Supporting Material Supplement

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1 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent Modeling of Fibrin Gels Based on Confocal Microscopy and Light-Scattering ata avide Magatti, Matteo Molteni, Barbara Cardinali, Mattia Rocco, and Fabio Ferri * ipartiento di Scienza e Alta Tecnologia, Università dell Insubria, Coo, Italy; and S.S. Biopolieri e Proteoica, IRCCS AOU San Martino-IST, Istituto Nazionale per la Ricerca sul Cancro, c/o CBA, Genova, Italy Subitted July 3, 0, and accepted for publication January 4, 03, corrected January 0, 07. *Correspondence: fabio.ferri@uninsubria.it avide Magatti s present address is Light in Light S.r.l., Coo, Italy. Barbara Cardinali s present address is S.S. Sviluppo Terapie Innovative, IRCCS AOU San Martino IST, Genova, Italy. Editor: Charles Wolgeuth. Supporting Material Suppleent Appendix A geoetrical properties of in silico fibrin gels In silico fibrin gels of different volue fraction φ can be generated by changing either the concentration n p of the nodal points at which the gel fibers are connected, and/or the fiber diaeter d. For any given gel, φ is expected to be proportional to d, ties n p, ties the average fiber length <L>. Thus, since the latter ones scales as > ~ ( < L n ) / 3 p, we get α φ = A d ( n ) (S) p where α = /3 and A is a diensionless constant that depends on the gel orphology, such as branching order and fiber length distributions. Fig. S reports the behavior of φ as a function of n p, for three series of in silico gels with d = 0.5 (blue squares), 0.0 (red circles) and 0.5 µ (green triangles). The solid lines are the best fits of the data to Eq. S, in which A and α were left as floating paraeters. The figure shows that all the data are nicely fitted by Eq. S with quite siilar

2 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent Figure S: Behavior of the in silico gel volue fractions φ as a function of the nodal points density n p for gels with fiber diaeters d = 0.5 (blue squares), 0.0 (red circles) and 0.5 µ (green triangles). The solid lines represent the best fits of the data to Eq. S. values of α and A, whose average values (averages over the three sets of data) are α= 0.64 ± 0.0 and A=.4 ± 0.0. The analysis of the geoetrical properties of the in silico gels at different volue fractions was carried out by changing the concentration of the nodal points while keeping fixed their diaeters. This is because these properties (such as fiber length distribution, distribution of nodal points branching order, and distribution of angles between fibers) depend only on the spatial arrangeent of the fiber constituting the gel, not on their diaeters. In this way, using a diaeter of d = 0.0 µ, we were able to vary the volue fraction over a range of about two decades, fro (see Fig. S, circles). The corresponding fiber length distributions are quite different (Fig. S), with gels at lower φ ade by of fibers uch longer than gels at higher φ. However, all the fiber distributions appear to be rearkably siilar in shape and suggest that all the Figure S: Fiber length distributions for a set of in silico gels with the sae diaeter d = 0.0 µ and different volue fraction φ. The distributions are highly dependent on φ (lower φ, longer fibers), but are rearkably siilar in shape, suggesting that all the gels are characterized by the sae orphology

