208
Biochimica et Biophysica Acta, 537 (1978) 208--225 © Elsevier/North-Holland Biomedical Press
BBA 38042 FIBRINOGEN-FIBRIN TRANSFORMATIONS CHARACTERIZED DURING THE COURSE OF REACTION BY THEIR INTERMEDIATE STRUCTURES A LIGHT SCATTERING STUDY IN DILUTE SOLUTION UNDER PHYSIOLOGICAL CONDITIONS
MICHAEL MUELLER and WALTER BURCHARD Institute for Macrornolecular Chemistry, University of Freiburg, Stefa~-Meier-Strasse 31, D-7800 Freiburg 1 (F.R.G.)
(Received April 13th, 1978) Summary Intermediate structures of h u m a n fibrin formed under physiological conditions were investigated by means of light scattering in the course of the polymer/network formation. Very low flbrinogen concentrations (c = 0.03--0.13 mg/ml) were used to lower the polymerization rate, and thrombin at five concentrations {0.0085--0.04 N.I.H./ml) was used for initiation. The light scattering data were evaluated from (i) a Zimm plot, (ii) a Holtzer plot, i.e., h R o / K c vs. h 2, and (iii) a Kratky plot, i.e., h2Ro/Kc vs. h 2. In the beginning of the polymerization process rod-like structures are formed. The dimensions of the rodlike monomeric unit in the fibrin polymer are 112 × 3.9 nm and agree with the dimensions of fibrinogen, which also was f o u n d to be a thin rod of 105 -+ 10 nm length and 3.9 nm diameter. The mass per unit length, obtained from the asymptote in the Holtzer plot, initially increases only slightly but for high thrombin concentrations increases steeply when a critical length of 1000 nm is exceeded. At this point also the total scattering behaviour changes considerably. The upturn in the Zimm plot and the occurrence of a m a x i m u m in the Kratky plot are clear indications for the onset of branching. At low thrombin concentrations the kink in the curve of Mw/Lw against Mw becomes smoothed out because of nonspecific side-by-side aggregation of fibrin strands. The results are discussed and compared with earlier findings by others, and lead to the following conclusions. (i) Fibrinogen is a polymer with some flexibility and can exist in conformations of a stretched rod 105 nm in length, a folded rod of 45 nm in length, and a banana-like conformation of 94 nm circumference. (ii) Under the conditions of the present work, fibrinogen has the thin stretched rod conformation, and has the same dimensions as the repeating unit in the fibrin polymer. (iii) After approx. 10--12 units, end-to-end aggregated m o n o m e r branching occurs. (iv) The end-to-end aggregation is promoted by the cleavage
209 of A peptides, branching is caused by the cleavage of B peptides while side-byside aggregation of strands is caused by nonspecific van der Waals interaction.
Introduction
The complex mechanism of blood clotting has been of interdisciplinary interest since the early days of systematic scientific research [1,2]. In modern physiology blood clotting is described as network formation caused by a stepby-step aggregation of activated fibrinogen followed by stabilization through chemical cross-links [3,4]. Penetration of the red blood cells through an external boundary accompanied with an unhampered exchange of water, electrolytes and other small molecules is most effectively prevented by a polymer network. Because of its function to tighten bleeding wounds, fibrin has special role in protein chemistry. Most other proteins are described by their metabolic or reinforcing functions, e.g. in connective tissue [5], defined by their specific amino acid sequences. Network formation, however, is not determined solely by amino acid sequence or spatial structure. While the study of network formation is well established in polymer science, the mechanism has been recognized as a predominantly statistical process [6] involving polymerization of small molecules which possess no particular conformation. The fibrin network differs from synthetic networks most in that its formation involves as m o n o m e r a macromolecule which has a special conformation arising from its particular primary structure. For approx. 20 years the extent to which the structure of the fibrinogen m o n o m e r determines the network structure of the fibrin has been discussed, but no definite answer has yet been found. The situation is even more unique, for in spite of enormous efforts, the structure of fibrinogen and of the fibrin-monomer remains elusive [7--14]. Here we are interested not so much in the fibrinogen structure but in the information gained from analysis of intermediate structures formed during the clotting process under physiological conditions. The advantage of the comparatively large size of the fibrin m o n o m e r and the resulting extremely large dimensions of the network and intermediate structures have long been recognized and several useful techniques have been applied, e.g., electron microscopy [7,8,15--17], viscometry [15,18,19], streaming birefringence [20], viscoelasticity [21--23] and light scattering methods [15,24-28]. The fact that really satisfactory conclusions could not be drawn demands a careful examination of the conception and performance of the earlier experiments. The following discussion is confined to investigations of solution properties. In the past, experiments have been carried out either at extreme pH values or extreme ionic strenght [3,15,20]; alternatively, a preformed clot has been partly dissolved in a medium of a certain urea or guanidine-HCl concentration [24,25], or in an aqueous solution containing 2% acetic acid [24]. All these conditions are far from physiological. It remains doubtful whether the observed intermediate structures are really formed during the clotting process; they could be more the result of the artificially modified clotting.
