INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION AND CONCLUSIONS CONCLUDING REMARKS REFERENCES

Light scattering techniques can provide a number of size- and shape-related solution parameters traditionally used for the characterization of macromolecules or particles (reviewed in Huglin, 1972; Schmitz, 1990; Harding et al., 1992). In the "static" mode, it is possible to recover, from the angular dependence of the time-averaged intensity of the scattered light, the weight-average molecular weight M_w and the z-average square radius of gyration R_g²_z of polydisperse macromolecular systems. This is usually done by recording with a moving detector the intensity of the scattered light at a number of angles in the plane of the incident beam, and then applying extrapolation procedures to zero scattering angle. Although this procedure is normally used to characterize macromolecular solutions in thermodynamic equilibrium conditions, it is also possible to follow the time evolution of M_w and R_g²_z for solutions away from equilibrium, such as during a polymerization reaction. However, in the traditional set-up described above, this is possible only for very slow reactions, because the detector can collect the scattered light only at one angle at a time, and several angles are needed, each with a finite collection time, to obtain statistically meaningful results. This bottleneck can be, now at least, partially bypassed by the advent of light scattering photometers of a different design, in which the scattered light is collected at many angles simultaneously by a number of fixed detectors. One commercially available instrument, the DAWN-DSP from Wyatt Technology Corp. (Santa Barbara, CA), was developed as an on-line detector for gel-permeation/size-exclusion chromatography, and is equipped with a solid glass cylindrical cell, with a flow-through small horizontal bore, around which eighteen photodiodes are positioned at fixed angles (see Fig. 1). The combination of a flow-through cell and multiangle laser light scattering (MALLS) detection renders this instrument very well suited for recovering the physicochemical parameters of macromolecular solutes away from equilibrium conditions, such as during polymerization and aggregation processes, nucleation and growth of crystals, etc.; when coupled with a rapid mixing device, these reactions can be followed with a time resolution of at least a few tenths of a second.

View larger version (38K): [in this window] [in a new window]   FIGURE 1   Schematic drawing of the DAWN-DSP-F photometer. (A) Head of the instrument with the eighteen fixed detectors. (B) Glass flow-through cell with one photodiode shown (both panels reproduced with permission of Wyatt Technology Corp.).

In this paper, we describe the experimental set-up for the stopped-flow/MALLS system used by us. Furthermore, because we are currently interested in the polymerization of rod-like macromolecular monomers, we have performed a series of numerical tests to ascertain if the data analysis routines available with the instrument software could recover from Zimm-like plots (Zimm, 1948) the standard parameters, M_w and R_g²_z, of synthetic data within a reasonable range of error. In addition, the experimental data were also reduced in the framework of the Casassa approximation (Casassa, 1955), from which the mass/unit length M_L of ensembles of long rods, polydisperse both in thickness and length, could be derived. To show the potential afforded by the simultaneous determination of M_w and R_g²_z, together with M_L, of evolving polymers, we report a set of data on a very important physiological reaction: the supramolecular polymerization of fibrinogen.

Fibrinogen (FG) is a high molecular weight (340,000), rod-like (~46-50 nm long) glycoprotein made up by three pairs of different polypeptide chains (2A, 2B, and 2) joined together by disulfide bonds to give a symmetric particle (see Mosesson and Doolittle, 1983; Doolittle, 1984). The N-terminal ends of all the chains are contained within a central E domain, from which they depart to form two triple coiled-coils connectors ending in two outer D domains, each containing the C-terminal ends of the B and  chains. The C-terminal ends of the A chains (>400 amino acids) do not enter the D domains and seem to fold back to form a fourth domain positioned above the central one (Mosesson et al., 1981; Erickson and Fowler, 1983; Medved' et al., 1983; Weisel et al., 1985; Spraggon et al., 1997). FG is activated by thrombin, which sequentially removes first a pair of small peptides called fibrinopeptides A (FPA), and later on a second pair (fibrinopeptides B, FPB) from the N-terminal ends of the A and B chains, respectively. The resulting fibrin monomers [()₂] polymerize forming rod-like fibrils, which, by branching and lateral aggregation, give rise to a three-dimensional network (see Mosesson and Doolittle, 1983; Doolittle, 1984; Mosesson, 1990; Blombäck, 1996). Two sets of complementary knob-hole interactions between sites uncovered by the removal of the fibrinopeptides in the central E domain (the knobs or A and B sites) with sites always available in the outer D domains (the holes or a and b complementary sites) have been convincingly demonstrated to play a major role in fibrin formation. Under physiological conditions, the A-a interactions are responsible for the linear elongation of the polymers into half-staggered, double-stranded "protofibrils", whereas the B-b interactions influence their subsequent lateral aggregation, (Laurent and Blombäck, 1958; Fowler et al., 1981; Weisel, 1986; Medved' et al., 1990; Weisel et al., 1993; Spraggon et al., 1997; Everse et al., 1998; and references therein). However, the specificity of these processes is highly dependent on the nature and concentration of the ions present in solution (see Di Stasio et al., 1998, and references therein). Whereas the general mechanism of fibrin formation is now quite well understood, several issues are still controversial. In particular, the early stages leading to the formation of the protofibrils, and the onset of branching, a fundamental requisite for the formation of a three-dimensional network, remain to be elucidated.