3 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 3 (a) (b) (c) Figure S3 Angles between fibers (a), branching order of nodal points (b) and rescaled lengths (c) distributions for the sae gels of Fig. S. gels are characterized by the sae orphology. This feature is illustrated in the three panels of Fig. S3: (a) the angle distribution between fibers linked at the sae nodal points is fairly broad, skewed toward sall angles, with an average angle <α> = 0 ± 37 degs; (b) the branching order distribution is characterized by an average value <k> = 4.0 ± 0.9; (c) the rescaled fiber length distribution is a sooth bell shaped curve with a relative standard deviation σ L /<L> ~ It should be pointed out that, while the distributions of Fig. S3 appear to be qualitatively consistent with confocal iages of fibrin gels (see Fig. a of the ain text), when copared with the distributions recovered fro electron icroscopy data the situation is quite different. For exaple, fro Ref. (), fibrin gels grown under physical-cheical conditions siilar to the ones of Fig. a exhibit branching order distributions characterized by average values such as <k> = 3.4 ± 0.8 (0.05 M gel, Table, row ) or <k> = 3. ± 0.3 (0.0 M gel, Table, row ), which are clearly saller than the ones reported in Fig. S3(b). At the sae tie, fiber length distributions appear to be uch broader, with relative standard deviations of the order of σ L /<L> ~ Unfortunately, no data are available for angle distributions. This discrepancy is due to the fact that our in silico gels were designed to reproduce the structure of real fibrin gels as observed with a confocal icroscope or recovered fro elastic light scattering data. In both cases, the spatial resolution is in the sub-µ range, while for electron icroscopy it is in the sub-n range. Thus for exaple, two (or three) close 3-branched nodal points would appear, when observed at a confocal icroscope, as a single nodal point characterized by a branching order equal to four (or five). At the sae tie the short fibers connecting these close nodal points would not be resolved, and the fiber length distribution would be characterized by a larger average length and saller relative width. 3

4 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 4 Nevertheless, we would like to point out that such discrepancies do not invalidate the accuracy of our in silico gels in reconstructing the structural and orphological features of real fibrin gels, which depend on length scales of the order of µ or larger. We tested this clai by investigating the effect of re-adjusting a posteriori the spatial arrangeent of the nodal points so to obtain the desired branching and fiber length distributions. We followed this procedure: whenever a nodal point has a branching order k higher than expected, a new nodal point is generated at rando in the proxiity of the previous one, at a distance saller than the resolution of the icroscope (typically 00 n). Then, the two fibers closer to the new points are disconnected fro the previous point and re-connected at the new one. A third (short) fiber is created, connecting the previous and new nodal points. In this way we have created a new nodal point with k = 3 and reduced the branching order of the previous point fro k to k-. The procedure is repeated for all the nodal points of the network until the desired branching and fiber length distributions are obtained. Once the new network was generated, we copared it with the original one and observed that the two networks present the sae structural features, such the sae width of the correlation function, the sae fractal diension, and the sae average pore size. Thus we conclude that our original in silico networks, although inaccurate as far as concerns branching order and fiber length distributions, are quite suitable for accurately reproducing the ain structural features real of real fibrin gels in the sub-µ length scale range. 4

5 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 5 Appendix B detailed description of the fibrin gel for factor R(q) In this appendix we first describe the ain features of the intensity distribution R(q) given by Eq. 4 of the ain text (reprinted below as Eq. S,) and then discuss its differences with respect to the original expression that was proposed in Ref. (). S(q) R( q) = K c F M [ ) q ξ β exp( 4πη ( d / ξ) [ + ( d / ξ) / / [ + ( q ξ / δ ) [ + ( q d / δ ) P seg ( q, d ) (S) A A A 3 A(q) Fig. S4(a) reproduces the behavior of R(q) vs. q, for ξ = 0 µ, β =, η =, =.30 and d = 0. µ. The corresponding function g 3 (r), obtained by inverting Eq. 3 is shown in Fig. S4(b). As shown in Fig. S4(a), R(q) is characterized by three different regies, deliited by the two wavevectors qp and qx. These regies are clearly deterined by the factors S(q), A(q), P seg (q) of which R(q) is the product, and are represented, for S(q) and A(q), by the two solid straight lines with slopes and, respectively, reported in the figure, and for P seg (q) by the straight line with slope 4. A detailed description of these three regies is reported in Appendix C. Figure S4 (a) Log-log plot of the scattering Intensity distribution R(q) generated according to Eq.4 with ξ = 0 µ, η = β =, =.30 and d = 0.µ. (b) correlation function g 3 (r) obtained fro R(q) by inverting Eq. 3. Note the presence of a iniu in g 3 (r) which is associated to the presence of a axiu in R(q). 5

