The elasticity of an individual fibrin fiber in a clot

Abstract

A blood clot needs to have the right degree of stiffness and plasticity to stem the flow of blood and yet be digestable by lytic enzymes so as not to form a thrombus, causing heart attacks, strokes, or pulmonary emboli, but the origin of these mechanical properties is unknown. Clots are made up of a three-dimensional network of fibrin fibers stabilized through ligation with a transglutaminase, factor XIIIa. We developed methods to measure the elastic moduli of individual fibrin fibers in fibrin clots with or without ligation, using optical tweezers for trapping beads attached to the fibers that functioned as handles to flex or stretch a fiber. Here, we report direct measurements of the microscopic mechanical properties of such a polymer. Fibers were much stiffer for stretching than for flexion, as expected from their diameter and length. Elastic moduli for individual fibers in plasma clots were 1.7 ± 1.3 and 14.5 ± 3.5 MPa for unligated and ligated fibers, respectively. Similar values were obtained by other independent methods, including analysis of measurements of fluctuations in bead force as a result of Brownian motion. These results provide a basis for understanding the origin of clot elasticity.

Blood clots play an essential role by stopping bleeding, but they can also cause heart attacks and strokes. Clots are formed when the enzyme thrombin cleaves fibrinogen to generate fibrin monomers, which polymerize to produce a three-dimensional network of fibers (1-8). Fibrin is stabilized by ligation,¶ the formation of intermolecular covalent bonds at specific sites with a transglutaminase, factor XIIIa, rendering the whole clot stiffer and resistant to fibrinolytic dissolution (9, 10). The viscoelastic properties of clots and their major constituent fibrin are normally finely tuned to optimize how they stop bleeding while also minimizing their effect in cardiovascular disease, because bleeding occurs if clot stiffness is too low; a decreased rate of fibrinolysis and increased thrombosis and thromboembolism are generally associated with stiff and friable clots, although such relationships are complex (10-14). Although much is known of fibrin assembly mechanisms (1-8, 15-18), the origin of clot viscoelasticity remains to be established.

The elasticity of a fibrin clot, like that of rubber-like polymers, is characterized by very large deformability with essentially complete recovery (19). However, the elasticity of the fibrin clot cannot be rubber-like, because it is not a random-coil network made up of thin, highly flexible strands; instead, it is a network made up of thick branching fibers. As an example of how unrealistic such rubber-like models are, it can be calculated from clot stiffness that there would be an average of only one fibrin molecule per branch point for a rubber-like model (20), yet electron micrographs show that the clots used for these experiments commonly have ≈1 million fibrin molecules between branch points (Fig. 1). As an alternative model, it has been suggested that the clot's flexibility may arise from bending of the fibers (21, 22), and some observations of fibers in deformed states by electron microscopy support this idea (23). To investigate the microscopic basis of clot elasticity, we characterized the mechanical properties of individual fibers of plasma fibrin clots, together with the bulk viscoelastic properties of the whole clot and the three-dimensional architecture of the fibrin network. Our findings indicate that the clot's elasticity arises from bending of the fibers rather than from properties espoused in a rubber-like model, which is incorrect by six orders of magnitude (20).

Fig. 1.

Confocal scanning light micrographs of plasma clots with and without ligation. These clots were prepared in the same way as those used for the laser tweezers measurements. The fibers are labeled with 5-nm colloidal gold and viewed in reflectance mode. This image is a two-dimensional projection from 150 optical sections made with Zeiss software. (A) Unligated clot formed with DDITS, an inhibitor of factor XIIIa. (B) Ligated clot formed without inhibitor of factor XIIIa. (Scale bar, 5 μm.)

