Towards understanding the effect of fibrinogen interactions on fibrin gel structure

Fibrin gelation involves the enzymatic conversion of the plasma protein fibrinogen to fibrin monomers which then polymerize to form the gel that is a major structural component of a blood clot. Because fibrinogen provides the material from which fibrin is made, it is generally regarded as promoting the gelation process. However, fibrinogen can bind to a site on a fibrin oligomer, preventing another fibrin oligomer from binding there, thus slowing the polymerization process. “Soluble fibrin oligomers,” which are mixtures of fibrin and fibrinogen, are found in the blood plasma and serve as biomarkers for various clotting disorders, so understanding the interplay between fibrin and fibrinogen during fibrin polymerization may have medical importance. We present a kinetic gelation model of fibrin polymerization which accounts for the dual and antagonistic roles of fibrinogen. It builds on our earlier model of fibrin polymerization that proposed a novel mechanism for branch formation, which is a necessary component of gelation. This previous model captured salient experimental observations regarding the determinants of the structure of the gel, but did not include fibrinogen binding. Here, we add to that model reactions between fibrinogen and fibrin, so oligomers are now mixtures of fibrin and fibrinogen, and characterizing their dynamics leads to equations of substantially greater complexity than previously. Using a moment generating function approach, we derive a closed system of moment equations and we track their dynamics until the finite time blow-up of specific second moments indicates that a gel has formed. In simulations begun with an initial mixture of fibrin and fibrinogen monomers, a sufficiently high relative concentration of fibrinogen prevents gelation; the critical concentration increases with the branch formation rate. In simulations begun with only fibrinogen monomers that are converted to fibrin at a specified rate, the rates of conversion, fibrinogen binding to oligomers, and branch formation together determine whether a gel forms, how long it takes to form, and the structural properties of the gel that results.

Figure 1

Schematic of monomers. Monomer $M$ and monomer $\widehat{M}$ with $S$ and $\widehat{S}$ half-domains, respectively.

Figure 2

Schematic of binding reactions in fibrinogen-fibrin branching model. The linking reaction in panel (a) and the binding reaction in panel (b) involve only $M$ monomers that represent fibrin and are identical to reactions in Ref. [6]. The reaction in panel (c) involves an $M$ monomer and an $\tilde{M}$ monomer representing fibrinogen and results in the occupation of an $S$ domain. The dangling $\widehat{S}$ domain cannot interact with any other species. (a) Link formation with $M$-type monomers. (b) Branch formation process with $M$-type monomers. (c) $\widehat{M}$-type monomer binding to $M$-type monomer, occupying an $S$ domain.

Figure 3

Schematic of oligomer, $C_{120}$. A trimer consisting of one $M$ and two $\widehat{M}$ monomers, representing a heterogeneous trimer with two free fibrinogen reaction sites and no free fibrin reaction sites.

Figure 4

(Case 1) Time evolution of nondimensional concentrations for varying $\kappa$ and $\varphi$ with fixed $\gamma =1$. (a, b) Initial composition parameter $\varphi =0.2$, and (a) $\kappa =1$, (b) $\kappa =10$. (c) $\kappa =10$ and $\varphi =0.7$. The inset in panel (c) shows that $v$ remains above 1 at large times.

Figure 5

(Case 1) Results for various branching rates $\kappa$ and initial composition parameter values $\varphi$, with $\gamma =1$. To the left of the black line, a gel forms. (a) Gel times, ${\tau }_{\text{gel}}$, and concentrations of (b) branch points $b$, (c) fibrinogen monomer ${\tilde{c}}_{020}$, and (d) fibrinogen in oligomer $\widehat{o}$ at ${\tau }_{\text{end}}$.

Figure 6

(Case 2) For various monomeric conversion rate ${\eta }_m$ and branching rate $\kappa$ values with fibrinogen binding rate $\gamma =1$, heatmaps of (a) ${\tau }_{\text{end}}$ and concentrations at ${\tau }_{\text{end}}$ of (b) branches $b$, (c) fibrinogen monomer ${\tilde{c}}_{020}$, and (d) fibrinogen in oligomer $\widehat{o}$. Black curves depict the gel/no-gel boundary.

Figure 7

(Cases 2 and 3) (Row 1) ${\tau }_{\text{end}}$, (Row 2) $\widehat{o}\left({\tau }_{\text{end}}\right)$, and (Row 3) $o\left({\tau }_{\text{end}}\right)$ as functions of branching rate $\kappa$ and conversion rate ${\eta }_m$. Columns from left to right: ${\eta }_o/{\eta }_m=0$ (Case 2), ${10}^{-5}, {10}^{-1}$, 1. Panel (a) shows the same data as Fig. 6 with a different colorbar range.