6 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 6 The behavior of g 3 (r) obtained by inverting Eq. 3 of the ain text is reported in Fig. S4(b). As expected for a fractal syste, it decays in the region d<<r<<ξ, as a power-law with an exponent α = 3. Notice the presence of a shallow iniu (see inset c) located at r ~6.7 µ, which is associated to the peak present in R(q). As entioned above, Eq. S was ipleented by starting fro the original expression for R(q) proposed in Ref. (). For the reader s convenience, such expression is here reported as Eq. S3, and re-written in ters of the paraeter η instead of γ [ γ = ( η π ) and with l = d as in Eq. S. With respect to Eq. S3, the ain differences appearing in Eq. S are: R(q) = K c F M q ξ [ β exp ( ) 4πη [ [ + ( q ξ / π) / + ( d / ξ) ( + q d / 3 ) (S3) A A 3 B(q) a) In the fractal correlation ter A (q) we replaced the constant π appearing in Eq. S3 with the function δ, whose dependence on (found heuristically by fitting the data of Fig. 7(a) is: δ = π 0.38 {[ tgh([ + (S4) / 0.0) / } According to Eq. S4, δ varies following a sigoidal shape centered around =.4, fro δ ~ π at = to δ ~. 76 at =. In this way the asyptotic behavior of R(q) in the fractal regie fits ore accurately the power spectra of the synthetic gels for any value of between < <. In the original Eq. S3, =. δ was fixed to δ = π, and was correct only for b) In the blobs structure factor A(q) we added the iddle ter A which takes into account the fact that the fractal correlation described by the ter A cannot persist down to length scales saller than d. Thus A is a negative correction that corresponds to the (noralized) 3 Fourier integral 3 ) of the fractal correlation function ( g( r) ~ r fro zero to d. By using Eq. 3 of the ain text we can work out this integral as 6

7 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 7 A (q) A (q = 0) = ~ 4π d 0 r 4π (3 ) d 0 r (3 ) [ + ( q d / δ r sin(qr) dr qr ) r dr / = F 3 [ { },{, + }, q d 4 (S5) where F [{a },{b,b },z is the hypergeoetric function p F q of order p = and q = (integral evaluated with Matheatica 7.0) that has been approxiated with the Fisher-Burford (3) function reported in the second line of Eq. S5. The behaviors of F and A as a function of q d for = are reported in Fig. S5, where one can appreciate that the noralized ter A approxiates quite accurately the hypergeoetric function both for q d << and for q d >>. Thus, A allows the correct noralization of the structure factor A(q) in q = 0, i.e. A(q = 0) = and, at the sae tie for q, it counterbalances exactly the A ter so that A (q ) = A (q ) and A( q ) = ( d / ξ). We tested and quantified the effectiveness of this correction in the case of infinitely long stiff cylinders ( = ), as shown in Fig. S6, where we report in panel (a) the behaviors of the scattering intensity distributions as predicted by Eq. S (red line) and Eq. S3 (gray line) with the theory (blue line) (4) as a function of the rescaled wavevector q d. The corresponding ratios between Eq. S, Eq. S3 and the theoretical behavior are shown in the inset Fig. S6(b). As one can easily notice, the atching between Eq. S and theory is rearkably better than Eq. S3, with deviations always saller than ~± 5%. - Figure S5 Coparison between the behaviours of the Hypergeoetric function F and the Fisher-Burford function versus the rescaled wavevector q d. 7

8 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 8 Figure S6 (a): Coparison between the scattering intensity distributions of infinitely long stiff cylinders as predicted by Eq. S (red line) Eq. S3 (gray line) and theory (blue line) as a function of the rescaled wavevector q d. (b): The ratios Eq. S/theory (red line) and Eq. S3/theory (blue line) versus q d show the uch better accuracy of Eq. S with respect to Eq. S3. c) Based on confocal iages of fibrin gels where fiber diaeters appear to be rather onodisperse in size (see Refs. (5,6), Fig. (a) of the ain text, and Figs. 5(a),(b) of Ref. (7)), or on electron icroscopy data (see Ref. () Table, rows and ) where the diaeter polydispersity appear to be relatively narrow (~0.5), we replaced the segent for factor B(q) of Eq. S3 (that was found heuristically on the basis of a supposed large diaeter polydispersity), with the for factor of onodisperse randoly oriented thick cylinders of diaeter d and length l = d, whose expression can be written as (4): P seg ( q, d) = P ( q, l = d) S ( q, d) (S6) rod sec where P rod (q, l) and S sec (q, d) are, respectively, the for factor of a thin rod and its diaeter cross section ter: P rod [ Si ( qd) sin( qd / ) ( q, d ) = (S7-a) qd qd / S sec ( J ( qd / ) qd / q, d) = [ (S7-b) In Eqs. S7 Si(x) and J (x) are the Sine integral defined as Bessel functions, respectively. Notice that the factorization of the two ters appearing in Eq. S6 is supposed to be accurate only when l>>d (4), a condition that is clearly not fulfilled by our 8 Si (x) [sin(t) / t dt and the first order = x 0