Materials and Methods

Preparation of Fibrin Clots. Clots were formed in specially designed microchambers (consisting of coverslips on microscope slides separated by spacers) by addition of calcium at a final concentration of 10 mM and α-thrombin (Enzyme Research Laboratories, South Bend, IN) at a final concentration of 0.9 units/ml to citrated human plasma. Clots were made in the presence or absence of 1 mM 1,3-dimethyl-4,5-diphenyl-2[2(oxopropyl)thio] imidazolium trifluoromethyl-sulfonate (DDITS), a highly specific, active site-directed inhibitor of factor XIII (24). Clots made in the presence of DDITS were soluble in acetic acid or urea and showed no γ dimers or α polymers on SDS/PAGE.

Confocal Microscopy and Analysis of Clot Structure and Measurement of Viscoelastic Properties of Clots. Confocal microscopy was carried out on plasma clots made in the same manner just described and then labeled with colloidal gold particles so that they could be imaged in reflection mode (14). Microchambers containing these plasma clots were connected to a reservoir, washed with buffer (0.15 M NaCl/0.01 M Tris·HCl, pH 7.4), and then washed again with the same buffer containing 5-nm colloidal gold particles at a final concentration of 2.5 × 10¹² particles per ml; excess beads not binding to fibrin were washed away. Clots prepared and viewed in this manner were identical in all measured parameters to fluorescein-labeled clots viewed in fluorescence mode, but the problem of fluorescence fading was avoided. The clots were visualized by scanning in reflection mode with a Zeiss LSM 510 confocal microscope with a ×63 water immersion objective. Generally, 150 optical sections were collected at intervals of 0.5 μm. Fiber diameters (n = 150) were measured on high magnification images by using Zeiss software. For the determination of branch point densities and fiber segment lengths, three-dimensional reconstructions were obtained by using the kheops software package (Noesis, Courtaboeuf, France).

Viscoelastic properties of clots made at the same time and under the same conditions were measured for clots between 12-mm glass coverslips by using a torsion pendulum in free oscillation, as described in refs. 25 and 26.

Optical Tweezers Experiments. The optical tweezers instrumentation used for this research was coupled to a light microscope, which was used in the differential interference contrast mode. Polystyrene latex beads with a diameter of ≈1 μm were attached to the fibers by perfusion into a fully formed clot in microchambers made by attaching a glass coverslip to a microscope slide with spacers. There was a gradient in the density of beads stuck to fibers with a maximum concentration at the point of introduction. The beads stuck to the fibers so firmly that they could not be detached from the fibers. An individual bead in the middle of a fiber was identified and trapped by the optical tweezers. A calibrated force was applied in the lateral direction in an oscillatory manner at ≈3 Hz. Images of the oscillating fibers were recorded with an SVHS video recorder attached to a charge-coupled device camera (Hamamatsu, Middlesex, NJ).

Analysis of Results. The images were digitized by using an AG-7750 SVHS player (Panasonic, Secaucus, NJ) attached to a PIXCI SV4 frame grabber. Approximately 170 frames of video were captured for each fiber, and frames showing the maximum displacement at the point load in each direction were averaged, with ≈18 frames for each image (Fig. 2). The averaged bitmap images were analyzed by using a program written in mathcad software (Mathsoft, Cambridge, MA). Transverse line profiles were measured along the length of the fiber, and the coordinates along the center of the fiber were determined. Third-order polynomial lines of best fit were determined for each fiber by fitting the coordinates in excel (Microsoft). The lines of best fit were used to calculate the maximum displacement as well as the extrapolation of the fiber end points where there was no movement. Fiber diameters were determined by modeling of the images by taking into account the effects of differential interference contrast imaging. In essence, a phase-shift image was convolved with a 0.15-μm shearing function introduced by the Wollaston prism of the microscope and the optical blurring characterized by the point-spread function of the objective lens. The elastic moduli were calculated by using the formula given in the legend of Fig. 3 (27), averaging the results from 10 experiments. Elastic moduli were also calculated from the results of stretching experiments, as described in Results.

Fig. 2.