Light scattering techniques have been applied in the past to the study of fibrin polymerization, but the constraint of measuring the scattered intensity at one single angle at a time was a major limitation. Different ways have been followed to overcome this problem, with a series of important papers appearing ~20 years ago (references to earlier work can be found in Sheraga and Laskowski, 1957; studies using mainly dynamic light scattering, like the recent one by Bauer et al., 1994, will not be dealt with here). In particular, the kinetic aspect of fibrin polymerization was studied by measuring the intensity of scattered light at right angle only (Hantgan and Hermans, 1979), or by determining M_w as a function of reaction time at relatively high ionic strength (0.5 M NaCl) after collecting, almost simultaneously, the light scattered at nine different angles (Visser and Payens, 1982). Alternatively, more structural data were collected after the process had been slowed down by using very low thrombin amounts (Müller and Burchard, 1978; Müller et al., 1981; Wiltzius et al., 1982a,b; Bauer et al., 1994) or by the addition of an inhibitor of fibrin polymerization (Knoll et al., 1984). All these studies have added important elements to the general picture, but because of the lack of both time- and true angle-resolved collection, they could not provide the necessary details for a better understanding of the earliest stages of the fibrinogen-fibrin conversion. In particular, the works of Müller and Burchard (1978) and of Wiltzius et al. (1982a) have suggested the presence of end-to-end polymers following enzyme activation, apparently at odds with the accepted mechanism for the initial events, the formation of half-staggered, double-stranded polymers. However, the most convincing evidence for this mechanism, apart from the biochemical data, has been mainly provided by electron microscopy (EM) (Fowler et al., 1981; Janmey et al., 1983b; Medved' et al., 1990; Weisel et al., 1993), which unfortunately suffers from poor time resolution and requires sample manipulation. Thus, providing accurate, time-resolved structural data from unperturbed solutions could help to clarify this issue.

The simultaneous determination with our stopped-flow/MALLS system of M_w, R_g²_z, and M_L for the growing fibrin polymers after activation by thrombin in near physiological conditions, but in the absence of added Ca²⁺ and in the presence of EDTA-Na₂, has confirmed many previous observations and has yielded some preliminary new intriguing results. Theoretical curves calculated for various polymerization mechanisms of bifunctional rod-like monomers are also presented to tentatively interpret the data.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION AND CONCLUSIONS CONCLUDING REMARKS REFERENCES

Protein purification and quality control

All chemicals were reagent grade from Merck (Darmstad, Germany), unless otherwise stated, and double-distilled water was used in the preparation of all the solutions. Lyophilized human fibrinogen (TF grade, IMCO, Stockholm, Sweden) was dissolved at 37°C at a nominal concentration of 20 mg ml¹ in 0.3 M NaCl, to which were added 10 units/ml KIR (serine proteases inhibitor, Richter, Milano, Italy). It was then dialyzed at 4°C for 18 h against two changes of TBS buffer (50 mM Tris, 104 mM NaCl, 1 mM EDTA-Na₂, KIR 10 u/ml, pH 7.4). The dialyzed solution was centrifuged at 30,000 × g for 30 min, divided into aliquots, and stored at 80°C. FG concentrations were determined from the absorbance at 280 nm using a specific absorption coefficient E of 1.51 ml mg¹ cm¹ (Mihalyi, 1968), after correcting for scattering contributions by subtracting the absorbance at 320 nm. A Beckman DU640 spectrophotometer (Beckman Analytical, Milano, Italy) was used for absorption measurements. Thrombin from human plasma (T-6884, lyophilized from Na citrate) was from Sigma-Aldrich (Milano, Italy). It had a nominal activity of ~2000 NIH units/mg protein, was reconstituted with water to a final nominal concentration of 1000 NIH units/ml, and was stored in small aliquots at 80°C.