9 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 9 segents. However, it should be recalled that in the blob odel associated to Eq. S, the segents are not copletely randoly oriented because they are assebled end-to-end to for straight fibers of average length <L> >> d. Conversely, the fibers are totally randoly oriented in space and polydisperse in length as well. Thus, the use of Eq. S6 is justified by the fact that P seg (q, d) appears only as a factor of the product A(q) P seg (q, d), which takes into account the overall scattering of long fibers (<L> >> d), randoly oriented and fractally arranged in space. Appendix C the three regies of R(q) In this appendix we describe in soe detail the behavior R(q) vs. q (Eq. S) in the three regies deliited by the two wavevectors qp and qx indicated in Fig. S4(a). These regies are clearly deterined by the factors S(q), A(q), P seg (q) of which R(q) is the product, and are represented by the three solid straight lines with slopes, and 4, respectively. (a) - Mesh regie (q q p ) The behavior for q q p is deterined by the paraeters η and β. The paraeter η = ξ / ξ 0 deterines the level of (linear) filling ratio or overlapping between the blobs whose average distance is ξ 0. Thus when η = the blobs just touch each other, while for η < they are close but not in contact, and for η > they overlap. The paraeter β deterines the aplitude of spatial correlations aong blobs. When β =, the blobs are strongly correlated, S(q) decays to zero as S( q 0) ~ q ξ / 4πη (see straight line with slope = in Fig. S4(a)), and the other two ters are constant, i.e. A(q) = P seg (q,d) =. Thus R(q) can be approxiated to: S( q ) = K c M ( q ξ / 4πη ) [ q << ξ, β = (S8) F The close packing condition (β =, η ~ ) produces a long-range order in the structure of the gel, which anifests itself in two ways: it produces a peak at q p in the scattering distribution R(q) (Fig. S4(a)) and digs a shallow dip in the correlation function g 3 (r) (Fig. S4(c)). The peak position q p is related to the iniu r in and to the zero crossing r zxs positions of the correlation function by: q p r in ~.9 ± 0. (β = ) (S9a) q p r zxs ~.0 ± 0. (β = ) (S9b) 9

10 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 0 which are alost independent, except for β, fro any other paraeter appearing in Eq. 4. Eqs. S9 were tested nuerically upon varying ξ in the range -50 µ, in the range.0-.0, η in the range Thus, by cobining Eqs. S9 and Eq. of the ain text, it is possible to relate the peak position to the average fiber length, giving: q p < L > ~.5 ± 0.08 (β = ) (S0) fro which it is possible to estiate, fro the peak position, the average fiber length <L>. It should be noticed that a relation siilar to Eqs. S9 cannot be established between q p and the blob size ξ, because, as shown in the next paragraph, the product q p ξ is highly dependent on and η. (b) - Fractal regie (q p << q << q x ) For q p << q << q x, the behavior of R(q) is deterined by the arrangeent of the segents inside the blobs. If, within this range, the condition ξ << q << d is satisfied, we are in the fractal regie, where the ass fractal diension characterizes the power-law decay of R(q). Indeed, in this region the spatial correlation aong blobs are not iportant (S(q) = ), the segents behaves as point scatterers (P seg (q,d) = ), and in the blobs structure factor the ain contribution coes fro the ter A (q) ~ ( / qξ δ ). Thus R(q) can be approxiated to: R( q ) F K M ( / ) = c qξ δ [ ξ << q << d (S) or, equivalently: R( q) π 4 3 = K cf N A ρ ( δ ) d q [ ξ << q << d (S) in which we have used Eq. 5 for M. Eqs. S and S show that, in the fractal regie, the scattered intensity decay as a power law with an exponent equal to the fractal diension and with an aplitude which depends, besides c F, on the paraeters ρ, and d, but not on ξ and η. Thus, provided that R(q) is easured in absolute units, one can recover fro the aplitude of the scattering intensity the product length ratio of the gel fibers) or the diaeter d if ρ is also known. 3 ρ d (which, when =, corresponds to the ass per unit 0