210
These doubts and recent progress in light scattering techniques, combined with a deepened understanding of the statistical process of network formation [29,30], have given the impetus to this new study of intermediate structures formed during the clotting process under physiological conditions. A somewhat specific physical analysis will be applied to interpretation of the light scattering data, and the basis of this somewhat unfamiliar analytical technique it outlined in some detail in a theoretical section. The previous studies focused mainly on the structure of rods which are formed at the beginning of the aggregation process [3,24--26]. The present study includes, in addition, the consideration of the branching and cross-linking process occurring at later stages and at the end of the clotting process. Materials and Methods Human fibrinogen was isolated and freed from plasma proteins according to the m e t h o d of Osbahr et al. [31] by Professor I. Witt, Kinderklinik, Freiburg. The purified protein from 500 ml blood was dialysed 72 h against 10 l aqueous 0.3 M KC1 solution at 2°C. The diffusate, usually containing 25--38 mg/ml, was freeze-dried and yielded 620--780 mg of the lyophylized product. The concentration was first determined by the Biuret-method, but later it was measured by absorbance at 280 nm using the specific extinction of a 0.1% solution in a 1-cm cell U- "£ Zv0.1% 1.51. ble m The measurements of the aggregation processes were carried out by means of light scattering and were repeated with fibrinogen from different blood donors, where the isolation technique of Osbahr et al. was used. The result was checked once again with a fibrinogen sample which was supplied by KABI, Stockholm and purified by Professor E.A. Beck and his coworkers, H~/matologisches Zentrallabor, Inselspital, Bern. Purification was performed in this case by DEAE-cellulose chromatography, and the fractions were freed from factor XIII. The sample was dialysed against 0.1 M KC1/0.05 M Tris-HC1 buffer (pH 7.4). The measurements were repeated with fibrinogen from different sources to ensure that samples were free from denaturation or other artifacts possibly introduced by different isolation techniques or different blood donors. In particular, the KABI/fibrinogen was isolated by a different technique and was more throughly purified than were other samples isolated by the Osbahr method. The light scattering measurements were carried out at a wavelength, k0, of 435 nm (blue mercury line) at 20°C using a light scattering photogionimeter manufactured by Fica, Paris. The angles of observation were in the range of 20--150 °. Fibrinogen concentrations of 0.03--0.13 mg/ml were used. Solutions were centrifuged for clarification at 20 000 rev./min in an ultracentrifuge Spinco L using a fixed angle rotor. After centrifugation the solutions were introduced into the light scattering cells by a special pipette. This clarification procedure proved to be insufficient for the extremely low fibrin concentrations used. Clarification of aqueous solutions is a well k n o w n problem which becomes very serious at the low protein concentrations used, as traces of dust already cause large disturbance at low angles of observation. Best results =
211 were obtained eventually by centrifugation of the solutions directly in the light scattering cells in a swinging bucket rotor at 20 000 rev./min using the method of Dandliker [32,33] adapted to our light scattering cells. Test thrombin (60 N.I.H. units) from Behringwerke Marburg was used for initiation. The five polymerization series were carried out with 0.0085, 0.014, 0.028, 0.036 and 0.040 N.I.H./ml thrombin concentrations. Theory and Evaluation of Measurements The method applied in this study is elastic light scattering from dilute solutions. This technique is widely used in polymer science for characterization of large molecules but has remained limited in most cases to the determination of the molecular weight, Mw, and mean-square radius of gyration, (S2)z [34]. (The indices w and z indicate the weight and z-average, respectively) *. However, the angular dependence of scattered light yields much more information on the shape of the molecule than merely that of (S~)z which is a measure of only the overall dimensions. Advantage can be taken of this information if at least one of the particle dimensions is comparable in size with the wavelength of the light. In such a case the scattered beams emerging from different points of the same particle exhibit phase differences which cause interference and, thus, attenuation of the scattered light intensity. This attenuation depends on the angle of observation and, in a characteristic manner, on the shape of the particle. The phase difference between the different scattered beams, coming from different points of the same particle, increases with increasing angle of observation but is zero at zero angle. Therefore, no diminution is observed at zero angle. The scattering intensity normalised to unit at zero angle is called the particle scattering factor P(h). P(h) = i(O)/i(O)
(1)
where 0 is the scattering angle, i(O) and i(0) the scattering intensities at angles 0 and 0, respectively. Clearly P(0) = 1. Theoretical considerations have shown that the particle scattering factor is a unique function of the value of the scattering vector, h, where h = (47r/k) sin 0/2
(2)
and can be calculated by Debye's general formula [35] P(h)
1 ~
~~ ,
~n ~sinhrij~
T
\---hr]-~-/
(3)
In these equations, X = X0/n0, the wavelength of the light in the medium, ~0 the wavelength in vacuum and no the refractive index of the medium; rij is the spatial distance between two scattering points i and ]; the angular brackets denote the average over all orientations and, if flexible particles are involved, the average over the spatial pair distance distribution W(rii) [36]. The double sum em* F o r d e f i n i t i o n o f t h e d i f f e r e n t averages, s e e t e x t b o o k s o f p o l y m e r s c i e n c e .
212 braces all pairs of the n scattering elements in a particle. The particle scattering factors for different particle structures have been calculated on the basis of Eqn. 3 [37,38]. From these calculations, P(h) exhibits a specific shape-dependent behaviour at large values of h. At low values of h Eqn. 3 can be expanded in a power series in terms of h 2 and gives for all shapes P(h) = 1 -- (1) h 2 + ...
(4)
where the coefficient
- 2n12 ~i ~j
(5)
is the mean-square radius of gyration. Scattering behaviour of rods
One of the intermediate fibrin structures discussed in the literature is described as a rod of two strands of end-to-end polymerized fibrin monomers aggregated in parallel [24,25]. The scattering behaviour of such rods is determined by the particle scattering factor [ 39]:
1 }xsiot _ \ sinx 2) ~ dt --~
P(h) = ~ o
(6)
which has for X > 0.8 a very characteristic asymptote [40] P(h)-~ 2X
1 2X 2
(7)
where X = (27r/X) L sin 0 / 2 = h L / 2
(8)
and L is the rod length. Eqn. 7 shows that the function hP(h) approaches a constant value of 7r/L if X > 5 [41], but this constant value can be found by a fit through Eqn. 7 at lower values of X where the constant asymptote is n o t y e t reached. The relationship for the particle scattering factor can be combined with the relationship for the intensity of the scattered light, which according to Debye [42] for small values of h 2 and low concentrations is Kc 1 smart 1 + (~)(S2)z h 2 R o - P(h) M w h:-~ Mw
(9)
The quantities in this equation are K = 4n:n~)(dn/dc):
(10)
Ro = i(O) r:/Io is the Rayleigh ratio of scattered light i(O) to primary light inten-
sity, I0, at a distance r from the scattering centre, N is Avogadro's number, no the refractive index of the solvent and d n / d c the refractive index increment.
213 Insertion of Eqn. 7 into Eqn. 9 gives [41,24]
hRo _ Mw _ 2Mw 7rKc Lw h~rL~
(11)
From this equation, the weight per unit length can be obtained from the constant of the a s y m p t o t e in the plot of hRo/(Kc) against h 2, which we shall call a Holtzer plot. The length itself may be found from the mean-square radius of gyration [43] z = ¼(z/3 + r~yl) -~ ~/12
(12)
which is found from the initial slope in a Zimm plot (see Eqn. 9) where Kc/Ro is plotted against h2; royI is the radius of a cylindrical rod and the arrow indicates the limit of a thin rod, i.e., rc2yl > 1. The I ~ a t k y plot of P(h)
(19)
[a91
h2(1 + C($2)zh2/3) h2P(h) = [1 + (1 + C)($2}zh2/6] 2
(20)
shows a m a x i m u m at 6 h2"': = ($2), (1 -- 3C)
(21)
and has a decaying a s y m p t o t e at large values of u 2 = (S2)zh 2, C 12 h2P(h) -+ (1 + C) 2 (S2)zh 2
(22)
This theoretical outline of the behaviour of the three significantly different structures suggests the following procedure for evaluating scattering measurements.
