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* Polymers and Complex Fluids, Department of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom; School of Biomedical Sciences, University of Leeds, Leeds, LS2 9JT, United Kingdom; Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany; and Institut Laue-Langevin, BP 156-38042, Grenoble Cedex 9, France
Correspondence: Address reprint requests to Thomas A. Waigh, E-mail: t.a.waigh{at}leeds.ac.uk.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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-solvent. Furthermore, measurements of the flexibility as a function of temperature demonstrate that titin-II and the I-band titin fragment experience a similar denaturation process; unfolding begins at 318 K and proceeds in two stages: an initial gradual 50% change in persistence length is followed by a sharp unwinding transition at 338 K. Complementary microrheology (video particle tracking) measurements indicate that the viscoelasticity in dilute solution behaves according to the Flory/Fox model, providing a value of the radius of gyration for titin-II (63 ± 1 nm) in agreement with static light scattering and small angle neutron scattering results. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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3.0–3.5 MDa) and is one of the major constituents of the striated muscles of vertebrates (Bray, 1992
; Wang et al., 2001). The protein is involved in a number of important regulatory mechanisms of muscle related to its development, structural organization, elasticity, and intracellular signaling (Tskhovrebova and Trinick, 2003; Granzier and Labeit, 2004). Titin's role in muscle elasticity is a major issue in current studies of the molecular mechanisms involved in muscle function. A titin molecule is >1-µm long and spans half of the sarcomere, the repeating structural and contractile unit of muscle (Fig. 1). More than half of the molecule is an integral part of different sarcomere elements, i.e., of the thick filament and the Z- and M-line regions. The rest of the molecule is not bound to the other sarcomere proteins, but functions as an elastic connection between the thick filaments and the Z-line region. When the sarcomere contracts or elongates this part of titin coils up or extends in proportion. As a result, mechanical properties of the molecule are directly related to the elastic properties of the sarcomere. A full picture of the viscoelasticity of titin, therefore, is of vital importance in modeling the dissipative processes involved in the functioning of striated muscles.
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). The interdomain linker sequences are estimated to contain between two and five peptide residues along most of the length of titin. Flexibility of the linker regions may provide some interdomain mobility resulting in global flexibility of this multimodular protein.
The flexibility and extensibility of titin, its individual segments, and expressed small fragments have been studied previously using a series of techniques: dynamic light scattering (Higuchi et al., 1993), atomic force microscopy (AFM) (Rief et al., 1997), optical tweezers (Kellemayer et al., 1997; Tskhovrebova et al., 1997), and transmission electron microscopy (Tskhovrebova and Trinick, 2001). The estimates of the persistence length obtained, which is a quantitative measure of bending stiffness, vary for the native protein by almost a factor of 10, ranging from 42 nm from in situ mechanical measurements for the physiologically elastic region (Linke et al., 1998), to 3 nm, from single-molecule experiments as an average estimate over the majority of titin's length (Leake et al., 2004). The exact reasons for such a discrepancy remain unclear. It is likely, however, that they reflect both the structural and mechanical differences in the titin segments studied, and the shortcomings of different methodical approaches. Structural modeling does predict that interdomain mobility will differ to some extent in different regions of the molecule (Amodeo et al., 2001; Fraternali and Pastore, 1999). Immunoelectron microscopy also suggests that non-Ig/Fn3 regions of titin are significantly more compliant than the regions composed from Ig and Fn3 domains (Granzier et al., 1996), and thus the relative presence of these regions in a particular segment of the molecule will affect the average estimate. Additional complications are related to the extent the behavior of the molecule is compatible with models for its conformation, subsequently used for fitting experimental curves. In particular, there are a number of questions related to the effects of torsional modes (Yoshizaki and Yamakawa, 1980) and polyelectrolyte ionic strength effects (Odijk and Houwaart, 1978), which could play a role in determining the elasticity of this biopolymer and are not considered in current simplified approaches.
In this work we have applied scattering techniques, static and dynamic light scattering (Berne and Pecora, 2000; Hohenadl et al., 1999) and small angle neutron scattering (King, 1999), as well as video particle tracking microrheology (Goodman et al., 2002; MacKintosh and Schmidt, 1999) for the comparative study of purified titin-II corresponding to the mainly physiologically inelastic A-band part of the molecule, bound in situ to thick/myosin filament, and a proteolytically isolated fragment from the physiologically elastic I-band part of titin. In comparison to electron microscopy and mechanical experiments on single molecules, these techniques have the advantage that they minimally affect the equilibrium conformation of the protein and give ensemble averages over a whole series of molecules, thus producing accurate representative measurements. The techniques provide direct information regarding the size, viscosity, and diffusion coefficient of the protein and estimates of the persistence length. In comparison with light scattering, small angle neutron scattering is significantly more sensitive to small length scales, allowing access to the direct signature of the chains' flexibility (Higgins and Benoit, 1994) and providing better discrimination between possible models of the chains' conformations.