Before stopped-flow experiments, fibrinogen was further purified from its aggregates by gel filtration chromatography by loading 2-3 ml of stock solution on a glass column (1.5 × 167 cm) filled with Sepharose CL-4B (Pharmacia Biotech, Uppsala, Sweden), eluted at 6 ml h¹ at room temperature. The pooled peak fractions were furthermore centrifuged at 38,000 × g for 1 h (20°C). The purity of FG was tested by polyacrylamide (PAA) gel electrophoresis in the presence of sodium dodecyl sulfate (SDS-PAGE) of samples either nonreduced or reduced with dithiothreitol (Sigma) according to Laemmli (1970), using electrophoresis-grade reagents from Bio-Rad (Hercules, CA). The results of the purification steps are shown in Fig. 2, where a typical chromatogram is reported, together with the SDS-PAGE analysis on a 5% PAA gel of unreduced samples of the starting material, of two peak fractions and of the final pooled material, all at the same nominal protein loading concentration (inset A, lanes 1-4, respectively). As can be seen in Fig. 2, the high molecular weight species present, albeit in low amounts, in the starting FG sample (inset A, lane 1), gave rise to detectable humps in the chromatogram. The asymmetric main chromatographic peak resulted also from the incomplete separation of the doublet corresponding to various monomeric FG species differing by the extent of C-terminal degradation of the -chains (see below), as evidenced by the electrophoretic analysis of two representative fractions (arrows; inset A, lanes 2 and 3). By taking only the main peak fractions (shaded area), nearly all the aggregates were removed (inset A, lane 4), leaving the FG doublet (more clearly seen when roughly one-fourth of sample was loaded on the gel as shown in inset A, lane 5). Rather than attempting a difficult and material-consuming separation between these two closely related species, we have performed a careful analysis of their composition. This was done by blotting on nitrocellulose sheets (Towbin et al., 1979) gels loaded and run with reduced peak FG samples, followed by immunostaining with the Y18 monoclonal antibody (a gift of Dr. W. Niuwenhuizen, Leiden, The Netherlands), which recognizes an epitope on the FPA (Koppert et al., 1985). Color was developed using a horseradish peroxidase-conjugated goat anti-mouse IgM secondary antibody (Southern Biotechnology Associates, Birmingham, AL) and 4-chloro-1-naphtol (Fluka Chemie, Buchs, Switzerland) as a substrate. In inset B of Fig. 2, a blot of a reduced pooled peak FG sample run on an 8% PAA gel and stained with amido black (lane 1) and its immunostained counterpart (lane 2) are shown. The immunostained blots were then scanned on a Mustek MFS 6000CX flatbed scanner using a 300 × 300 dots-per-inch resolution, and the area of each band was quantified with the One-Dscan software (Scanalytics, CSPI, Billerica, MA) with Gaussian deconvolution of overlapping peaks. The molecular weight of each A chain species (identified on the left side of lane 2 in inset B) was deduced from the relative migration of each band, and checked against the calculated molecular weights of FG fragments derived from the potential plasmin cleavage sites in the A-chains C-terminal portion, as will be reported in more detail elsewhere (Lai M. E., G. Franzoni, A. Profumo, C. Cuniberti, and M. Rocco, in preparation).

View larger version (39K): [in this window] [in a new window]   FIGURE 2   Elution profile of a fibrinogen solution (c = 16.4 mg ml1; 2 ml loaded) from a 1.5 × 167-cm Sepharose CL-4B chromatography column, monitored at  = 276 nm. (A) The nonreduced SDS-PAGE analysis on 5% PAA gel of the starting material (lane 1), of two fractions (lanes 2-3) collected at the points indicated as A2 and A3 (arrows) in the main figure, and of the pooled material (shaded area) (lanes 4-5). Loaded amounts: lanes 1-4, 9 µg; lane 5, 2.5 µg. (B) A Western blot of the reduced pooled material after SDS-PAGE separation on a 8% PAA gel is shown stained with amido black (lane 1) or immunostained after reaction with the Y18 monoclonal antibody (lane 2).

Multiangle laser light scattering (MALLS)

Static light scattering experiments were carried out with the multiangle photometer DAWN-DSP-F (Wyatt Technology Corp., Santa Barbara, CA) equipped with a 5-mW He-Ne laser source ( = 632.8 nm), a K5 glass flow cell, and Peltier control of the temperature, which was kept at 25.0 ± 0.2°C. The true scattering angles seen by the fixed detectors in the DAWN-DSP-F depend on the refractive index of both the cell (n = 1.52064) and the solvent used (Wyatt Technology, 1997), and, under our conditions, 17 angles ranging from 4.7° to 158.1° were theoretically available. However, the two lower angles, positioned at 4.7° and 14.8°, are very noisy when aqueous solvents are used, and the practical range of the instrument was thus limited to fifteen angles from 21.9° to 158.1°.

The index of refraction of the buffer was measured at 589.3 nm and at 25°C by an Abbé refractometer and then interpolated at 632.8 nm at 25°C by using the same ² dependence calculated for water from literature data (Lange, 1967). A value of 1.334 was found for TBS. For the fibrinogen solutions, a refractive index increment (dn/dc) of 0.192 ml mg¹, (Carr et al., 1977) obtained by interpolation from literature data (Schulz and Ende, 1963), was used.