215
Zimm plot [53] Molecular weight and mean-square radius of gyration can be obtained from a plot of Kc/Re against h ~ = (4~r/~)2sin2 0/2, first suggested by Zimm [52], where the intercept of this curve is 1/Mw, and the slope is (S2)z/(3Mw). (Strictly speaking, due to interparticle interactions these values are apparent quantities which depend on the polymer concentration c, but for very dilute solutions such as for fibrin the difference between the apparent and true quantities is negligibly small.) The shape of the curves in this plot gives a first hint of whether rod-like structures are present, whether randomly coiled or randomly branched structures exist, or whether significant deviations from both models exist. Fig. 1 shows the feature of the angular dependence of the three models discussed.
Mass per unit length from rod.like structures; Holtzer plot [41] A plot of hRe/(Kc) against h 2 will give a clear indication for a rod-like structure. Fig. 2 shows that a constant asymptote is obtained for rods only; its value is 7rMw/Lw.
Kratky plot [47] For all angular distributions of scattered light which do not show the rod asymptote, the Kratky plot will be most instructive. Fig. 3 shows the significantly different behaviour of the three models (randomly branched coils, nonrandomly branched coils, rods) at large deviations. Application of these three plots will allow exclusion of a large number of possible structures, and in most cases only a limited number of feasible conformations will remain. Further specification can then be achieved by quantitative comparison of the angular dependences measured and derived theoretically from special models. For further details of evaluating light scattering data the article by Geiduscheck and Holtzer [64] may be consulted.
c 20
1.0 .c
J:
,=..
b 10
0.6 O.4
0.2
'
I~
2~
.Xs'>,
30
0.1 0
,
10l
=
2|0
=
3=0
=
h'(S')z
Fig. 1. Reciprocal particle scattering factor p~l (h) as a f u n c t i o n o f h2(S2)z, (Zimrn plot), h = (4~r/X) sin 8/2: a, ro d; b, r a n d o m l y b r a n c h e d coil; c, coil branched under restriction. Fig. 2. The particle scattering factor m u l t i p l i e d b y h(S2) I/2 (Holtzar plot) as f u n c t i o n o f h2($2) z. a, rod; b, r a n d o m l y branched coil; c, coil b r a n c h e d u n d e r restriction,
216 3
a
2
b
%
O0
ItO
'
210 h ~ csh2
(25)
Therefore, the slope in the logarithmic plot of Fig. 6c could be equivalent to the mean-square radius of gyration of the crosssection of a rod which, for a cylindric crosssection, is related to the radius, reyl, b y the equation r e1y2
=
2cyl
(26)
The extrapolated intercept gives, as before, a constant asymptote, the mass per unit length. The Kratky plots shown in Fig. 6(a--c) reveal the occurrence of a maximum at times when the sudden change in the scattering behaviour has taken place. This maximum is more an indication for a nonrandomly-formed branched structure than for an excessive increase of rod diameter [45]. This tentative interpretation is supported b y Figs. 7 and 8 where ($2>~ and M w / L w are plotted against Mw. A fairly steep increase of (S~>z is observed in the beginning and a very flat curve at Mw > 2 • 106. The mass per unit length shows opposite behaviour and increases more rapidly b e y o n d Mw = 6 • 106 for high thrombin concentration. It must be stressed here that the values of this latter part are apparent values only, since in that region the decaying a s y m p t o t e of Eqn. 25 is observed, and this decay will occur for branched structures, also. The characteristic kinks in the curves of Figs. 8 and 9 are not observed at all thrombin concentrations. When the thrombin concentration is lowered, the
219 10 e
~'E
6 4
=
2
E IO s
.o ,.,..-~,.~.-o ~ °
6
4.: 3
.~o" ~,~ 6 4 2 10 s 6
,,'41
~2
2 10 5
•
2
i SJ 4 6 10 0 2
~ 2
104 6 4 ,
4 6 10 ,/ 2
4 6 10 e 2
4 6 10 8 2
~'~w; e;mot
4 6 10 7 2
,
4 6 10 s 2
Mw;glm°t
Fig. 7, Mean-square radii o f g y r a t i o n , (S2)z, of fibrin i n t e r m e d i a t e s as f u n c t i o n o f t h e m o l e c u l a r w e i g h t , M w. T h e n u m b e r s i n d i c a t e cuxves o b t a i n e d at d i f f e r e n t t h r o m b i n c o n c e n t r a t i o n s , 1, 0.0085 N.I.H./ml; 3, 0.028 N.I.H./ml; 4, 0.036 N.I,H./rnL e, fib~nogen m o n o m e r . Fig. 8. Mass per u n i t l e n g t h , Mw/L w. as a f u n c t i o n o f the m o l e c u l a r w e i g h t , M w . T h e n u m b e r s i n d i c a t e curves o b t a i n e d at d i f f e r e n t t h r o m b i n c o n c e n t r a t i o n s . 1, 0.0085 N.I.H./ml; 2, 0.014 N.I.H./ml; 3, 0.028 N.I.H./ml; 4, 0.036 N,I.H./ml; 5, 0.04 N.I.H./mL
kink becomes smoothed out and eventually disappears completely at the very low thrombin concentrations used for the activation of fibrinogen. Furthermore, the striking maximum in the Kratky plot becomes less pronounced and could not be observed for the run with the lowest thrombin concentration. Instead the asymptote typical of randomly branched or cross-linked polymers is obtained. Discussion The fibrinogen monomer Most physical investigations on polymers in the blood-clotting process have dealt with the structure of fibrinogen, which is the nonactivated monomer for the fibrin polymer. In spite of great efforts to determine the structure, and the
21
1°0
.A
!
4
I
I
8
!
,I
12
I
16
h2,104; nm-2 Fig. 9. H o l t z e r p l o t o f t h e s c a t t e r i n g data o f f i b r i n o g e n in c o m p a r i s o n w i t h t h e o r e t i c a l curves o f rods w i t h l e n g t h s of 45 and 105 n m , r e s p e c t i v e l y .