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EXPERIMENTAL |
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). SDS-PAGE shows that the size of the purified protein in these preparations is smaller than the size of intact protein present in muscles. This is due to the loss of an N-terminal segment containing the Z-line and part of the I-band regions, due to the action endogenous proteinases during the purification. This preparation is therefore usually referred to as titin-II. The ß-connectin preparation used in previous light scattering studies is similar (Higuchi et al., 1993). The titin-II includes the full length A-band part together with the C-terminal portion of the I-band part of the native molecule. The samples were kept in glycerol at –20°C and subsequently dialyzed for 24 h against selected buffers. The buffer recipes for titin-II in solution were: 1), buffer A (pH 7.4–7.5, Debye screening length 
) contained 0.5 M KCl, 10 mM TRIS, 1.0 mM DTT, 2.0 mM EGTA, 1.0 mM NaN3; and 2), buffer B (pH 7.4, 
) contained 0.5 M NaCl, 20 mM TRIS, 0.3 mM DTT, and 1.0 mM EGTA.
A proteolytic fragment from the physiologically elastic part of titin (referred to further in the text as the "I-band titin fragment") was prepared as described by Houmeida et al. (2003). From transmission electron microscopy measurements, the fragment length was 100 ± 20 nm. The fragment included the tandem Ig segment of the molecule starting at the Ig domain I20 (according to the annotation given by Labeit and Kolmerer, 1995) from the I-band region near the end of the thick filament (Fig. 1).
Measurements were performed at room temperature, except in dynamic light scattering experiments where the temperature dependence was studied.
Static/dynamic light scattering
An ALV-5000 goniometer was used for both static and dynamic light scattering experiments with a fast correlator (from 12.5 ns to 1000 s) and an argon ion laser (488 nm, 100 mW). The intensity of the scattered light was calibrated with respect to a toluene standard. The temperature control was accurate to ±0.1°C, and was provided by water circulating around the toluene bath and cuvette.
The dynamic light scattering/static light scattering (DLS/SLS) measurements were performed in the range of angles between 30 and 120° with an angular step of 10°. The collection time for each angle was 10–15 min, depending on the signal/noise ratio.
Titin-II at a concentration of 1.41 mg/mL in buffer B was centrifuged at 40,000 g at a temperature of 288 K for 2 h to remove dust contaminants and aggregates, as reported previously (Higuchi et al., 1993). After centrifugation, the solution was carefully transferred into the scattering cell. The elastic fragment at a concentration of 1.60 mg/mL in buffer B was filtered with polyethersulfone membranes (low binding protein filters) with 0.8-µm pore size to minimize the presence of aggregated protein.
Analysis of static light scattering data
The radius of gyration of titin-II was calculated from fits of the Debye function (P(x)) (Berne and Pecora, 2000) for the flexible polymers to the measured intensity versus scattering vector curves:
 | (1a) |
 | (1b) |
is the scattered angle, n is the index of refraction of the solvent and 
is the wavelength. L indicates the contour length of the protein, lp is the persistence length, and Rg is the radius of gyration.
Analysis of dynamic light scattering data
In the dynamic light scattering experiments the normalized time autocorrelation function g2(q,t) of the scattered intensity (I(q,t)) is measured:
 | (2) |
The latter can be expressed in terms of the field autocorrelation function g1(q,t) (i.e., the time autocorrelation function of the scattered electric field) through the Siegert relationship (Berne and Pecora, 2000)
 | (3) |
0.51 for the equipment employed.
Standard Contin analysis of the correlation curves showed a single relaxation time in the decay rate distribution versus time. Therefore, diffusion coefficients (D) were calculated from the field correlation functions g1(q,t) fitted with a single exponential decay.
 | (4) |
is the relaxation rate and t is time.
Small angle neutron scattering
The experiments were carried out on two different large-scale facilities, KWS1 (FRJ2) (www.neutronscattering.de.) and D11 (ILL) (www.ill.fr/lss/grasp/grasp_main.html), at the Forschungszentrum Jülich (KWS1) and the Institute Laué Langevin (Lindner et al., 1992), respectively.