An individual fibrin fiber with attached bead ≈1 μm in diameter that is being flexed by the optical tweezers. The curvature of the fiber is not obvious, because deflections were small. The fiber and bead are seen in differential interference contrast mode. In this experiment, the bead was oscillating at 3 Hz, and 170 frames (5.7 s) of video were digitized, so 18 frames with the bead at its maximum position upward were averaged to generate this image. (Scale bar, 1 μm.)

Fig. 3.

Elastic curves of a bent fiber and diagram showing the measured parameters used for calculation of the elastic modulus. (A) Plot of the coordinates of the center of a flexed fiber from transverse line profiles of averaged bitmap differential interference contrast images along the length of the fiber at maximum deflection. For each experiment, four fiber segments were measured on either side of the bead (represented here as a dashed circle) at maximum deflection in either direction. (B) The same fiber curve with the vertical displacement exaggerated to better show the fitting of each segment to a third-order polynomial curve. The curves of best fit were used to calculate the maximum displacement as well as the extrapolation to the fiber “end points” where there was no movement. (C) Diagram of fiber with attached bead that is being flexed by optical tweezers. The movement of bead and fiber in the vertical direction here is also exaggerated for the sake of illustration of the parameters measured. The elastic modulus of a fiber was calculated from the formula for the relationship between loading and deflection of a simple beam (27) E = (pab/6IyL)(L² - b² - a²), where E is elastic modulus (in Pa), p is force applied (in pN), a is distance from left-hand end point to bead (in nm), b is distance from right-hand endpoint to bead (in nm), y is deflection at the point load (in nm), L is effective length of fiber segment (in nm), and I is second moment of circular plane area (in nm²), which is the observed shape of fibrin fibers, I = (πr⁴)/4, where r is fiber radius (in nm).

With the laser tweezers, spring constants for a fiber can be determined from statistical analysis of the noisy position data as a result of thermal motion of a trapped bead bound to a fiber. Because the bead is in thermal equilibrium with the surrounding fluid, the Boltzmann equipartition theorem says that the average potential energy associated with the trapped bead for each dimension, which is equal to one-half times the Boltzmann constant times the absolute temperature, is also equal to one-half times the spring constant times the time-averaged variance in that dimension (28). Because we can very accurately (to 1 nm) measure the statistical variance of the particle's x and y coordinates with the quadrant detector in the back focal plane of the objective lens of the laser tweezers and we know the temperature, we can determine the spring constant and then use the equation mentioned in the last paragraph to calculate the elastic moduli.

Statistical Analysis. Means and standard deviations were calculated in the conventional manner. Group differences in continuous variables were determined by one-way ANOVA. A risk of error of 0.05 was accepted to evaluate the statistical significance. Equality of variances between groups was first evaluated by the F test. When the overall F statistic was significant, the Bonferroni test was used to ascertain these differences in both normal and logarithmic scales.

Results

In the present investigation, plasma fibrin clots were made in the presence and absence of a highly specific, active site-directed, synthetic inhibitor of factor XIIIa, DDITS (24), so that there is no ligation of fibrin; in its absence, there is complete ligation of the γ chains and extensive ligation of the α chains of fibrin, resulting in a considerable increase in stiffness of the clots (29-31). The structures of plasma clots identical to those used for optical tweezers experiments were characterized by laser scanning confocal microscopy (Fig. 1) (14). There was no significant difference between unligated and ligated clots (P > 0.05) in fiber diameters, branch point densities, fiber densities, and distances between branch points measured from three-dimensional reconstructions of optical sections of the clots (Table 1). Measurements of the bulk viscoelastic properties of the clots showed that ligation increased the shear storage modulus (G′), or stiffness, of the clots 2.3-fold and decreased the loss tangent, or fraction of energy dissipated in nonelastic processes, of the clots 0.74-fold (Table 2).

Table 1. Structure of plasma fibrin clots determined by confocal microscopy.