220 different techniques applied, several contradicting models are still under discussion. These are: (i) the three bead model of Hall and Slayter [7] with distances o f a p pr o x . 25 nm be t w een the beads: (ii) the cylinder model of Bachman et al. [11] with dimensions of 45 × 9 nm; and (iii) the spherical or banana-like model o f Marguerie et al. [13,14] with an average diameter of 29--30 nm. The three bead mo d el o f Hall and Slayter [7] has been recognized to be the result of d e h y d r a t i o n in the course of preparation for electron m i croscopy [11] and will n o t be discussed further. Assuming a rod-like structe, one obtains with the molecular weight of Mw = 340 000 +- 34 000 and a mean-square radius of gyration of (S:)z = 920 -+ 160 nm 2, measured b y the present light scattering study, a rod length of Lz = 105 +10 nm and a mass per unit length of Mw/Lz = 3230-+ 630 g . tool -1- nm -1, where the subscripts w and z refer to weight and z-averages, respectively. With this length the co n s t a nt a s y m p t o t e in the Holtzer plot is n o t y e t fully attained when using visible light of k0 = 435 nm. The correct a s y m p t o t i c value is found, however, by means of Eqn. 11. Fig. 9 shows the experimental curve for fibrinogen in comparison with theoretical curves of monodisperse rods with lengths of 45 and 105 nm, respectively. One realizes that the short rod model as suggested by Bachman et al. [11] b y no means fits the experimental curve. A rod of lenght 105 nm would lead to good agreement o f the scattering curve at large scattering angles, but would show slight deviations at lower angles. The fit with 105 or 45 nm rod length gives values for the cons t a nt a s y m p t o t e of Mw/Lw of 3220 and 7555 g . mo1-1- nm -1, respectively. Therefore, f r om our experimental findings, the Bachman model can be ruled out. It is n o t e w o r t h y that also Lederer and Hammel [12] could n o t find the high value of 7555 g • mo1-1 • n m -1 for the mass per unit length from their X-ray small angle scattering measurements, but f o u n d only values in the range 4 2 0 0 - - 6 1 5 0 g • mo1-1 • n m -1. Considering as an alternative a globular structure as suggested by Marguerie et al. [13,14] one should observe a slight u p t u r n in the Zimm plot at large scattering angles; instead, a slight d o w n t u r n is observed. F u r t h e r m o r e , if the sphere radius is calculated f r om the measured mean-square radius of gyration one obtains r = 39.2 nm for the c o m p a c t sphere model and r = 30.3 nm for the hollow sphere mode, while Marguerie's model has a radius of only 14--15 nm [13,14]. This also leads to the conclusion that this model cannot fit our experimental results. T h er ef o r e, o u r measurements correspond well to a long and thin rod-like structure of the fibrinogen with dimensions of L = 105 nm and a cylinder diameter, d, o f 3.9 nm. These data raise t he question of w h e t h e r the apparently contradicting models and t he findings f r om ot her laboratories can still be interpreted u n d er a c o m m o n scheme. It appears conceivable to us t hat fibrinogen can exist in different isomeric structures which are reversibly interchangeable. This conjecture is supported by the fact t hat the circumference of Marguerie's model is 88--94 nm, which is close to the rod length observed in our experiments. Fig. 10 demonstrates the different isomeric structure which we tentatively suggest as an explanation for the embarrassing diversity of experimental findings. When discussing these structures we have to take into careful consideration
221
-.19om1.13
~---- 30
13
nm.----~]
~ P
A
B
C 3.9 nm
T
|
,,
IB
94nm
Fig. 1 0 . T h r e e d i f f e r e n t i s o m e r i c s t r u c t u r e s o f f i b r i n o g e n . A0 t h e b a n a n a - l i k e m o d e l o f M a r g u e r i e ; B. the f o l d e d r o d o f B a c h m a n n ; C, the stretched r o d f o u n d in the present s t u d y for f i b r i n o g e n a n d t h e m o n o m e r unit in the fibrin polymer.
the conditions of ionic strength and Ca 2÷ concentration. Marguerie et al. [13, 14] performed all their measurements with solutions of a certain, defined, Ca 2÷ concentration. In our experiments no calcium was added, and there can be present only traces originating from the twice-distilled water or from the chemical reagents used to prepare the buffer. This may be the reason for the formation of the unfolded thin rod structure which we observed. Evidently the folded structure found b y Marguerie et al. is stabilized b y Ca 2÷, and the fibrinogen molecule unfolds if the Ca 2÷ concentration falls below a critical value. In fact, Marguerie et al. suggest that Ca 2÷ forms a chelate complex with the hystidin residues and carboxyl groups of the two equivalent chain ends in the symmetric monomer, which leads to the observed spherical structure [53]. The observation of the more c o m p a c t cylindric form for fibrinogen might be a result of the higher concentration applied in the small angle X-ray scattering experiments. The transition to the cylindrical isomer may be caused b y intramoleculax association, possibly hydrophobic and H-bond interactions, between the long rod-like sections of the molecule which do n o t belong to the disulfide knot. This structure m a y be the thermodynamically stable conformation at high concentrations and under the extreme conditions which exist in the preparation for electron microscopy. Our suggestion would imply hinges of some flexibility near the disulphide knot which closely resemble those in the antibody structure. Similar flexible
222 isomeric structures of fibrinogen were previously discussed by Marguerie and coworkers [54]. We believe that the astounding difficulties in determining the spatial structure of fibrinogen may have their origin in the fact that only definite structures have been considered for interpretation while the simultaneous presence of different isomeric structures has hitherto not been taken into consideration. It appears desirable now to reconsider all structural investigations performed in the past in light of this new aspect. For instance, the low value of the mass per unit length found by Lederer may have a simple explanation.
The rod-like fibrin intermediates The data obtained by light scattering lead unambiguously to the conclusion that the fibrin m o n o m e r , i.e., the repeating subunit in the fibrin polymer, has the same rod-like structure as the fibrinogen m o n o m e r with dimensions of 105 × 3.9 nm. This conclusion may be demonstrated with an example. At a molecular weight, Mw, of 1.25 • 106 we find ($2)~ = 9.98 • 103 nm 2 and a mass per unit length, M~/L~, of 3855 g • mo1-1 • nm -1. These data correspond to an increase of molecular weight by a factor 3.67, and if the lateral increase by a factor of 1.19 is taken into account one finds an end-to-end aggregation of 3.08 m o n o m e r units. The rod length obtained from
Biochimica et Biophysica Acta, 537 (1978) 208--225 © Elsevier/North-Holland Biomedical Press
BBA 38042 FIBRINOGEN-FIBRIN TRANSFORMATIONS CHARACTERIZED DURING THE COURSE OF REACTION BY THEIR INTERMEDIATE STRUCTURES A LIGHT SCATTERING STUDY IN DILUTE SOLUTION UNDER PHYSIOLOGICAL CONDITIONS
MICHAEL MUELLER and WALTER BURCHARD Institute for Macrornolecular Chemistry, University of Freiburg, Stefa~-Meier-Strasse 31, D-7800 Freiburg 1 (F.R.G.)
