Condens. Matter 2008, 20, 155101–155111. (46) Starger, J. M.; Brown, W. E.; Goldman, A. E.; Goldman, R. D. J. Cell Biol. 1978, 78, 93–109. (47) Steinert, P. M.; Idler, W. W.; Zimmerman, S. B. J. Mol. Biol. 1976, 108, 547–567. (48) Steinert, P. M.; Marekov, L. N.; Parry, D. A. D. J. Biol. Chem. 1993, 268, 24916–24925. (49) Storm, C.; Pastore, J. J.; MacKintosh, F. C.; Lubensky, T. C.; Janmey, P. A. Nature 2005, 435, 191–194. (50) Timasheff, S. N.; Grisham, L. M. Annu. Rev. Biochem. 1980, 49, 565–91. (51) Tomski, S. J.; Murphy, R. M. Arch. Biochem. Biophys. 1992, 294, 630–638. (52) Wickert, U.; Mu¨cke, N.; Wedig, T.; Mu¨ller, S. A.; Aebi, U.; Herrmann, H. Eur. J. Cell Biol. 2005, 84, 379–391.
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Figure 3. Temporal evolution of the mean lengths for vimentin filaments at two concentrations. To evaluate if a certain model under consideration is appropriately describing the assembly, the length distributions are needed. After translating the length of an individual filament determined in nanometers into the corresponding number of ULFs contained in this filament, the number of filaments composed of i ULFs, the so-called filaments of degree i, is counted. This number is then divided by the total number of filaments to obtain the proportion of filaments of degree i. Model. To study the assembly dynamics of elongated objects or filaments, it is important to know their length distribution. The length of the filaments influences the association rate constants because only some parts of the filaments are reactive (e.g., their ends). Several short filaments would contribute much more to the polymerization than one long filament. To investigate IF assembly, the temporal evolution of Fi, the concentrations of filaments of degree i, is described. The state variable F1 represents the concentration of filaments of degree 1. In other words, F1 is the concentration of ULFs. At the initial time, it is assumed that the smallest structural stage is the ULF. Here the only reactions considered are •
ULF-ULF, that is, k0 d1, 1 F1 þ F1 f F2
•
ULF-filament, that is, k0 d1, i F1 þ Fi f Fi þ1
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Figure 4. Temporal evolution of the length distribution of vimentin filaments for an initial concentration of 0.1mg/mL. • and filament-filament, that is, k0 di, j Fi þ Fj f Fi þj
where i and j are the filament lengths in ULFs or the polymerization degree of the filaments. The constants k0 di,j represent the rates at which the reactions proceed. In the in vitro IF assembly situation, there is no significant filament disassembly, and reactions are assumed to be irreversible.30 In addition, because no protein is added or removed during individual experiments, the model describes a closed system. Model hypotheses are based on phase 2a-c presented in Figure 1. On the basis of the above hypotheses, the model is written as follows: N -1 X dF1 ¼ k0 -2F1 d1, i Fi dt i ¼1
0
j -1 X
dFj ¼ k0 @ di, j -i Fi Fj -i -2Fj dt i ¼1
N -j X
! ð1Þ Figure 5. Similar distributions are obtained for c = 0.1 and
1
0.02 mg/mL, but the assembly is 5 times faster for c = 0.1 mg/ mL than for c = 0.02 mg/mL.