11 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent The crossover between the esh and fractal regies takes place at peak position q p, which is related to ξ by the correspondence: q p = α ( ) [4π η δ ξ (S3) where α( ) is a nuerical correction which takes into account the difference between the actual peak position and the value obtained as the intercept between the asyptotic behaviors given by Eqs. S8 and S, and is represented by the two leftost solid straight lines of Fig. S4(a). The ter α( ) varies linearly between < <, fro α() =.3 to α() =.05. As an iportant reark related to Eq. S, it should be pointed out that the peak position q p depends, bedsides, on both ξ and η. Thus, the paraeters ξ and η are expected to be highly correlated (different values of ξ and η give rise to the sae q p ) and difficult to be recovered fro the data fitting. (c) - Segent regie (q d >>) For q d >> there is a crossover to the segent regie, in which R(q) depends only on the segent for factor P seg (q,d), independently fro the segents spatial arrangeent. Under these conditions, in Eq. S we have S(q) ~ and recalling that the segent olecular weight is A( q) ~ ( d / ξ ). Thus, aking use of Eq. 5 and 3 seg = N A ( π / 4) ρ d, we obtain: R q) = K c P ( q, d ) [ q d >> (S4) ( F seg seg which corresponds to the (incoherent) su of the intensities scattered by all the segents inside the gel. It is worth pointing out that, although P seg (q,d) describes the for factor of a segent with a onodisperse diaeter, it decays on average as ~ q 4, exactly as expected for systes with a very large diaeters polydispersity where a fully developed Porod regie (R(q) ~ q 4 ) is to be observed. This behavior can be seen by further approxiating the two factors appearing in Eqs. S7 as (4) P rod ( q, d) ~ π /( qd) and S 3 sec ( q, d) (8/ π)cos ( qd / 3π / 4) /( qd / ) =, iplying that: P seg 64 ( q, d) = cos ( qd / 3π / 4) [ q d >> (S5) 4 4 d q whose q 4 decay is highly odulated by the deep oscillations with periodicity q = π/d due to the cos (x) ter. These oscillations can be reduced by the presence of a diaeter polydispersity, but in order to wash the out, the level of polydispersity has to be so high (δd ~ d) that is clearly inconsistent with confocal (5, 6) or electron icroscopy data ().

12 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent Appendix fitting procedure The data fitting (both real and in silico) to the function given by Eq. S is rather difficult because such a function depends on six paraeters, naely ρ, β, η,, ξ, and d, and soe of the are highly correlated as well. All these paraeters, with the exception of ρ, affect the shape of R(q) in different regions of the q-range, i.e. β, η, ξ affects the esh regie, η,, ξ the fractal regie, and d the segent regie. The paraeter ρ, conversely, affects only the aplitude of R(q) because appears in Eq. S only as a constant factor inside the olecular weight M ter (see Eq. 5). In order to siplify the fitting procedure we first fixed β= for all the data fitting (consistently with the experiental data for which R(q=0) ~ 0 (8)) and then, as done in Ref. (), we exploited the close packing condition, which allows us to establish a relationship between the reaining fitting paraeters. Indeed, if the blobs are close packed (β=, η ~ ), the average concentration around each blob is equal to the saple concentration. Thus, since the average distance between the blobs is ξ 0 and the volue available for each blob is c ~ (M / N A ) / according to: ~ ξ 0, the saple concentration can be written as ξ. Therefore, by using Eq. 5 it can be easily worked out that ξ scales with c F 3 F ξ [ c / ρ η [ π/4 d (S6) where we have used η = ξ/ξ 0. Eq. S6 can be used as sort of constraint between paraeters when fitting the data to Eq. S, and in particular is quite useful for the recovery of ξ and η that are expected to be strongly correlated, as shown in Eq. S3. So the idea behind the fitting procedure is to first recover the paraeters ρ, and d that can be reliably recovered fro the fitting, and then use the two Eqs. S3 and S6 as a syste in which ρ, and d are known and ξ and η are the unknowns. Practically, the fitting procedure was developed as following: a) Fit the CLS+LAELS data of Fig 7(b) with β =, η = fixed and all the other paraeters (ρ,, d, ξ) floating. In this way ρ,, and d were reliably recovered, but not ξ, which depends on the value assigned to η. The ain result of this fitting was to show that the fiber density is alost constant over the different concentrations, with an average value ρ = 0. ± 0. g/c 3.