(Received April 13th, 1978) Summary Intermediate structures of h u m a n fibrin formed under physiological conditions were investigated by means of light scattering in the course of the polymer/network formation. Very low flbrinogen concentrations (c = 0.03--0.13 mg/ml) were used to lower the polymerization rate, and thrombin at five concentrations {0.0085--0.04 N.I.H./ml) was used for initiation. The light scattering data were evaluated from (i) a Zimm plot, (ii) a Holtzer plot, i.e., h R o / K c vs. h 2, and (iii) a Kratky plot, i.e., h2Ro/Kc vs. h 2. In the beginning of the polymerization process rod-like structures are formed. The dimensions of the rodlike monomeric unit in the fibrin polymer are 112 × 3.9 nm and agree with the dimensions of fibrinogen, which also was f o u n d to be a thin rod of 105 -+ 10 nm length and 3.9 nm diameter. The mass per unit length, obtained from the asymptote in the Holtzer plot, initially increases only slightly but for high thrombin concentrations increases steeply when a critical length of 1000 nm is exceeded. At this point also the total scattering behaviour changes considerably. The upturn in the Zimm plot and the occurrence of a m a x i m u m in the Kratky plot are clear indications for the onset of branching. At low thrombin concentrations the kink in the curve of Mw/Lw against Mw becomes smoothed out because of nonspecific side-by-side aggregation of fibrin strands. The results are discussed and compared with earlier findings by others, and lead to the following conclusions. (i) Fibrinogen is a polymer with some flexibility and can exist in conformations of a stretched rod 105 nm in length, a folded rod of 45 nm in length, and a banana-like conformation of 94 nm circumference. (ii) Under the conditions of the present work, fibrinogen has the thin stretched rod conformation, and has the same dimensions as the repeating unit in the fibrin polymer. (iii) After approx. 10--12 units, end-to-end aggregated m o n o m e r branching occurs. (iv) The end-to-end aggregation is promoted by the cleavage
209 of A peptides, branching is caused by the cleavage of B peptides while side-byside aggregation of strands is caused by nonspecific van der Waals interaction.
Introduction
The complex mechanism of blood clotting has been of interdisciplinary interest since the early days of systematic scientific research [1,2]. In modern physiology blood clotting is described as network formation caused by a stepby-step aggregation of activated fibrinogen followed by stabilization through chemical cross-links [3,4]. Penetration of the red blood cells through an external boundary accompanied with an unhampered exchange of water, electrolytes and other small molecules is most effectively prevented by a polymer network. Because of its function to tighten bleeding wounds, fibrin has special role in protein chemistry. Most other proteins are described by their metabolic or reinforcing functions, e.g. in connective tissue [5], defined by their specific amino acid sequences. Network formation, however, is not determined solely by amino acid sequence or spatial structure. While the study of network formation is well established in polymer science, the mechanism has been recognized as a predominantly statistical process [6] involving polymerization of small molecules which possess no particular conformation. The fibrin network differs from synthetic networks most in that its formation involves as m o n o m e r a macromolecule which has a special conformation arising from its particular primary structure. For approx. 20 years the extent to which the structure of the fibrinogen m o n o m e r determines the network structure of the fibrin has been discussed, but no definite answer has yet been found. The situation is even more unique, for in spite of enormous efforts, the structure of fibrinogen and of the fibrin-monomer remains elusive [7--14]. Here we are interested not so much in the fibrinogen structure but in the information gained from analysis of intermediate structures formed during the clotting process under physiological conditions. The advantage of the comparatively large size of the fibrin m o n o m e r and the resulting extremely large dimensions of the network and intermediate structures have long been recognized and several useful techniques have been applied, e.g., electron microscopy [7,8,15--17], viscometry [15,18,19], streaming birefringence [20], viscoelasticity [21--23] and light scattering methods [15,24-28]. The fact that really satisfactory conclusions could not be drawn demands a careful examination of the conception and performance of the earlier experiments. The following discussion is confined to investigations of solution properties. In the past, experiments have been carried out either at extreme pH values or extreme ionic strenght [3,15,20]; alternatively, a preformed clot has been partly dissolved in a medium of a certain urea or guanidine-HCl concentration [24,25], or in an aqueous solution containing 2% acetic acid [24]. All these conditions are far from physiological. It remains doubtful whether the observed intermediate structures are really formed during the clotting process; they could be more the result of the artificially modified clotting.