dj , i Fi A,
i ¼1
j ¼ 2, :::, N -1 ð2Þ N -1 X dFN di, N -i Fi FN -i ¼ k0 dt i ¼1
! ð3Þ
The parameter k0 is an intrinsic bimolecular rate constant for the association of two filament ends, and di,j is a length-dependent proportionality factor (see below). To mimick experimental conditions, the model is considered with the initial conditions F1(0) = c, "i > 1, and Fi(0) = 0, where c is the initial concentration of ULFs. For a given initial concentration c, N is the total number of ULFs present in the experiment: N = cVNA, with V being the volume and NA being Avogadro’s number. The parameter N can also be interpreted as the maximal possible length of a filament for a given initial concentration c. The total association rate constant between a filament of degree i and a filament of degree j is then k0 di,j. This assumes that the association is diffusion-controlled with the objects in the correct relative orientations of their reactive regions to interact. Thus, the rate depends on the size and shape of the reactants and the viscosity of the solution.24,42 8820 DOI: 10.1021/la900509r
Early on, Hill24 studied the length dependence of the rate of annealing of two linear aggregates in solution. In that work, linear aggregates were considered to be rodlike objects, with their ends being the only reactive region. On the basis of statistical arguments that included the loss of degrees of freedom in the ends of the rod upon association, Hill concluded that for long rods the association rate was proportional to [i ln j + j ln i]/[ij(i + j)], where i and j represented the number of elements in the two linear aggregates. The proportionality factor di,j governing the association rate of two filaments of size i and j is then obtained as di , j : ¼
i ln j þ j ln i ijði þ jÞ
ð4Þ
Because Hill’s treatment of the association of rigid rods is based upon an expression of the hydrodynamic friction factor that breaks down for short objects, we approximate the lowest-order parameters d1,1 = 0.27 and di,1 = d1,i as di,1 = (0.4i + ln(i))/ (i(i + 1)). In the model, the aggregation is assumed to be driven only by diffusion control (Brownian collisions). Only end-to-end annealing is possible, and filaments are considered to be rigid objects. Hence, all objects, from ULFs to filaments of degree N, are assumed to have the same properties except for Langmuir 2009, 25(15), 8817–8823
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their length. Therefore, only objects differing in length are considered. Distribution. To compare model responses to data, length distributions over time are needed. To obtain the length distribution of filaments, the proportions of filaments of degree i, P(X = i)(t), must be defined: Fi ðtÞ i ¼ 1, 3 3 3 , N, PðX ¼ iÞðtÞ ¼ N P Fi ðtÞ i ¼1
Note that "t g 0, ΣN i=1 P(X = i)(t) = 1. The expected value of X, LX(t), which is also the mean length of filaments expressed in the number of ULFs, is
LX ðtÞ : ¼ E½XðtÞ ¼
N X
N P
iPðX ¼ iÞðtÞ ¼ i ¼1 N P i ¼1 i ¼1
iFi ðtÞ Fi ðtÞ
The feature LX(t) is also interpreted as the mean degree of polymerization. Model Validation. To validate the model, experimentally obtained length distributions of IFs are used as a comparison criterion between model responses and data. A value of the intrinsic rate constant k0 has to be determined to obtain the best fitting of data distributions. For distribution fitting, a genetic algorithm is used to minimize the global error between the data and the distributions obtained from model responses. This has the advantage that it can be easily translated into “parallel code”. Because the formation of ULFs is neglected, the early time points are considered with a lower weight. At each time, the error is computed as the sum of the squared residues, giving a measure of the distance between the observations and the model. Because the calculation of the global error can use different weighting vectors, several values of k0 are obtained.
Results and Discussion We have demonstrated previously that the length of filaments can be accurately determined by measuring their contour length after deposition onto different kinds of supports such as glow-discharged carbon-coated copper grids for electron microscopy and mica for atomic force microscopy.30 In the same study, we also investigated two ways to terminate assembly (i.e. fixation with glutaraldehyde versus vigorous dilution). Therefore, we wanted to investigate whether the length of different filaments has a significant influence on the binding to the surface. In particular, we employed six different deposition times (i.e., 30 s to 5 min for filaments obtained 15 min after the initiation of the assembly).30 Here, 90% of the filaments exhibited a length between 60 and 1300 nm. Notably, for this wide length spectrum of filaments, we did not find any discrimination in the binding of long filaments versus short filaments. These experiments further indicated that the dissociation of IFs is negligible, as in the case of terminating assembly by vigorous dilution in which the filament mean length did not differ at all from the values obtained by instantaneous fixing with glutaraldehyde without dilution. Moreover, it is important to note that for the atomic force microscopy measurements no fixing, staining, washing or drying steps were employed (i.e., adsorbed filaments were indeed observed in solution). Therefore, because the values for the length of individual filaments are the same with both types of measurements, filaments are obviously not significantly altered with respect to their length in the assembly regimes employed. Langmuir 2009, 25(15), 8817–8823
Early on, we routinely carried out assembly at room temperature. However, we learned when IF proteins from various species (i.e., fish to man) were investigated that temperature optima of assembly differ considerably. Moreover, the window of the “permissive” temperature of assembly for every species is indeed very distinct.5,15,16,43 In these studies, it was shown that by raising the temperature slightly above that for the optimum assembly, heavy non-IF aggregation is a consequence. For instance, trout vimentin has its optimal temperature of assembly at 24 C. At 26 C, it still formed IFs to some extent, but at 28 C, “open” non-IF structures prevailed.17 The viscosity generated by these structures is very low, similar to that gained with the mutant HumVim(SAT), whose assembly is staled at the ULF stage.19 In summary, temperature has a more complex influence on IF assembly than encountered in a standard biochemical reaction because it influences the conformation of subdomains within these proteins considerably and thereby the multiple interactions occurring during assembly. For this reason, we now carry out assembly experiments at the body temperature of the respective organism from which the IF protein is derived . Data Acquisition. Vimentin forms stable tetrameric complexes at low ionic strength. Upon raising the ionic strength by the addition of a concentrated salt solution, they self-assemble instantaneously and completely