KWS1 (FRJ2)
A combination of three camera lengths (2, 8, and 20 m) was used, providing a momentum transfer q between 10–3 Å–1 and 0.2 Å–1. The raw two-dimensional data were corrected for the empty cell scattering and D2O background. The detector sensitivity corrections and the transformation to absolute scattering cross sections were made with a Lupolene standard (d
/d
= 1.78155 cm–1).
The specimens were dialyzed against D2O buffer to provide the correct contrast for the small angle neutron scattering (SANS) experiments, and ultraviolet absorption was used to calibrate the sample concentrations after dialysis. The samples in D2O were loaded in flat Helma cells with a 2-mm thickness of specimen to provide the optimal pathlength. The sample temperature was maintained at 296 K during the experiment.
D11 (ILL)
The camera lengths used were 2 and 8 m to insure overlap between the separate measurements and access to all the relevant length scales. Momentum transfers could be accessed in the range between 2.5 x 10–3 and 2 x 10–1 Å–1. The software package Grasp was used combined with the MATHLAB 5.1 software to analyze the SANS data (D11) of the fragment. The flat field correction was taken from a 1-mm H2O sample. The empty cell background was subtracted; the beam center was calibrated with respect to the direct beam and absolute normalization was achieved relative to the water standard.
The scattering cross section (d/d) for neutrons is defined as:
 | (5) |
is the contrast factor between the polymer and the matrix solvent, calculated on the basis of tabulated values of the amino acid coherent scattering lengths (Jacrot, 1976); P(q) and S(q) indicate the intramolecular form and the intermolecular structure factors, respectively (Higgins and Benoit, 1994). In the specific case of the dilute regime S(q) 
1.
The radius of gyration of titin-II was evaluated by Guinier analysis of the SANS data using Eq. 6, which is valid in the dilute regime, where the interparticle interactions are negligible, and in the limit of qRg < 1 (King, 1999):
 | (6) |
Video particle tracking microrheology
Microrheology experiments were performed on titin-II in buffer A in the range of concentrations 0.12–0.50 mg/mL, using poly(amino) probe beads (Sigma Aldrich, St. Louis, MO) of 0.472 µm diameter. The experimental temperature was held constant at 296 K. Probe particles were tracked using an Olympus OH2 microscope and a modified version of the IDL tracking software of Weeks and co-workers. (http://glinda.lrsm.upenn.edu/weeks/idl/tracking.html). A 100x oil immersion lens was used to focus into the sample a few micrometers underneath the coverslip. The poly(amino) beads were chosen to reduce adsorption of titin-II onto the probes and improve phase stability of the colloid/protein mixtures. The displacement of the particle centroids was tracked in the focal plane of the objective at a rate of 25 frames per second.
Individual time-averaged mean-square displacements were calculated from the two-dimensional trajectories, and the viscosity was measured at a series of concentrations. Care was taken to provide a correct thermal equilibration of the specimens and removal of the effects of convection before recording a movie.
The viscosity of the buffer was measured to be 0.92 cP at 296 K.
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RESULTS |
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). The temperature dependence of the persistence length is shown in Table 1 and Fig. 3 b. The persistence length remained constant up to 318 K (40°C), and then sharply changed its value to 1 nm, assuming the contour length of the unfolded titin-II polypeptide is 10 µm.
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). The polydispersity index for both the whole molecular and fragments were on the order of 0.30, independent of angle and temperature (below the unfolding temperature). The diffusion coefficients were calculated from the gradient of the relaxation rate () versus q2 plots in the angular range 30–120°. The hydrodynamic radius was then evaluated from the diffusion coefficients using the Stokes-Einstein relationship:
 | (7) |
S is the solvent viscosity, and Rh is the protein's hydrodynamic size.
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):
 | (8) |
S is the solvent viscosity, and 
is defined as 1/(2lp), where lp is the persistence length.
At 298 K, the persistence length was estimated to be 16 nm for titin-II and 10 nm for the fragment. The calculations were done with the assumptions that the contour length of titin-II and the I-band titin fragment were 1 µm and 100 nm, respectively. The values of lp of both proteins at different temperatures are collected in Tables 1 and 2. The temperature dependence of the persistence length for titin-II, calculated according to Eq. 8, is shown in Fig. 3 b (solid symbols). Initial experiments examined the effect of reducing the buffer concentration to physiological strengths (0.2 M KCl) and showed no effect on the resultant persistence length of the titin-II molecules with DLS and SLS.