210
These doubts and recent progress in light scattering techniques, combined with a deepened understanding of the statistical process of network formation [29,30], have given the impetus to this new study of intermediate structures formed during the clotting process under physiological conditions. A somewhat specific physical analysis will be applied to interpretation of the light scattering data, and the basis of this somewhat unfamiliar analytical technique it outlined in some detail in a theoretical section. The previous studies focused mainly on the structure of rods which are formed at the beginning of the aggregation process [3,24--26]. The present study includes, in addition, the consideration of the branching and cross-linking process occurring at later stages and at the end of the clotting process. Materials and Methods Human fibrinogen was isolated and freed from plasma proteins according to the m e t h o d of Osbahr et al. [31] by Professor I. Witt, Kinderklinik, Freiburg. The purified protein from 500 ml blood was dialysed 72 h against 10 l aqueous 0.3 M KC1 solution at 2°C. The diffusate, usually containing 25--38 mg/ml, was freeze-dried and yielded 620--780 mg of the lyophylized product. The concentration was first determined by the Biuret-method, but later it was measured by absorbance at 280 nm using the specific extinction of a 0.1% solution in a 1-cm cell U- "£ Zv0.1% 1.51. ble m The measurements of the aggregation processes were carried out by means of light scattering and were repeated with fibrinogen from different blood donors, where the isolation technique of Osbahr et al. was used. The result was checked once again with a fibrinogen sample which was supplied by KABI, Stockholm and purified by Professor E.A. Beck and his coworkers, H~/matologisches Zentrallabor, Inselspital, Bern. Purification was performed in this case by DEAE-cellulose chromatography, and the fractions were freed from factor XIII. The sample was dialysed against 0.1 M KC1/0.05 M Tris-HC1 buffer (pH 7.4). The measurements were repeated with fibrinogen from different sources to ensure that samples were free from denaturation or other artifacts possibly introduced by different isolation techniques or different blood donors. In particular, the KABI/fibrinogen was isolated by a different technique and was more throughly purified than were other samples isolated by the Osbahr method. The light scattering measurements were carried out at a wavelength, k0, of 435 nm (blue mercury line) at 20°C using a light scattering photogionimeter manufactured by Fica, Paris. The angles of observation were in the range of 20--150 °. Fibrinogen concentrations of 0.03--0.13 mg/ml were used. Solutions were centrifuged for clarification at 20 000 rev./min in an ultracentrifuge Spinco L using a fixed angle rotor. After centrifugation the solutions were introduced into the light scattering cells by a special pipette. This clarification procedure proved to be insufficient for the extremely low fibrin concentrations used. Clarification of aqueous solutions is a well k n o w n problem which becomes very serious at the low protein concentrations used, as traces of dust already cause large disturbance at low angles of observation. Best results =
211 were obtained eventually by centrifugation of the solutions directly in the light scattering cells in a swinging bucket rotor at 20 000 rev./min using the method of Dandliker [32,33] adapted to our light scattering cells. Test thrombin (60 N.I.H. units) from Behringwerke Marburg was used for initiation. The five polymerization series were carried out with 0.0085, 0.014, 0.028, 0.036 and 0.040 N.I.H./ml thrombin concentrations. Theory and Evaluation of Measurements The method applied in this study is elastic light scattering from dilute solutions. This technique is widely used in polymer science for characterization of large molecules but has remained limited in most cases to the determination of the molecular weight, Mw, and mean-square radius of gyration, (S2)z [34]. (The indices w and z indicate the weight and z-average, respectively) *. However, the angular dependence of scattered light yields much more information on the shape of the molecule than merely that of (S~)z which is a measure of only the overall dimensions. Advantage can be taken of this information if at least one of the particle dimensions is comparable in size with the wavelength of the light. In such a case the scattered beams emerging from different points of the same particle exhibit phase differences which cause interference and, thus, attenuation of the scattered light intensity. This attenuation depends on the angle of observation and, in a characteristic manner, on the shape of the particle. The phase difference between the different scattered beams, coming from different points of the same particle, increases with increasing angle of observation but is zero at zero angle. Therefore, no diminution is observed at zero angle. The scattering intensity normalised to unit at zero angle is called the particle scattering factor P(h). P(h) = i(O)/i(O)
(1)
where 0 is the scattering angle, i(O) and i(0) the scattering intensities at angles 0 and 0, respectively. Clearly P(0) = 1. Theoretical considerations have shown that the particle scattering factor is a unique function of the value of the scattering vector, h, where h = (47r/k) sin 0/2
(2)
and can be calculated by Debye's general formula [35] P(h)
1 ~
~~ ,
~n ~sinhrij~
T
\---hr]-~-/
(3)
In these equations, X = X0/n0, the wavelength of the light in the medium, ~0 the wavelength in vacuum and no the refractive index of the medium; rij is the spatial distance between two scattering points i and ]; the angular brackets denote the average over all orientations and, if flexible particles are involved, the average over the spatial pair distance distribution W(rii) [36]. The double sum em* F o r d e f i n i t i o n o f t h e d i f f e r e n t averages, s e e t e x t b o o k s o f p o l y m e r s c i e n c e .
212 braces all pairs of the n scattering elements in a particle. The particle scattering factors for different particle structures have been calculated on the basis of Eqn. 3 [37,38]. From these calculations, P(h) exhibits a specific shape-dependent behaviour at large values of h. At low values of h Eqn. 3 can be expanded in a power series in terms of h 2 and gives for all shapes P(h) = 1 -- (1) h 2 + ...
(4)
where the coefficient
- 2n12 ~i ~j
(5)
is the mean-square radius of gyration. Scattering behaviour of rods
One of the intermediate fibrin structures discussed in the literature is described as a rod of two strands of end-to-end polymerized fibrin monomers aggregated in parallel [24,25]. The scattering behaviour of such rods is determined by the particle scattering factor [ 39]:
1 }xsiot _ \ sinx 2) ~ dt --~
P(h) = ~ o
(6)
which has for X > 0.8 a very characteristic asymptote [40] P(h)-~ 2X
1 2X 2
(7)
where X = (27r/X) L sin 0 / 2 = h L / 2
(8)
and L is the rod length. Eqn. 7 shows that the function hP(h) approaches a constant value of 7r/L if X > 5 [41], but this constant value can be found by a fit through Eqn. 7 at lower values of X where the constant asymptote is n o t y e t reached. The relationship for the particle scattering factor can be combined with the relationship for the intensity of the scattered light, which according to Debye [42] for small values of h 2 and low concentrations is Kc 1 smart 1 + (~)(S2)z h 2 R o - P(h) M w h:-~ Mw
(9)
The quantities in this equation are K = 4n:n~)(dn/dc):
(10)
Ro = i(O) r:/Io is the Rayleigh ratio of scattered light i(O) to primary light inten-
sity, I0, at a distance r from the scattering centre, N is Avogadro's number, no the refractive index of the solvent and d n / d c the refractive index increment.
213 Insertion of Eqn. 7 into Eqn. 9 gives [41,24]
hRo _ Mw _ 2Mw 7rKc Lw h~rL~
(11)
From this equation, the weight per unit length can be obtained from the constant of the a s y m p t o t e in the plot of hRo/(Kc) against h 2, which we shall call a Holtzer plot. The length itself may be found from the mean-square radius of gyration [43] z = ¼(z/3 + r~yl) -~ ~/12
(12)
which is found from the initial slope in a Zimm plot (see Eqn. 9) where Kc/Ro is plotted against h2; royI is the radius of a cylindrical rod and the arrow indicates the limit of a thin rod, i.e., rc2yl > 1. The I ~ a t k y plot of P(h)
(19)
[a91
h2(1 + C($2)zh2/3) h2P(h) = [1 + (1 + C)($2}zh2/6] 2
(20)
shows a m a x i m u m at 6 h2"': = ($2), (1 -- 3C)
(21)
and has a decaying a s y m p t o t e at large values of u 2 = (S2)zh 2, C 12 h2P(h) -+ (1 + C) 2 (S2)zh 2
(22)
This theoretical outline of the behaviour of the three significantly different structures suggests the following procedure for evaluating scattering measurements.