Spectroscopic grade toluene (Uvasol, Merck) directly filtered in the flow-through scattering cell through 25 mm diameter, 0.22 µm pore-size PTFE syringe filters (Acrodisc CR, Gelman Sciences, Ann Arbor, MI), was used for the absolute calibration of the MALLS photometer. A refractive index of 1.4912 (Timmermans, 1965) and a Rayleigh ratio of 1.414 × 10⁵ cm¹ (Forziati, 1950; Forziati et al., 1950) at 632.8 nm and 25°C were used, respectively. Normalization of the fifteen photodiodes, to correct for their different responses, posed a very difficult problem. According to manufacturer instructions, when the flow-cell is used with aqueous solvents in the micro-batch mode, normalization of the detectors should be done by injecting a solution of a monodisperse low molecular weight polymer, filtered through 0.02-µm pore-size syntherized syringe filters (ANOTOP 25, Whatman, Maidstone, UK), and then assuming isotropic distribution of the scattered light. We tested both dextran (M_w = 40,000, a gift of Dr. D. Friscione of Alfatech, Genova, Italy) at a concentration of 3.3 mg ml¹, and poly(ethylene glycol) (M_w = 2700-3300, cat. 81230, Fluka Chemie), at a concentration of 18 mg ml¹, both filtered as stated above after centrifugation for 30 min at 38,000 × g, but this procedure was found to be unsatisfactory, especially at the lower angles. Alternatively, the detectors of the MALLS photometer can be normalized in the chromatography mode, by injecting into a size-exclusion (SE) chromatography system a solution of a globular protein of small R_g, and using only the monomer's peak slices from the chromatogram (Wyatt Technology, 1997). Bovine serum albumin (BSA, A-0281, Sigma; R_g = ~2 nm) was dissolved in TBS at 5 mg ml¹, filtered through cellulose acetate 0.2 µm pore-size microcentrifuge filters (SPIN-X, Corning Costar, obtained through Sigma-Aldrich) and 200 µl were injected on a size-exclusion high pressure liquid chromatography (HPLC) set-up consisting of a 6 × 40 mm TSK-GEL PW_XL guard column (TosoHaas, Stuttgart, Germany) and three 7.8 × 300 mm TSK-GEL analytical columns (G5000PW_XL, G4000PW_XL, G3000PW_XL, TosoHaas) connected in series and operated at 0.3 ml min¹ from a System Gold HPLC system (Beckman) composed of 126 Solvent Module and 166 UV/VIS Concentration Detector. To further remove dust and particulate as much as possible, two on-line stainless steel filters (pore size 0.5 µm and 0.2 µm) were placed in series before the guard column, and another 0.2 µm pore size on-line stainless steel filter was placed right after the columns. The concentration of the eluted BSA was measured at 280 nm by the UV/VIS concentration detector module placed before the MALLS detector, using an extinction coefficient E of 0.66 ml mg¹ cm¹ (Edwards et al., 1969).

Stopped-flow MALLS: experimental set-up

The experimental stopped-flow MALLS apparatus used to follow the time evolution of fibrin polymerization is schematically represented in Fig. 3. It is composed of three parts: the injection device (a RX-1000 rapid mixing device from Applied Photophysics, Leatherhead, UK), the MALLS photometer, and a 100-µl, 10-mm pathlength square Suprasil quartz flow-through cuvette (Hellma, Müllheim, Germany) directly connected to the exit of the light scattering cell. The cuvette can be disconnected and placed inside a spectrophotometer for absorbance measurements.

View larger version (27K): [in this window] [in a new window]   FIGURE 3   Experimental set-up for the stopped-flow/MALLS experiments (see Materials and Methods for details).

The injection device consisted of two glass Accudil syringes (Hamilton, Bonaduz, Switzerland) simultaneously driven by a metal plate, which can be operated either manually or via a pneumatic drive. Each of the two 2.5-ml syringes is screwed on a Teflon block containing a three-way T-valve, with an inlet luer port and an outlet Teflon line exiting from it. The two syringes were filled via the inlet ports through additional three-way nylon luer-lock T-valves (Sigma), to which 25-mm-diameter, 0.22-µm-pore-size Millex-GP filters were attached, permitting the refill operations to be carried out with all-polypropylene syringes without introducing any dust or contaminants. Priming/normalizing operations can also be carried out through these three-way valves, without affecting the outlet lines. These lead to a machined Teflon "reverse Y" mixer (the two inlet lines enter in a V conformation, and the outlet line exits from the apex of the V from the same side as the inlet lines; a small mixing chamber is machined by slightly prolonging the outlet bore from the V apex). From the mixer, the outlet leads to a solvent-resistant four-way valve (SV-4, Pharmacia Biotech), from which the flow can be switched either to a waste line or to the flow cell of the MALLS photometer. The remaining port of the four-way valve is directly connected by a small PTFE tubing with a female luer lock fitting to a 20-ml polypropylene syringe to which a 25-mm-diameter, 0.22-µm Millex-GP filter is attached. At the exit of the MALLS cell, an inert three-way T-valve (HV3-3, Hamilton) allows the solutions to pass either through the UV cuvette or go directly to waste. Before the mixer, another three-way inert T-valve is inserted on one of the two mixing lines (not shown in Fig. 3), to allow the priming of the lines and of the mixing chamber when the glass syringes' content is changed, and their washing after the reaction is started. Immediately after every injection, the three-way valve after the MALLS cell is switched to an all-closed position, to avoid disturbing the solution in the cell by drainage/outgassing effects.