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):
 | (9) |
2); Pcr(q) is the scattering function for a cylindrical shape ([2J1(qR)/(qR)]2), which approximates locally the finite cross section of the molecules; 
q1 and p1 are empirical constants, and PDebye (q) is the Debye scattering function (Eq. 1a; see earlier text). Such an analysis has the advantage over the Kratky-Porod graphical method (Kratky, 1982), because it includes the effects of the cross section of the molecules. The graphical method does however clearly demonstrate the semiflexible nature of titin-II and the I-band titin fragment with a q–2 scaling (Gaussian coil) at large length scales and q–1 scaling (rodlike) at small length scales (see Fig. 8 a, inset). Fits of Eq. 9 to experimental data are shown in Fig. 8, a and b. This analysis provided values of the radius of gyration for titin-II and the I-band titin fragment of 50 and 17 nm, respectively. From these values the persistence lengths were estimated to be 10 and 9 nm for titin-II and the I-band titin fragment, assuming a contour length of 1 µm and 100 nm, respectively, and a cross-radius of 2 nm for both proteins.
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/d vs. q2) as it is shown in Fig. 9. This analysis provided a value of 56 ± 5 nm. Using this estimate, the persistence length of 9 ± 2 nm of titin-II was calculated from the relation given for Gaussian coils (Eq. 1b), assuming a contour length of 1 µm. The Guinier analysis could not be applied to the fragment, because the condition qRg < 1 was not satisfied in the data set.
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1. Using the estimated value of the radius of gyration from Guinier analysis and a nominal value of 3.0 MDa for the molecular weight of the protein, we evaluated the overlap concentration (c*), i.e., the concentration above which intermolecular interactions could not be neglected:
 | (10) |
7 mg/mL and the measurements were in the dilute regime. We deduce that the SANS experiments were at sufficiently low concentrations to neglect interchain interference, because there is good agreement of the radius of gyration with results from other techniques (Table 3). The impact of concentration would be even smaller on higher q features such as the persistence length.
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r2
),
 | (11) |
The mean-square displacement of the poly(amino) beads as a function of time for 0.12–0.50 mg/mL titin-II concentrations showed a pure viscous behavior (Fig. 10) (i.e., r2t). Subsequently the viscosity was calculated from the Stokes-Einstein relationship (Eq. 7), where Rh is replaced by the radius of the probe particle.
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], was then used as a means to estimate the size of titin-II in solution. [] was determined from the intercept of a plot of the specific viscosity versus concentration according to the following relationship (Burchard, 1999; Goodman et al., 2002):
 | (12) |
Mw is the protein molecular weight, NA is the Avogadro number, S is the solvent viscosity, and R is the protein size. Microrheology measurements provided a value for the specific viscosity [] of 400 ± 40 mL/g, which is in agreement with the 410 mL/g estimated from previous light scattering data for ß-connectin (Fujime and Higuchi, 1993). In Fig. 11 the specific viscosity is plotted versus concentration with a fit of Eq. 12.
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S extrapolated from the experimental data fitted with Eq. 12 was 1.2 ± 0.1 cP, an accuracy of 20% on the measured value for the solvent. A value of 65 ± 5 nm was found for R. In Table 3 the main results obtained from the different experimental techniques for the persistence length are collected. The persistence length of titin-II, calculated as for Gaussian coils (Eq. 1b), gave a value of 12 ± 2 nm assuming a contour length of 1 µm. |
DISCUSSION |
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), and its proteolysis during the preparation results in a loss of 1/3 of the length, producing what is known as titin-II or ß-connectin. This corresponds to the mainly physiologically inelastic thick filament-bound part of titin formed by a linear chain of immunoglobulin and fibronectin-3 domains. Similar preparations (ß-connectin of chicken breast muscle) were studied previously by means of dynamic light scattering by Higuchi et al. (1993) that provided estimates for the persistence length and other physical parameters of this part of the molecule. We analyzed the scattering properties of titin-II and of its fragment from the physiologically elastic I-band part to obtain a closer insight into the relationship between structure and flexibility of the molecule. The molecular homogeneity of the proteins was ensured by high-speed centrifugation before the experiments, and was confirmed by estimates of its molecular weight based on the parameters evaluated from the scattering data.