Fibrin clot	Fiber diameter, nm	Branch point density, per 103 μm3	Fiber density, per 103 μm3	Distance between branch points, μm
Clots without ligation	286 ± 48	1.2 ± 0.7	4.2 ± 0.2	23 ± 9
Clots with ligation	290 ± 56	1.4 ± 0.5	5.1 ± 0.1	20 ± 9

No significant difference was observed between parameters with or without ligation.

Table 2. Viscoelastic properties of the plasma fibrin clots used for laser tweezers experiments.

Fibrin clot	G′ storage modulus, Pa	G″ loss modulus, Pa	G″/G′ loss tangent tan δ
Clots without ligation	13.2 ± 3.6	2.4 ± 0.6	0.19
Clots with ligation	31.0 ± 8.8	3.4 ± 0.5	0.14

All parameters are significantly different with and without ligation; P < 0.01.

Stiffnesses of individual fibers were measured so that elastic moduli could be calculated. Latex beads perfused into the clots became attached to fibers and were trapped for use as handles to move individual fibers in a sinusoidal oscillation, and the required force was measured (Fig. 2). For measurements of flexural elasticity, beads near the middle of fibers were chosen to minimize movement of adjacent fibers. In other experiments, beads were also moved in the direction parallel to the fiber axis to stretch the fibers. Generally, fibers moved in synchrony with the beads, because we chose beads that were in the same plane as the fibers to minimize rotation of the beads about the fiber, but in some cases, depending on the position and attachment of a bead on the fiber, there was some rotation of the beads. Such experiments were disregarded as much as possible. The amount of movement was generally proportional to the force applied. Video images were digitized, and frames at the extremes of the sinusoidal displacements were averaged. Line profiles across the fiber were generated to determine the displacement of the fiber accurately. Ligated fibers were considerably stiffer than unligated fibers.

The elastic modulus or Young's modulus, a material property of fibrin fibers independent of specific clot or fiber geometry, can be calculated from the theory of deflection of a simple beam (Fig. 3) (27) along with measurements of the fiber diameter and the elastic curve characterizing the fiber shape under deformation. It should be noted, however, that one limitation of these results is that the formula is based on the assumption that fibers are made up of a homogeneous, isotropic, linearly elastic material. Although this assumption is clearly not true for fibrin or any biological polymer, such elastic moduli are commonly used as the best available approximation to characterize these materials.

To measure the diameters of the fibers from the differential interference contrast images of the clots, fiber images were modeled by using microscope parameters obtained from differential interference contrast images of plastic beads of known diameter (Table 3). The low contrast of the fibers was enhanced by averaging images from the maximum displacement from several oscillation cycles (Fig. 2). It should be noted that the precision of the measurement of the position of an edge in these images is considerably better than the resolution of the light microscope (32). The calculated average diameters of 284 ± 44 nm and 274 ± 34 nm of fibers from differential interference contrast images of clots without and with ligation, respectively, were remarkably similar to the diameters measured from confocal micrographs (Table 1).

Table 3. Properties of individual plasma fibrin fibers determined from laser tweezers flexion experiments.

Fibrin clot	Fiber diameter d, nm	Fiber segment length L, μm	Elastic modulus E, × 106 Pa
Clots without ligation	284 ± 44	9.2 ± 1.8*	1.7 ± 1.3*
Clots with ligation	274 ± 34	14.7 ± 2.5*	14.5 ± 3.5*

^*Significant difference between parameters with and without ligation; P < 0.001.

Elastic curves for maximally deformed fibers were determined by fitting the points along the fibers with a third-order polynomial (Fig. 3 A and B). The shapes of the curves suggested that the results were best modeled as a simply supported beam under a concentrated load. From the light microscope images, it was possible to determine the distances of the two effective end points from the load and the maximum deflection, the parameters necessary for calculation of the elastic modulus. The elastic moduli of individual unligated and ligated fibers (Table 3) are strikingly different, 1.7 ± 1.3 and 14.5 ± 3.5 MPa, respectively. Because each fiber is connected to the clot network, it is difficult to know the beam end conditions or even where the effective fiber ends are, because they are not generally at the branch points in a plasma clot. Therefore, we also did calculations for other end conditions and obtained somewhat similar results, but models with fixed or restrained ends appear to be less likely.