At the beginning of an experiment, the whole system was filled and washed with TBS buffer several times until reasonably dust-free, and then an injection was made and the baselines of the MALLS detectors recorded, with a data point every 0.25 seconds, in a new file on a Pentium PC computer directly connected to the A/D board of the DAWN-DSP-F, using the software ASTRA 4.60.07 (Wyatt Technology). The baseline UV spectrum was also recorded from the 100-µl cuvette. Then syringe A was filled (with normalization/priming) with a fibrinogen solution at twice the desired final concentration, syringe B was refilled with buffer, and a second injection was made, allowing the recording on the same file of the intensity of the light scattered by fibrinogen, as well as measurement of its real concentration in the mixture from the UV cuvette reading. Subsequently, syringe A was refilled with fibrinogen, syringe B was filled (with normalization/priming up to the valve before the mixer) with a thrombin solution (at nominal 0.25 NIH units/mg FG, final concentration), and the experiment was started by the injection of the reaction mixture in the flow cell. The measuring cell was then locked by turning the 4-way valve in the direction of the washing syringe, and the mixer and the outlet lines were immediately washed with TBS by connecting another polypropylene syringe with a 0.22-µm filter to the three-way valve attached just in front of the mixer (not shown on Fig. 3). The mixing operation and solution transfer were made by hand-driving the metal plate, and took less than 3 s. The complete intensity data were collected in the same file where blank and fibrinogen data were recorded. Because we were interested in the early stages of fibrin polymerization, the reaction was usually stopped before the intensity of the scattered light saturated the detectors, well before a recognizable clot was formed in the cell. Therefore, up to three reactions could be recorded on a single file, and, between them, the flow-through scattering cell was carefully washed via the external syringe until back to starting conditions.

Light scattering theory

For infinitely dilute macromolecular solutions, the excess intensity of scattered light recorded as a function of the angle  between the primary and the scattered beam, expressed as the Rayleigh ratio R (cm¹), is given by the sum of the contributions from the single particles,

(1)

where c is the sample concentration (g cm³), and K is the optical constant (cm² mol g²), equal to 4²n²(dn/dc)²/(N_A⁴), n being the refractive index of the solvent, dn/dc the refractive index increment of the solute in that particular solvent,  the wavelength in vacuo and N_A the Avogadro number. M_i is the molecular weight of the ith particle, w_i its corresponding weight fraction, and the adimensional function P_i() represents the particle scattering factor, which is normalized to unity for  = 0, i.e., P( = 0) = 1. Expressions of P() for particles of various geometries are available in the Rayleigh-Gans-Debye (RGD) approximation (see Huglin, 1972), and, in this paper, we will be dealing with either rigid rod-like particles or semiflexible worm-like chains. For a rod-like particle of length L, the form factor is given by

(2)

with x = qL, q being the scattering wavevector given by q = (4n/) sin(/2). For semiflexible worm-like chains, we can refer to the expression developed by Koyama (1973),

(3)

where L_c is the contour length of the chain, l_k is the Kuhn statistical segment length (equal to two times the persistence length l_p), and the function  is defined elsewhere (Koyama, 1973). It is worth noticing that, when the ratio  = l_p/L_c 1, the chain is infinitely stiff and behaves as a rigid rod, with Eq. 3 reducing to Eq. 2.

In any case, even when the expression of P() is not known, the RDG theory predicts that the behavior of the scattered intensity distribution becomes independent of particle shape as  approaches 0. Under this limiting condition, it is convenient to follow the data treatment first suggested by Zimm (1948):

(4)

Thus, the weight-average molecular weight,

(5)

and the mean square z-average radius of gyration,

(6)

can be obtained from the intercept, and from the ratio between the slope and the intercept, respectively, of the plot of Kc/R versus sin²(/2). Strictly speaking, measurements at various concentrations should be performed and the data extrapolated to zero concentration, but, because of the low fibrinogen concentration used by us (between 0.1 and 0.35 mg/ml), this dependence is neglected here. Equation 4 is applicable only to a limited range of dimensions or angles, i.e., when the second term inside the right-hand side parentheses is 1. When this term approaches unity, the Zimm plot deviates from a linear behavior (for example, a value of 0.2 gives deviations of about a few percents), and its curvature depends on the particular structure of the particles being studied. Thus, no direct information on the parameters M_w and R_g²_z can be easily recovered in this case, unless smaller scattering angles become available.

In the case of polydisperse solutions of rod-like (or worm-like) particles, a second treatment derived by Casassa (1955) can be used. The Casassa method is applicable in a range of angles and particle sizes that is opposite from that of the Zimm plot: for rod-like particles (l_p/L_c 1), the condition is qL_c > 3.8 (and not >1.5 as stated in Wiltzius et al., 1982b), whereas, for semiflexible worm-like chains, two conditions must be satisfied, qL_c > 3.8 and ql_p > 1.9 (see also Koyama, 1973; Yamakawa and Fujii, 1974). When the above requirements are met, (R)¹ is linearly related to sin(/2) and, from the slope of the plot of Kc/R versus sin(/2),

(7)

one can recover the average mass for unit length M_L = _i M_iw_i/L_c(i), where L_c(i) represent the contour length of the ith chain.