into ULFs within seconds. Hence, we consider ULFs to represent the principal IF assembly unit (i.e., the minimal IF). The process of IF elongation proceeds by a filament-forming mechanism that is driven by the collision of extended macromolecular assemblies and their longitudinal annealing via filament ends.18,30 Because IFs are nonpolar, either end of an IF can anneal to the end of another filament. Another notable property of IF proteins is that they exhibit a relatively low critical concentration for assembly that is far below that routinely used for in vitro assembly experiments. Conventionally, in vitro assembly has been performed in a concentration range of 0.1 to 1.0 mg/mL,1,6,7,12,25,26,47 which is close to the concentration estimated for vimentin in fibroblast cells (i.e., 1.6 mg/mL or 0.15 fmol per cell).13 However, the formation of ULFs by vimentin has been observed at protein concentrations as low as 0.005 mg/mL;17 therefore, a critical concentration would have to be even lower. The time to reach a certain mean filament length is inversely proportional to the protein concentration (Figure 3). Hence, at a protein concentration of 0.2 mg/mL of vimentin, filaments with a mean length of 480 nm or approximately 11 ULF units are encountered at 600 s, whereas at half this concentration assembly filaments exhibit the same mean length after 1200 s. Because the mean length values do not give sufficient information about the species present at the individual time points, we determined the filament length distribution over time (i.e., at 10, 30, 60, 300, 600, and 1200 s). With these data at hand, our goal was to establish a mathematical model that would facilitate the prediction of the time-dependent length distribution for vimentin IFs during early assembly. When analyzing the filament species formed at a protein concentration of 0.1 mg/mL, we observed that the measured distributions of filament lengths exhibit exponential decays at 10 and 30 s. At 10 s, most of the measured particles are ULFs, and only approximately 15% of the filamentous structures are longer than ULFs. At 30 s, only half of the filaments are ULFs. The other half represents filaments consisting of 2, 3, or 4 ULFs in decreasing amounts. From 60 s on, the distribution deviates from an exponential decay (Figure 4a). We have now observed asymmetric distributions with peak values of about 2 ULFs (60 s), in between 3 and 4 ULFs (300-600 s) and 5 ULFs (1200 s) (Figure 4b). DOI: 10.1021/la900509r
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Figure 6. Model versus data. The best fit of the length distribution is obtained by using a global error in the six time points with a weighting vector [1 1 5 10 20]. The model is considered with k0 = 1.7 μM-1 s-1.
Moreover, when we compared the assembly data collected at 600 s for 0.1 mg/mL but now at one-fifth of the protein concentration (i.e,. 0.02 mg/mL), a nearly identical filament length profile was obtained at 3000 s corresponding to the inverse linear relationship of protein concentration and time needed to obtain a certain mean filament length (Figure 5). Application of the Model. We investigated whether the system (1)-(3) presented above would successfully describe the time-dependent length distributions in a general way. Model responses are determined by only two parameters: the proportionality factors di,j and the intrinsic rate constant k0. In this model, the proportionality factors di,j are, for every pair of interacting filaments, directly dependent on the length of two filaments under consideration. Therefore, di,j determines the shape of the filament length distribution for every individual mean length encountered in the course of assembly. Note that in our approach di,j values are dimensionless. The intrinsic rate constant, k0, is intended to modify the speed of filament formation and can therefore be used as a fitting parameter. With these tools, we now explored whether the length distributions derived from the assembly experiments agreed with the curves predicted by the model. Depending on weighting vectors used, different values for k0 are found. In the range of 1.5 to 2.6, we obtained curves that fit the experimental data well. As an example, we show a fit with k0 equal to 1.7 μM-1 s-1 (Figure 6). It describes the measured assembly distributions of early time points very well. The fitting obtained at 1200 s is less optimal but still very good, and the observed deviation of measured and calculated length profiles may reflect a problem in the model due to the fact that it deals with stiff rods whereas vimentin filaments are highly flexible with a measured persistence length of ∼1 μm roughly corresponding to F25.32 In our previous model,30 we had to introduce different binding constants to fit the shape of the curve 8822 DOI: 10.1021/la900509r
for assembly times below 60 s appropriately. With respect to those results, we could not exclude the existence of a short delay in ULF-to-ULF association, but with the new equation system (1)-(3), it is not necessary to make different assumptions for the measured filament lengths at the different time points. Here, for a given experimental condition, we want k0 to be a global parameter in order to provide a criterion by which to compare the different assembly conditions and IF proteins. Nevertheless, allowing different values of the parameter k0 at different time points would provide a tool for investigating the changes in the reaction speed over the time and understanding if the dependence of the aggregation on the object size varies over time. In future studies, we will investigate this very interesting point in more detail. In the next step, we wanted to learn what the model would predict for the persistence of certain shorter filament species (e.g., F5, F10, and F15). From electron microscopy images obtained at 1 h of assembly, the impression persists that only very long filaments are present at this time. However, the distributions predicted by the model show that after extended times relatively distinct numbers of these short filaments should be present (Figure 7). Indeed, they can be found on close inspection. However, most of them may be obscured in the very dense networks usually obtained under the conditions used for electron microscopy, and their quantification may not be that easy, though we will follow up on this issue in future experiments. Furthermore, if and how the ULF concept may be applied to the in vivo situation of IF formation are of great interest. Previously, in one cellular situation (i.e., the attachment of trypsinized fibroblasts to a solid support), a large number of dot-like vimentin structures were observed in spread out areas.39 These dots apparently “fuse” over time as observed by live cell Langmuir 2009, 25(15), 8817–8823
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Figure 7. Temporal evolution of some concentrations of filaments of degree 5 (F5), filaments of degree 10 (F10), and filaments of degree 15 (F15) with an initial concentration of c = 0.1 mg/mL.