Persistence length
The range of scattering vectors used in this neutron study satisfy the condition 2
/q lp for lp between 10 and 20 nm, indicating that the experiments contain direct information on the semiflexibility of the molecules in this regime. The SANS data from titin-II and the I-band titin fragment allows robust self-consistent measurement of the persistence length.
Fits of a modified Sharp Bloomfeld model shown in Fig. 8, a and b, provided more accurate values of lp, than the Kratky/Porod graphical method (Fig. 8 a, inset), because they take better account of the chain's statistics (semiflexible nature) and include the cross section of the chain (Pedersen and Schurtenburger, 1996). Combining the results from DLS/SLS and SANS measurements the ratio between the hydrodynamic size and the radius of gyration was calculated, which allows information on the molecular geometry to be made (Rubinstein and Colby, 2003). Rg/Rh was evaluated as a function of temperature. The naïve theoretical calculation for a flexible Gaussian chain in a -solvent gives the expected value of Rg/Rh from Zimm theory is 1.5, whereas the accepted experimental value (Rubinstein and Colby, 2003) is measured to be 1.30. Our combination of SLS and DLS measurements at ambient temperature provides a value of 1.3 ± 0.1, in agreement with the accepted experimental value and with modern molecular dynamics simulations (Oono, 1983). This gives an additional proof of the flexible Gaussian nature of the titin-II molecules at large length scales. Within error we find no significant change in the persistence length between the titin I-band fragment and the titin-II molecules. We conclude that the I-band section of the titin molecule beyond its attachment point with the thick filament, is a semiflexible chain of 10.0 ± 0.3 nm persistence length (see Table 3), which globally acts in a Gaussian manner. Small differences in the persistence length derived from the different experimental techniques are probably due to the different weighting of the techniques to molecular specific features, e.g., torsional modes of rotation, hydrophilic/hydrophobic interdomain interactions, etc. Explanation of these subtle effects is a goal for future improved modeling.
Evidence for association was found for noncentrifuged samples as the forward scattering in SLS experiments dropped during temperature ramps, indicating fragmentation of the aggregates. In this behavior the molecules resemble the behavior of sticky charged hydrophobically modified polyelectrolytes (Di Cola et al., 2004).
Effect of temperature
The flexural rigidity (
) of the titin-II was calculated as a function of the temperature according to the following relationship:
 | (13) |
is calculated to be 5 and 7 x 10–20 dynecm2 for titin-II and the I-band titin fragment, respectively, close to the value found for ß-connectin ( = 6 x 10–20 dynecm2). The flexibility of titin-II depends on the temperature in a manner that is not explained by the standard temperature dependence expected for the modulus of a wormlike chain (Odijk and Houwaart, 1978). We deduce that a series of structural changes occur during its denaturation.
In Fig. 3 b the temperature dependence of the persistence length is shown; lp was calculated according to Eqs. 1.b and 8 assuming a contour length L of 1000 nm in the temperature range below 318 K; above this threshold the protein starts to denature. The denaturation process involves a change in the length of the molecules, due to the unfolding of titin domains. L in the fully unfolded state is expected to be 10 times the length in the native state. Thus a value of 10 µm was qualitatively assumed above 318 K, providing an upper bound for lp. Temperature measurements showed a gradual rather than a sharp transition between coil and uncoiled states (Fig. 3, a and b), which could reflect the unfolding of the domains in transient intermediate steps. Generally, large proteins composed of multiple structural domains lead to complex unfolding curves, because the independent domains unfold statistically depending on random localized thermal perturbations (Creighton, 1993).
It is important to highlight that at a temperature of 333 K the q2 dependence of the decay rate was no longer observed; thus the value of the diffusion coefficient was calculated by the relaxation time at an angle of 90°. This only provides an estimate of the protein hydrodynamic size in the unfolded state.
The persistence length of the unfolded titin domains is measured to be at least 10 times shorter than that of the native state, indicating a high flexibility in the denatured state of the molecule (e.g., Rief et al., 1997). The DLS measurements offer another piece of experimental evidence for this feature of the protein dynamics.
Dynamic light scattering can be used as a molecular probe of the thermal denaturation of titin. The results are in agreement with the denaturation temperature (333 K) previously found for bovine and porcine titin using differential scanning calorimetry (Pospiech et al., 2002). A new result is that there is a gradual process of unwinding/decrease in persistence length, which occurs as a precursor to the DSC endotherm, in the range of temperatures between 318 and 333 K.