Data analysis and simulations

The collected MALLS intensity profiles were analyzed first using the software provided by the instrument's manufacturer, ASTRA version 4.60.07 (Wyatt Technology). Briefly, the baselines were set, including a "fake" baseline for the UV detector signal, generated to reproduce the actual fibrinogen concentration as measured from the UV cuvette, and "peaks" regions were selected (the software was originally developed for analyzing chromatography data, and, although a microbatch option is provided, none is unfortunately available for the treatment of kinetic data). The selected regions were then analyzed via the Debye plot option of the software, which is a Zimm-like plot without the concentration dependence of the data. The Kc/R versus sin²(/2) data for each time slice were then fitted with either a first-degree or a third-degree polynomial, to recover M_w and (R_g²_z)^1/2 from the intercept and from the ratio between the initial slope and the intercept, respectively, of the fits. Each complete set of time slices containing the M_w and (R_g²_z)^1/2 data, together with their calculated standard deviations, can be downloaded to an ASCII file by the software.

For the analysis of the data by the Casassa equation, the Rayleigh ratios R calculated at the various angles had to be downloaded from the software as an ASCII file, and were processed using the ORIGIN 4.0 software (Microcal Software Inc., Northampton, MA). Unfortunately, because the ASTRA software does not download the standard deviations associated with the R, these had to be computed again from the baseline values, and may slightly differ from those calculated by ASTRA (see United States Patent 5,528,366, 1996).

To ascertain the validity of the data analysis based on the polynomial fittings of the Zimm plots, we carried out several numerical simulations trying to match as much as possible the experimental conditions. First, we assumed to have a polydisperse dilute solution of either rod-like or worm-like particles characterized by a mass distribution typical of bifunctional polycondensation models in which the monomers are rods of length L₀ = 50 nm and diameter d₀ ~ 3 nm, for a volume of 357 nm³ corresponding to that calculated from the amino acid sequence of anhydrous FG. The 50-nm FG length was chosen over the 46-48-nm length deduced from EM data (Hall and Slayter, 1959; Mosesson et al., 1981; Erickson and Fowler, 1983; Weisel et al., 1985), because, from our SE-HPLC-MALLS solution data, an R_g of 14.5-15 nm was consistently found for monomeric FG (Bernocco, 1998; see Results). A density of 1.395 g ml¹ was assumed, so that the molecular weight of the monomers is M₀ = 3.0 × 10⁵ g mol¹ (see Results for the reason for this assumption). The most-probable size distribution predicted by the bifunctional polycondensation model with sites all having equal reaction probability, expressed in terms of the weight fraction w_i(p) of the ith polymer made of i monomers, is the Flory distribution (Flory, 1936, 1942, 1953):

(8)

where the parameter p represent the conversion degree of the monomers, related to the number-average polymerization degree  by the relation  = 1/(1  p).

Size distributions were also calculated in the framework of the theory developed by Janmey (Janmey, 1982; Bale et al., 1982). In this case, it was supposed that the rate of removal of the two FPA from each FG molecule is not equal, but that the second is released faster than the first by a factor Q. This leads to a distribution of unreactive (AB)₂, monofunctional A(B)₂, and bifunctional (B)₂ species whose time-dependent concentrations (referred to hereafter as [A₂₂]_t, [A₂]_t, and [₂]_t, respectively) can be calculated, given the initial fibrinogen [F₀] and thrombin [Th] concentrations, from the following set of two differential and one linear equations (Janmey, 1982):

(9)

(10)

(11)

where K₂ (7.3 × 10⁷ M[(NIH units ml¹)s]¹) is the rate constant for the liberation of one bovine FPA by bovine thrombin and K'_M is a Michaelis-Menten constant, which takes into account the competition between bovine A and B chains for bovine thrombin (Martinelli and Scheraga, 1980; Janmey, 1982; Bale et al., 1982),

(12)

where K_M = 9.2 × 10⁶ M, K_MB = 11.3 × 10⁶ M (Martinelli and Scheraga, 1980), and [B] is the concentration of intact B chains. For the early stages of the reaction, it can be safely assumed that [B] = 2[F₀]. To match our experimental conditions, we set [F₀] = 1.2 × 10⁶ M, resulting in K'_M = 1.12 × 10⁵ M, and values of [Th] between 0.05 and 0.17 NIH units ml¹ were chosen. After substitution of Eq. 11 in Eqs. 9 and 10, numerical integration with a Runge-Kutta method using LabVIEW 5.1 (National Instruments, Austin, TX) yielded the concentration of the reacting species as a function of discrete time steps.