The elastic properties were also measured from stretching experiments by pulling on a bead in the direction along the fiber axis. Assuming a homogeneous, isotropic, linearly elastic material, the fiber elastic modulus can be calculated from the formula for stretching, E = 4pL/πd²y, where E is elastic modulus, p is force applied, L is effective length of fiber segment, d is fiber diameter, and y is deflection at the point load. The length, L, empirically was not the same as in the flexion experiments, because the length in stretching experiments was generally the distance between the end points, whereas the length in flexion experiments was usually less than that. From stretching experiments, the elastic modulus for unligated fibers was 1.9 ± 1.8 MPa, whereas that for ligated fibers was 11.5 ± 5.1 MPa (Table 4). The experimental errors for these results include difficulties in measurement of the relatively short displacements (y) for large forces (p) and the fact that there was some movement of the branch points during these experiments. Because the fibers are not homogeneous and isotropic, these determinations are not identical to those from flexion experiments.

Table 4. Elastic moduli E of fibrin fibers in MPa.

Mode of force application	Unligated	Ligated
Bending	1.7	14.5
Stretching	1.9	11.5
Brownian motion	2.6	23.1

Fiber stiffness was also determined for ligated and unligated fibers by an independent method. Because the force on a bead can be very accurately measured with a quadrant photodiode conjugate to the back focal plane of the condenser, the fluctuations in the bead force as a result of Brownian motion can be followed. The variance of the Brownian force is proportional to the stiffness of the fiber, using the Boltzmann equipartition theorem (28). The elastic modulus calculated for fibers in unligated clots was 2.6 ± 1.5 and 23.1 ± 10.2 MPa for fibers in ligated clots (Table 4), similar within the standard deviations to the more direct measurements just described, confirming the utility of this innovative use of the laser tweezers. Limitations of these statistical determinations include the presence of shot noise in the images and the fact that the potential that the bead explores is assumed to be parabolic in shape but may not be. These fluctuation measurements also revealed the time course of the increase of fiber stiffness, in that the magnitude of the fluctuations decreased by a factor of ≈9 over a period of ≈45 min as the fibers were being ligated (Fig. 4), whereas the fluctuations or stiffness remained constant for clots in the presence of the factor XIIIa inhibitor.

Fig. 4.

Time course of changes in variance in position of a fibrin fiber in a clot determined from measurement of bead fluctuations as a result of Brownian motion. The magnitude of the fluctuations of a bead attached to a fiber is inversely related to the stiffness of the fiber. Circles, clot formed in the presence of factor XIIIa inhibitor DDITS (little change in fluctuations over time); squares, clot formed in the absence of DDITS (ligation by factor XIIIa occurs over time, resulting in a decrease in fluctuations or increase in stiffness).

Discussion

Atomic force microscopy has been used to mechanically manipulate individual fibrin fibers and determine the rupture force of fibers as a function of their diameter (33). The experiments reported here used laser tweezers to pull on fibers and measure their elastic moduli. Although there do not seem to be any other similar studies of the microscopic mechanical properties of any polymer networks in the literature, the Young's moduli of other microscopic biological objects have been determined, including microtubules (0.5-1.5 GPa) (34, 35), actin filaments (≈2.6 GPa) (34), and bacterial flagellar filaments (10 GPa) (36) (Table 5), all of which have considerably higher elastic moduli than do fibrin fibers. In contrast, the elastic modulus of fibrin fibers is similar to the bulk elastic moduli of elastin, resilin, abductin, and some rubber polymers. It should be reemphasized that the elastic moduli are independent of the dimensions of the biological structures.

Table 5. Rigidity comparisons.