Viscoelasticity
Video particle microrheology examines the thermal motion of colloidal particles embedded in a material to extract the bulk rheological properties. Compared with conventional rheology and scattering techniques, only small amounts of material are required (order of µL) (MacKintosh and Schmidt, 1999).
The microrheology experiments allowed a robust measurement of the radius of gyration assuming the non-free-draining Flory/Fox model (Edwards and Doi, 1986; Goodman et al., 2002), for which the viscosity of a dilute suspension of flexible polymers increases linearly with concentration. The observed difference between the measured and the extrapolated value of the buffer viscosity in the microrheology experiments (s) could be explained by a small degree of adsorption of the protein onto the poly(amino) probe beads. Thus, a correction was made taking into account that an adsorbed layer with thickness h is formed onto the beads of size a. We assumed h was independent of the titin-II concentration in the range investigated and we note that this assumption only has a small effect on the calculated radius of gyration.
m is defined to be the viscosity measured with single particle tracking and is given by the Stokes-Einstein relationship (Eq. 7). However in the expression of the diffusion coefficient (Do) we now take into account that the dimension of the bead is (a + h):
 | (14) |
Thus the comparison between the three equations (Eqs. 7, 12, and 14) leads to the correct form of the measured viscosity m:
 | (15) |
Equation 15 provided a value for R of 63 ± 1 nm. The sizes found from microrheology measurements are compared in Table 3 with values of Rg calculated by Guinier analysis of both SANS and SLS data. Moreover, these experimental results compare well with the theoretical predictions for the end to end distance (R) of a completely flexible Gaussian chain in a -solvent, i.e., R = (2lp)(Lc/2lp)1/2. For a single titin-II molecule this leads to an estimate of 58 nm for Rg = (R2/6)1/2.
Action of titin in vivo
It is interesting to consider the statistical mechanics of the Gaussian titin chains in a pore, because this approximates to their behavior in vivo contained between actin filaments. A fundamental question is how the pore geometry will change the conformation of the flexible titin molecules. This has been previously explained in the synthetic polymer literature using the concept of thermal blobs (Cifra and Bleha, 1999; Daoud and de Gennes, 1977; Rubinstein and Colby, 2003). A rough schematic diagram of the molecular arrangement of the I-band part of titin in a pore formed by hexagonally packed actin filaments is shown in Fig. 12. We take an approximation for the diameter of the pore (Db) from the x-ray measurements of Millman (1998), i.e., Db 40 nm. The use of the cylindrical pore to approximate a hexagonal actin cage is justified, because the excluded volume of the blobs in the neighboring pores acts as a steric wall, prohibiting the escape of titin blobs through the bars of the actin cages.
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; Daoud and de Gennes, 1977; Grosberg and Khokhlov, 1994):
 | (16) |
N is the number of Kuhn segments (25), a is the Kuhn segment length (30 nm with 10 protein domains in a segment), and Db is the pore size. Using values from the scattering experiments this implies an increase in the ambient unstretched length of the chain (R|| 600 nm compared to Rg 60 nm) when compressed inside the 40-nm pore. The end-to-end length is thus roughly 10 times longer due to steric interaction with the surrounding actin filaments.
The entropic force f exerted on the ends of the chain by the internal conformational fluctuations can be evaluated according to (Rubinstein and Colby 2003):
 | (17) |
is an exponent that equals 1/2 for a -solvent (indicated by the measured ratio of Rg/Rh) (Rubinstein and Colby, 2003). We thus calculate that the force of an individual titin molecule is 5 pN at a temperature of 298 K.
Note that there is no change in the elasticity of the stretched titin chain confined to a pore, because the size of the tension blob (
= (Rx/(Nb1/))/–1 is 8 nm, much smaller than the steric blob size defined by the interaction with the walls of the pore (40 nm) (Rubinstein and Colby, 2003). Here Rx is the length of sarcomere section containing the I-band segment of titin molecule, N is the number of monomers in the free 600-nm length section, and b is the monomer length.
These calculations assume no repulsion/attraction of the titin molecules by the walls of the pores. A more sophisticated future analysis would require the consideration of the effect of more than one titin molecule in a pore, better definition of the effect of the persistence length, adsorption, and charge effects, but should be considered a future goal for the complete understanding of the molecules in vivo.
No change in the hydrodynamic radius and thus the persistence length of the free solution state titin-II was found for Debye screening lengths in the range 0.43–0.61 nm.
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CONCLUSIONS |
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ACKNOWLEDGEMENTS |
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Submitted on October 21, 2004; accepted for publication March 8, 2005.
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