The time-dependent size distribution of polymers formed by the random association of a mixture of monofunctional and bifunctional units was derived by Janmey (1982) apparently starting from the work of Flory on linear polycondensations (Flory, 1936; 1942) and on polyesters degradation (Flory, 1940; ref. 14 in Janmey, 1982, is misquoted). Thus, we could calculate the time-dependent conversion degree p_t as

(13)

whereas the weight-fractions w_i(p_t) were then calculated as

(14)

(15)

with

(16)

If the solution is so dilute that the particles are not interacting, the intensity scattered by the overall sample can be expressed by Eq. 1, where P_i() is the form factor of the ith polymeric chain of mass M_i = iM₀. Four different polycondensation polymers models were considered: rod-like, either end-to-end single-stranded (RLSS) or half-staggered double-stranded (RLDS) stiff chains; and worm-like, either end-to-end single-stranded (WLSS) or half-staggered double-stranded (WLDS) semiflexible chains. For each of the four models M_w is the same and can be easily calculated at any p value by using Eq. 5 and Eqs. 8 or 14-16, whereas R and R_g²_z differ from model to model and can be determined after P_i() and the radius of gyration R_g(i) of the ith polymer are known. These were computed as follows.

The RLSS polymers were considered to be circular cylinders of length L_i = iL₀ and diameter d equal to the monomer diameter d₀. Thus, P_i() was simply given by Eq. 2 and

(17)

The RLDS polymers were considered to be a rigid assembly of monomers aligned according to the well known half-staggered geometry (Ferry, 1952; Hall and Slayter, 1959), each monomer being a cylinder of length L₀ and diameter d₀. Because the relative distances among monomers are fixed (and known), R_g(i) can be calculated by using standard formulas, and it is straightforward to show that

(18)

where R_g²(0) is the square radius of gyration of the monomer (calculated with Eq. 17), and r_ij is the distance between the monomers centers of mass. As to P_i(), the ith polymer was assimilated to a rod of diameter d_i and length L_i so that its radius of gyration (calculated according to Eq. 17) is equal to that given by Eq. 18 and its molecular weight is equal to the expected value M_i = iM₀. This approximation might be rather rough for the first oligomers, but works fairly well as soon as the polymers start to grow, with the values of d_i and L_i, which asymptotically tend to d_i = d₀ and L_i = (i/2)L₀, respectively.

For most of the WLSS polymers, for reasons that will be explained in the Results section, the length L_m of the fibrin monomer units inside the polymers was considered to be different from that of the nonactivated, rod-like fibrinogen monomers. It was assumed to be L_m = 75 nm, and, to conserve the monomer volume, its diameter was fixed at d_m = 2.45 nm. The contour length of the ith chain was L_c(i) = iL_m (for i > 1) and L_c(1) = L₀ = 50 nm for the monomer. The persistence length of the chain was set to be equal to the monomer length (l_p = 75 nm), or twice this value (l_p = 150 nm), regardless of L_c(i). Only in one particular case, all the units were considered as for the RLSS case, with L₀ = L_m = l_p = 50 nm. Thus, P_i() was calculated according to Eq. 3, whereas R_g(i) was calculated from (Benoit and Doty, 1953),

(19)

where _i = l_p/L_c(i). Note that, as for Eq. 3, when _i 1, the chain behaves as a rigid rod, and Eq. 19 reduces to the formula of an infinitely thin rod, R_g²(i) = (L_i²/12).

Finally, the WLDS polymers were considered to be semiflexible chains obtained by assembling monomers of length L₀ and diameter d₀ according to the same half-staggered geometry used for the RLDS model. The ith polymer was characterized by a contour length L_c(i) and a diameter d_i, which were determined by supposing that the polymer is infinitely stiff, and then by using the same procedure of the RLDS model. The persistence length l_p was taken to be equal for all the polymers, and values between 50 and 200 nm were chosen. For the WLDS model, the form factor P_i() and radius of gyration R_g(i) were calculated by using Eq. 3 and Eq. 19, respectively.

Thus, using Eqs. 1-3, 5, 6, 8-19, we have generated various R data sets for the 15 different scattering angle values associated with the detectors of the MALLS photometer. Some discrete values between 0.01 and 0.6 for the degree of conversion p were chosen, and the sample concentration was assumed to be 0.33 mg/ml, corresponding to one of our experimental conditions. In generating the distributions, a cut-off value of 1 × 10⁴ was used for w_i(p), to reasonably limit the number of polymers present contributing to the scattered light intensity. Then, both statistical and systematic noise were added to the data. As mentioned in the previous section, the statistical noise was estimated from the fluctuations of the baseline signals provided by the buffer solution before injecting the sample in the cell. These fluctuations were found to be of the order of 1.5 × 10⁷ cm¹ for the detectors at the lower two angles and the one at the largest angle. In between, they were somewhat smaller, of the order of 5 × 10⁸ cm¹. These values correspond to ~1% rms fluctuations of the intensity scattered by the buffer, which, in turn, is ~10-fold lower than the intensity scattered by the fibrinogen monomers (at the above-reported concentration) before polymerization is induced. Superimposed to the statistical noise, a little amount of systematic noise could also be added to the data for all the angles. Its level was chosen to be different from angle to angle, with the first and last ones being noisier because they are most susceptible to normalization errors, misalignments, and sensitive to stray light. For these two angles, the rms level was set equal to +0.5% of the signal level, whereas, for all the remaining angles, it was set to a relative value of 0.05%, with alternating positive and negative values.