215
Zimm plot [53] Molecular weight and mean-square radius of gyration can be obtained from a plot of Kc/Re against h ~ = (4~r/~)2sin2 0/2, first suggested by Zimm [52], where the intercept of this curve is 1/Mw, and the slope is (S2)z/(3Mw). (Strictly speaking, due to interparticle interactions these values are apparent quantities which depend on the polymer concentration c, but for very dilute solutions such as for fibrin the difference between the apparent and true quantities is negligibly small.) The shape of the curves in this plot gives a first hint of whether rod-like structures are present, whether randomly coiled or randomly branched structures exist, or whether significant deviations from both models exist. Fig. 1 shows the feature of the angular dependence of the three models discussed.
Mass per unit length from rod.like structures; Holtzer plot [41] A plot of hRe/(Kc) against h 2 will give a clear indication for a rod-like structure. Fig. 2 shows that a constant asymptote is obtained for rods only; its value is 7rMw/Lw.
Kratky plot [47] For all angular distributions of scattered light which do not show the rod asymptote, the Kratky plot will be most instructive. Fig. 3 shows the significantly different behaviour of the three models (randomly branched coils, nonrandomly branched coils, rods) at large deviations. Application of these three plots will allow exclusion of a large number of possible structures, and in most cases only a limited number of feasible conformations will remain. Further specification can then be achieved by quantitative comparison of the angular dependences measured and derived theoretically from special models. For further details of evaluating light scattering data the article by Geiduscheck and Holtzer [64] may be consulted.
c 20
1.0 .c
J:
,=..
b 10
0.6 O.4
0.2
'
I~
2~
.Xs'>,
30
0.1 0
,
10l
=
2|0
=
3=0
=
h'(S')z
Fig. 1. Reciprocal particle scattering factor p~l (h) as a f u n c t i o n o f h2(S2)z, (Zimrn plot), h = (4~r/X) sin 8/2: a, ro d; b, r a n d o m l y b r a n c h e d coil; c, coil branched under restriction. Fig. 2. The particle scattering factor m u l t i p l i e d b y h(S2) I/2 (Holtzar plot) as f u n c t i o n o f h2($2) z. a, rod; b, r a n d o m l y branched coil; c, coil b r a n c h e d u n d e r restriction,
216 3
a
2
b
%
O0
ItO
'
210 h ~ csh2
(25)
Therefore, the slope in the logarithmic plot of Fig. 6c could be equivalent to the mean-square radius of gyration of the crosssection of a rod which, for a cylindric crosssection, is related to the radius, reyl, b y the equation r e1y2
=
2cyl
(26)
The extrapolated intercept gives, as before, a constant asymptote, the mass per unit length. The Kratky plots shown in Fig. 6(a--c) reveal the occurrence of a maximum at times when the sudden change in the scattering behaviour has taken place. This maximum is more an indication for a nonrandomly-formed branched structure than for an excessive increase of rod diameter [45]. This tentative interpretation is supported b y Figs. 7 and 8 where ($2>~ and M w / L w are plotted against Mw. A fairly steep increase of (S~>z is observed in the beginning and a very flat curve at Mw > 2 • 106. The mass per unit length shows opposite behaviour and increases more rapidly b e y o n d Mw = 6 • 106 for high thrombin concentration. It must be stressed here that the values of this latter part are apparent values only, since in that region the decaying a s y m p t o t e of Eqn. 25 is observed, and this decay will occur for branched structures, also. The characteristic kinks in the curves of Figs. 8 and 9 are not observed at all thrombin concentrations. When the thrombin concentration is lowered, the
219 10 e
~'E
6 4
=
2
E IO s
.o ,.,..-~,.~.-o ~ °
6
4.: 3
.~o" ~,~ 6 4 2 10 s 6
,,'41
~2
2 10 5
•
2
i SJ 4 6 10 0 2
~ 2
104 6 4 ,
4 6 10 ,/ 2
4 6 10 e 2
4 6 10 8 2
~'~w; e;mot
4 6 10 7 2
,
4 6 10 s 2
Mw;glm°t
Fig. 7, Mean-square radii o f g y r a t i o n , (S2)z, of fibrin i n t e r m e d i a t e s as f u n c t i o n o f t h e m o l e c u l a r w e i g h t , M w. T h e n u m b e r s i n d i c a t e cuxves o b t a i n e d at d i f f e r e n t t h r o m b i n c o n c e n t r a t i o n s , 1, 0.0085 N.I.H./ml; 3, 0.028 N.I.H./ml; 4, 0.036 N.I,H./rnL e, fib~nogen m o n o m e r . Fig. 8. Mass per u n i t l e n g t h , Mw/L w. as a f u n c t i o n o f the m o l e c u l a r w e i g h t , M w . T h e n u m b e r s i n d i c a t e curves o b t a i n e d at d i f f e r e n t t h r o m b i n c o n c e n t r a t i o n s . 1, 0.0085 N.I.H./ml; 2, 0.014 N.I.H./ml; 3, 0.028 N.I.H./ml; 4, 0.036 N,I.H./ml; 5, 0.04 N.I.H./mL
kink becomes smoothed out and eventually disappears completely at the very low thrombin concentrations used for the activation of fibrinogen. Furthermore, the striking maximum in the Kratky plot becomes less pronounced and could not be observed for the run with the lowest thrombin concentration. Instead the asymptote typical of randomly branched or cross-linked polymers is obtained. Discussion The fibrinogen monomer Most physical investigations on polymers in the blood-clotting process have dealt with the structure of fibrinogen, which is the nonactivated monomer for the fibrin polymer. In spite of great efforts to determine the structure, and the
21
1°0
.A
!
4
I
I
8
!
,I
12
I
16
h2,104; nm-2 Fig. 9. H o l t z e r p l o t o f t h e s c a t t e r i n g data o f f i b r i n o g e n in c o m p a r i s o n w i t h t h e o r e t i c a l curves o f rods w i t h l e n g t h s of 45 and 105 n m , r e s p e c t i v e l y .