13 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 3 b) Fit the LAELS data with β =, η =, ρ = 0. g/c 3 fixed and, d, and ξ floating. The fixed value for ρ was necessary because the liited q-range of the LAELS data does not allow the siultaneous recovery of ρ and d. The ain result of this fitting was the retrieval of, and d for all the gels corresponding to the LAELS data (see Fig. 4 of Ref. ()). c) Insert in the syste of Eqs. S3 and S6 the values of ρ,, and d found at point b) as well as the peak position q p and concentration c F, and solve for η. In this way, the values of η are found aside for an arbitrary constant factor associated to the ~ appearing in Eq. S6. d) eterine the actual values of η by refitting the high concentration gels (c F 0.35 g/l or φ ) with η floating (β =, ρ = 0. g/c 3 fixed). For these saples, the q-peak is ore pronounced and the siultaneous recovery of ξ and η is actually reliable. e) Rescale the η values obtained at point c) by a constant factor so that, for the high concentration gels, they are ade equal to the ones found at point d). At this stage the dependence of η on φ is deterined. f) Find the dependence of ξ on φ for all the investigated gels by using Eq. S6 and the constant factor found at point e). SUPPORTING REFERENCES. Baradet, T. C., J. C. Haselgrove, and J. W. Weisel, 995. Three diensional reconstruction of fibrin clot networks fro stereoscopic interediate voltage electron icroscope iages and analysis of branching. Bioph. J. 68: Ferri, F., M. Greco, G. Arcóvito, M. e Spirito, and M. Rocco. 00. Structure of fibrin gels studied by elastic light scattering techniques: ependence of fractal diension, gel crossover length, fiber diaeter, and fiber density on onoer concentration. Phys. Rev. E. 66: Fisher M. W. and Burford R. J Theory of critical-point scattering and correlations. I. The Ising odel. Phys. Rev. A 56: Pedersen, J. S., and P. Schurtenberger Cross-section structure of cylindrical and polyerlike icelles fro sall-angle scattering data. I. Test of analysis ethods. J. Appl. Cryst. 9: Bloback, B, K. Carlsson, K. Fatah, B. Hessel and, R. Procyk Fibrin in huan plasa: gel architecture governed by rate and nature of fibrinogen activation. Throb. Res., 75:

14 Magatti et al. Modeling of fibrin gels - Supporting Material Suppleent 4 6. Collet, J. P., C. Lesty, G. Montalescot, and J. W. Weisel ynaic changes of fibrin architecture during fibrin foration and intrinsic fibrinolysis of fibrin-rich clots. J. Biol. Che. 78: Molteni, M.,. Magatti, B. Cardinali, M. Rocco, and F. Ferri. 0. Fast two-diensional "bubble" analysis of biopolyer filaentous networks pore size fro confocal icroscopy thin data stacks. Subitted to Bioph. J. 8. Ferri, F., M. Greco, G. Arcóvito, F. Andreasi-Bassi, M. e Spirito, E. Paganini and, M. Rocco. 00. Growth kinetics and structure of fibrin gels. Phys. Rev. E 63:340. 4