Finally, for each polymerization model, fine-spaced sets of M_w and R_g²_z data were generated by varying p in the range 0-0.9. These data were analyzed as R_g²_z versus M_w plots and compared with the corresponding graphs obtained from the experimental data.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION AND CONCLUSIONS CONCLUDING REMARKS REFERENCES

Basic performance of the stopped-flow MALLS set-up

In Fig. 4 are reported, as a function of time, the normalized raw scattering intensities, collected by the photodiodes placed at the various scattering angles, of a fibrin polymerization reaction (c = 0.11 mg ml¹) initiated by thrombin with stopped-flow mixing and allowed to proceed until saturation of the lower angles detectors (only the first 1200 s are shown). As can be seen, the traces are quite smooth even at the lowest angle collected, ~22° (no mathematical smoothing was applied). In the inset, the fibrinogen "baseline" (negative time points) and the first 40 s after thrombin addition are highlighted on an expanded scale, to show the behavior at the very early stages. The sudden increase of noise on the lowest angle detector (inset, upper trace) appearing at 25 s happened occasionally due to the opening of the valves in preparation for injection of the fibrinogen/thrombin mixture. It can be also observed that, after only 5-7 s of polymerization, there is already a small but appreciable change in the intensity of scattered light from the value of the unreacted fibrinogen zone. Overall, these traces are indicative of the performance of the set up, especially for what concerns dust contamination and instabilities due to the stopped-flow mixing, and clearly show that, even at such low protein concentrations, it is possible to recover good data points with a 0.25-s time resolution.

View larger version (38K): [in this window] [in a new window]   FIGURE 4   Plot of the scattered intensities as a function of time collected by the fifteen detectors of the DAWN-DSP-F photometer for the polymerization of fibrinogen (c = 0.11 mg ml1) induced by thrombin (0.25 nominal NIH units/mg FG). On the right y-axis are indicated the scattering angles seen by each detector. In the inset, the scattered intensities collected in the time range from 30 s before to 50 s after the injection of the reaction mixture (vertical arrow) are shown; the traces at negative times are due to the scattering of unreacted fibrinogen.

Next, we turned our attention to the problem of photodiodes normalization, to correct for their different responses. Whereas small errors in this procedure can be tolerated when dealing with either chromatograms or batch measurements on nonevolving samples, this was found not to be the case for polymerization studies. In particular, considering the peculiarities of the fibrinogen-fibrin conversion (see below), it was crucial to perform this operation as carefully as possible. To begin with, every batch of fibrinogen used in the polymerization studies was also analyzed by SE-HPLC with MALLS detection, as described for BSA in the Materials and Methods section, and using BSA for normalization. We consistently found an (R_g²_z)^1/2 of ~15 nm across the peak corresponding to monomeric fibrinogen (Bernocco, 1998), as will be reported in more detail elsewhere (Bernocco S., C. Cuniberti, and M. Rocco, in preparation). Incidentally, this (R_g²_z)^1/2 value compares well with the 14.2 ± 0.5-nm value obtained by small-angle x-ray scattering on bovine fibrinogen solutions (Lederer, 1972). Next, the BSA normalization obtained from SE-HPLC was introduced in the stopped-flow data files, but the results were uneven. Therefore, the stopped-flow data files were renormalized using the unreacted fibrinogen zone, using the 15-nm (R_g²_z)^1/2 value derived from the SE-HPLC separations. When checked against the original BSA normalization coefficients, the new coefficients were within 1% of the original values for angles above 40°, whereas changes of up to 5% were observed for the three lower angles. However, the improvement in the quality of data for the polymerization runs was quite good, especially on the early stages (data not shown). At longer times, when the scattering intensity had already risen by a factor of ten, the two normalizations performed similarly. Although this procedure can be questioned on an absolute scale, we feel that it can be justified at least from the point of view of the internal consistency of each data file. It is possible that, at these low macromolecular concentrations, small, flow-resistant spots in the cell borehole play a relevant role, especially at low scattering angles.

Stopped-flow MALLS data processing and evaluation

Experimental Zimm and Casassa plots

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View larger version (29K): [in this window] [in a new window]   FIGURE 5   (A-D) Experimental Zimm and (E-H) Casassa plots obtained for the polymerization of fibrinogen (c = 0.33 mg ml1) induced by thrombin (0.25 nominal NIH units/mg FG), taken at four different reaction times: (A, E) 3 s; (B, F) 20 s; (C, G) 70 s; and (D, H) 120 s. (A-D) The continuous lines are third degree polynomial fittings, whereas the dotted lines are linear fittings through all the data points (A and B) or through only the first three data points (C and D). (E-H) The continuous lines are linear fittings and the open symbols denote the points not included in the regression.

View this table: [in this window] [in a new window]   TABLE 1   Parameters extrapolated from the different polynomial fittings of the Kc/R versus sin2(/2) data and from the linear fittings of the Kc/R versus sin(/2) data presented in Fig. 5.

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Simulated Zimm-plots, tests of the polynomial fittings, and comparison with the experimental data