imaging employing green fluorescent protein-tagged vimentin in short filaments, called “squiggles”, and these squiggles further longitudinally anneal into long filaments.53 In the course of these studies, attempts have been made to visualize assembly precursors of IFs by electron microscopy.39 However, because of the small size of these precursors, they were not unambiguously identified. Nevertheless, recent life cell imaging results make it very likely that the IF assembly mechanism proposed by us for the in vitro assembly is highly relevant to the in vivo situation.14 Last, the current observations and the model derived from the length distributions may enable us to gain more insight into the development of viscosity as well as the establishment of networks in an assembling filament population. As previously published, measurements with a capillary viscometer demonstrated that the bulk viscosity of assembling vimentin solutions is reached 15 min after the initiation of assembly.19 In addition, using bulk rheology measurements we found that the storage modulus (G0) also increases very quickly.44 Depending on the protein and salt concentrations, network formation starts by 2 min and is complete by 30-60 min.10 It will be of very high interest to investigate how the filament length distribution and the storage modulus (i.e., the viscoelasticity of the system) correlate. (53) Yoon, M.; Moir, R. D.; Prahlad, V.; Goldman, R. D. J. Cell Biol. 1998, 143, 147–157.
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Several studies have investigated the filament length distributions of systems that are similar to our system (e.g., filaments that fuse end on and where no nucleation is needed). If disassembly is allowed, then the equilibrium length distributions will be found in an exponential mode.9,35,45 If there is no disassembly, then peak distributions comparable to those observed in this study will prevail.11,51 Here we present a model for the assembly of the cytoplasmic IF protein vimentin that substantially differs from models proposed for actin or tubulin assembly. We have employed a theory for the aggregation of linear macromolecules, as previously proposed by Terrell L. Hill (1983) in a more general context, to study the temporal evolution of the IF length distributions obtained by the measurement of vimentin IFs assembled in vitro. Because of the exhaustive, fast formation of ULFs, which takes place in the second range, we neglected nucleation. Moreover, in the case of IFs this is not a typical nucleation reaction (i.e., the formation of uniquely structured complexes that allow the addition of monomeric subunits), but it is instead the quantitative formation of “minimal” filaments (i.e., ULFs) that represent the basic building blocks of longitudinal filament growth. Notably, as ULF formation proceeds in the second range, the longitudinal annealing takes about 1 min under standard conditions. Both the formation of ULFs and the longitudinal elongation reaction engage a multitude of strong ionic and hydrophobic interactions such that IF formation is considered to be practically irreversible in vitro.30,33 The associations are driven by Brownian collisions. Every collision is assumed to result in growth. Objects are assumed to be rigid and reactive only at their ends. The model emphasizes the dependence of the association rate constants on the object size and shape. Moreover, an intrinsic rate constant, in terms of k0, is included to tune the fit of the experimental observations for the various time points (i.e., k0 implicitly includes other phenomena, which may influence the association kinetics). With this constant, we can now compare the elongation reaction of different IF proteins belonging to the vimentin assembly group in a quantitative manner.20 Last, this model may be of significant value to describe kinetic defects encountered with mutants of these IF proteins that cause severe human diseases.2,34,40 Acknowledgment. H.H. acknowledges grants from the German Research Foundation (DFG; HE 1853/4 - 3, 5 - 2 and BA 2186/3 - 1). S.P. is supported by an NSERC Discovery Grant. We thank the anomynous reviewers and the editor for their helpful suggestions.
DOI: 10.1021/la900509r
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