Biological polymer	Elastic modulus E, N/m2	Moment of inertia I, m4	Flexural rigidity EI, Nm2
Fibrin	1.45 × 107	3.0 × 10−28	4.35 × 10−21
	1.7 × 106		5.1 × 10−22
Microtubules	1.2 × 109	1.8 × 10−32	2.15 × 10−23
Actin	2.6 × 109	2.8 × 10−35	7.3 × 10−26

The two values given for fibrin are for ligated and unligated clots from bending experiments.

In that respect, the flexural rigidity of a typical fiber in a plasma clot is the best parameter to use for comparisons of the load-bearing properties of biological structures, because it takes into account the dimensions of the structures but is more independent of the experimental conditions than is a spring constant. Calculation of this parameter indicated that fibrin fibers are much stiffer than microtubules, which are much stiffer than actin filaments (Table 5); much of the difference in flexural rigidity comes about from large differences in moments of inertia, arising from the larger diameters of the fibrin fibers. These studies demonstrate that the microscopic properties of a supramolecular structure in its biological context can be determined by using the laser tweezers.

The present investigation provides a foundation for understanding the great differences between the mechanical properties of clots with dissimilar structures (19). As with any large, complex structure, knowing the properties of individual components as measured here, it should be possible to calculate the bulk mechanical properties of an entire clot, although this project will be a large one. Empirical relationships between clot structures and their rheological properties will aid this endeavor. For example, clot rigidity increases with an increase in both fiber thickness and the number of branch points, but, generally, large diameters and lengths are associated only with minimal branching, and increases in fiber length are associated with increases in fiber diameter (31). Thus, maximal rigidities are established in clots that display a balance between large fiber sizes and extensive branching. Additional studies on the correlations between clot structure and bulk viscoelastic properties are needed to provide a better understanding of the relationships between the microscopic properties of fibers determined in this study and the macroscopic mechanical properties of clots.

Our results provide quantitative evidence for the origin of clot elasticity. It is apparent from basic principles that the bending of fibers is likely to be largely responsible for the elasticity of fibrin clots. Any structure that is made up of elements that have length-to-diameter ratios of 30-50, such as the fibers in a fibrin clot, will predominantly deform under load by bending, rather than stretching the elements. For example, for a simply supported circular cylindrical rod of length L and diameter d of a material that is homogeneous, isotropic, and linearly elastic, the ratio of the displacement from flexion to that from stretch is (1/3)(L/d)². For L/d = 30, the minimum case here, the flexural displacement is 300 times stretch displacement. This conclusion for a homogeneous structure is reinforced because of the anisotropy of the fibrin fiber, because it has been observed by electron microscopy and x-ray diffraction that the molecular length of fibrin(ogen) is constant under many conditions, whereas images of individual fibrin(ogen) molecules commonly show various degrees of bending, most likely arising from the hinges in the molecules identified from x-ray crystallography (37). This mechanism is also consistent with the observations that fibrin fibers are twisted (38), meaning that they are under tension and are very straight, as indicated by the persistence length calculated here, but can be bent more easily than they can be stretched (23). The facts that the results are best fit by a simply supported beam and that the apparent fiber ends are not at branch points suggest that there may be “hinges” along the fibers, which could arise from nonuniformity along their length, perhaps regions where the fibers are thinner or disordered (and hence weaker points) but kinks were not observed.

In conclusion, to our knowledge, this study presents the first determination of the microscopic mechanical properties of a polymeric network, information necessary for understanding the molecular origin of the bulk mechanical properties of these materials. The study of the fibrin clot has particular biological and clinical significance because the mechanical properties of clots are essential for their functions in hemostasis and also are largely responsible for the pathology of thrombosis, but the origin of the elasticity of the fibrin clot has been a mystery for >50 years. The mechanical properties of individual fibers in a clot have been determined by using optical trapping as a technique of microscopic manipulation, providing a basis for understanding the molecular origins of the physical properties of clots.