220 different techniques applied, several contradicting models are still under discussion. These are: (i) the three bead model of Hall and Slayter [7] with distances o f a p pr o x . 25 nm be t w een the beads: (ii) the cylinder model of Bachman et al. [11] with dimensions of 45 × 9 nm; and (iii) the spherical or banana-like model o f Marguerie et al. [13,14] with an average diameter of 29--30 nm. The three bead mo d el o f Hall and Slayter [7] has been recognized to be the result of d e h y d r a t i o n in the course of preparation for electron m i croscopy [11] and will n o t be discussed further. Assuming a rod-like structe, one obtains with the molecular weight of Mw = 340 000 +- 34 000 and a mean-square radius of gyration of (S:)z = 920 -+ 160 nm 2, measured b y the present light scattering study, a rod length of Lz = 105 +10 nm and a mass per unit length of Mw/Lz = 3230-+ 630 g . tool -1- nm -1, where the subscripts w and z refer to weight and z-averages, respectively. With this length the co n s t a nt a s y m p t o t e in the Holtzer plot is n o t y e t fully attained when using visible light of k0 = 435 nm. The correct a s y m p t o t i c value is found, however, by means of Eqn. 11. Fig. 9 shows the experimental curve for fibrinogen in comparison with theoretical curves of monodisperse rods with lengths of 45 and 105 nm, respectively. One realizes that the short rod model as suggested by Bachman et al. [11] b y no means fits the experimental curve. A rod of lenght 105 nm would lead to good agreement o f the scattering curve at large scattering angles, but would show slight deviations at lower angles. The fit with 105 or 45 nm rod length gives values for the cons t a nt a s y m p t o t e of Mw/Lw of 3220 and 7555 g . mo1-1- nm -1, respectively. Therefore, f r om our experimental findings, the Bachman model can be ruled out. It is n o t e w o r t h y that also Lederer and Hammel [12] could n o t find the high value of 7555 g • mo1-1 • n m -1 for the mass per unit length from their X-ray small angle scattering measurements, but f o u n d only values in the range 4 2 0 0 - - 6 1 5 0 g • mo1-1 • n m -1. Considering as an alternative a globular structure as suggested by Marguerie et al. [13,14] one should observe a slight u p t u r n in the Zimm plot at large scattering angles; instead, a slight d o w n t u r n is observed. F u r t h e r m o r e , if the sphere radius is calculated f r om the measured mean-square radius of gyration one obtains r = 39.2 nm for the c o m p a c t sphere model and r = 30.3 nm for the hollow sphere mode, while Marguerie's model has a radius of only 14--15 nm [13,14]. This also leads to the conclusion that this model cannot fit our experimental results. T h er ef o r e, o u r measurements correspond well to a long and thin rod-like structure of the fibrinogen with dimensions of L = 105 nm and a cylinder diameter, d, o f 3.9 nm. These data raise t he question of w h e t h e r the apparently contradicting models and t he findings f r om ot her laboratories can still be interpreted u n d er a c o m m o n scheme. It appears conceivable to us t hat fibrinogen can exist in different isomeric structures which are reversibly interchangeable. This conjecture is supported by the fact t hat the circumference of Marguerie's model is 88--94 nm, which is close to the rod length observed in our experiments. Fig. 10 demonstrates the different isomeric structure which we tentatively suggest as an explanation for the embarrassing diversity of experimental findings. When discussing these structures we have to take into careful consideration
221
-.19om1.13
~---- 30
13
nm.----~]
~ P
A
B
C 3.9 nm
T
|
,,
IB
94nm
Fig. 1 0 . T h r e e d i f f e r e n t i s o m e r i c s t r u c t u r e s o f f i b r i n o g e n . A0 t h e b a n a n a - l i k e m o d e l o f M a r g u e r i e ; B. the f o l d e d r o d o f B a c h m a n n ; C, the stretched r o d f o u n d in the present s t u d y for f i b r i n o g e n a n d t h e m o n o m e r unit in the fibrin polymer.
the conditions of ionic strength and Ca 2÷ concentration. Marguerie et al. [13, 14] performed all their measurements with solutions of a certain, defined, Ca 2÷ concentration. In our experiments no calcium was added, and there can be present only traces originating from the twice-distilled water or from the chemical reagents used to prepare the buffer. This may be the reason for the formation of the unfolded thin rod structure which we observed. Evidently the folded structure found b y Marguerie et al. is stabilized b y Ca 2÷, and the fibrinogen molecule unfolds if the Ca 2÷ concentration falls below a critical value. In fact, Marguerie et al. suggest that Ca 2÷ forms a chelate complex with the hystidin residues and carboxyl groups of the two equivalent chain ends in the symmetric monomer, which leads to the observed spherical structure [53]. The observation of the more c o m p a c t cylindric form for fibrinogen might be a result of the higher concentration applied in the small angle X-ray scattering experiments. The transition to the cylindrical isomer may be caused b y intramoleculax association, possibly hydrophobic and H-bond interactions, between the long rod-like sections of the molecule which do n o t belong to the disulfide knot. This structure m a y be the thermodynamically stable conformation at high concentrations and under the extreme conditions which exist in the preparation for electron microscopy. Our suggestion would imply hinges of some flexibility near the disulphide knot which closely resemble those in the antibody structure. Similar flexible
222 isomeric structures of fibrinogen were previously discussed by Marguerie and coworkers [54]. We believe that the astounding difficulties in determining the spatial structure of fibrinogen may have their origin in the fact that only definite structures have been considered for interpretation while the simultaneous presence of different isomeric structures has hitherto not been taken into consideration. It appears desirable now to reconsider all structural investigations performed in the past in light of this new aspect. For instance, the low value of the mass per unit length found by Lederer may have a simple explanation.
The rod-like fibrin intermediates The data obtained by light scattering lead unambiguously to the conclusion that the fibrin m o n o m e r , i.e., the repeating subunit in the fibrin polymer, has the same rod-like structure as the fibrinogen m o n o m e r with dimensions of 105 × 3.9 nm. This conclusion may be demonstrated with an example. At a molecular weight, Mw, of 1.25 • 106 we find ($2)~ = 9.98 • 103 nm 2 and a mass per unit length, M~/L~, of 3855 g • mo1-1 • nm -1. These data correspond to an increase of molecular weight by a factor 3.67, and if the lateral increase by a factor of 1.19 is taken into account one finds an end-to-end aggregation of 3.08 m o n o m e r units. The rod length obtained from