Figure 8

(Case 2 and Case 3) For various monomeric conversion rate ${\eta }_m$ and branching rate $\kappa$ values, concentrations at ${\tau }_{\text{end}}$ of (a, b) branches $b$ and (c, d) free fibrin reaction sites $r$. Left column shows results for Case 2 with $\gamma =1$ and ${\eta }_o=0$. Right column shows results for $\gamma =1$ and ${\eta }_o={\eta }_m$.

Figure 9

(Case 3) Concentration of branches formed through reactions involving exactly three fibrin monomers (solid), exactly two fibrin monomer (dash-dotted), exactly one fibrin monomer (dashed), and oligomers only (dotted) for $\gamma =1, \kappa =100, {\eta }_m=1$. The line colors denote ${\eta }_o=0$ (blue) and ${\eta }_o=1$ (purple).

Figure 10

(Case 3 and fibrin-only case) For various ${\eta }_m$ and $\kappa$ values, (a, b) ${\tau }_{\text{end}}={\tau }_{\text{gel}}$, and concentrations at ${\tau }_{\text{end}}={\tau }_{\text{gel}}$ of (c, d) fibrin in oligomer, (e, f) branches, and (g, h) free reaction sites. Left column shows results for Case 3 with $\gamma =1$ and ${\eta }_o={\eta }_m$. Right column shows results for $\gamma =0$, corresponding to Fibrin-only polymerization. Panel (a) shows the same data as Fig. 7, with a different colorbar range and panel (e) shows the same data as in Fig. 8.

Figure 11

(Case 2) Branch concentration, branch reaction rates, and cumulative production and consumption of free reaction sites for $\gamma =1, \kappa ={10}^3$ and varying conversion parameter ${\eta }_m$. Time courses of (a) branch concentration, (b) branch formation rate, and (c) concentrations of branches made in reactions involving (solid) two or more monomers or (dotted) two or more oligomers. Conversion rate ${\eta }_m={10}^{-2}$ (blue), ${10}^{-1}$ (red), ${10}^0$ (green), ${10}^1$ (orange), ${10}^2$ (purple). Time courses (d, e, f) of the integral of the individual terms in Eq. (26), which are related to fibrinogen conversion, fibrinogen binding, branch formation, and link formation.

Figure 12

(Gillespie simulations) Rate constants $\kappa =100,\gamma =1,{\eta }_m=0.1,{\eta }_o=0.1$ for Case 3. Number of monomers $N=4\times {10}^4$, Volume $v=4\times {10}^4$. (a) Largest oligomer vs time, (b) weight-average oligomer size vs time, (c) deterministic (black) and stochastic concentrations of fibrin in oligomer (blue), free reaction site (red), branch (yellow), and fibrinogen in oligomer (green) vs time, and (d) concentration of all clusters, $M_{000}$ (blue), clusters with at least one free reaction site (red), and at least two free reactions sites (yellow).

Figure 13

(Case 3) Number-average functionality vs time from deterministic (black) and stochastic [29, 30] simulations (color). The green curve shows the number-average functionality, $R/M_{000}$ in the fibrin-only case (green), where $\gamma =0,{\eta }_m=0.1,\kappa =100$. For the other curves, $\gamma =1,{\eta }_o={\eta }_m=0.1, \kappa =100$. Number-average functionality in stochastic simulations is shown for all oligomers $f_A$ (blue), for oligomers with at least one free reaction site $f_A^{\left(1\right)}$ (red), and for oligomers with at least two free reaction sites $f_A^{\left(2\right)}$ (yellow). Vertical lines denote gel times found by solving the ODE system Eqs. (23, 24, 25, 26, 27, 28, 29, 30, 31, 32) until $v\to 0$.

Figure 14

(Case 1) Fibrinogen binding rate $\gamma$ and initial composition parameter $\varphi$ are varied with branching rate $\kappa =10$. To the left of the black line, a gel forms. (a) ${\tau }_{\text{end}}$, and concentrations at ${\tau }_{\text{end}}$ of (b) branch points $b$, (c) fibrinogen monomer ${\tilde{c}}_{020}$, and (d) fibrinogen in oligomer $\widehat{o}$ at ${\tau }_{\text{gel}}$ or at ${\tau }_{\text{end}}={10}^{10}$ with no conversion.

Figure 15

(Case 1) Time course of $r\left(\tau \right)$ and ${\tilde{c}}_{020}$ for (a) $\varphi =1/4$ and (b) $\varphi =1/2$ with $\kappa =10$. Dotted and dashed black lines correspond to plotting the approximate value for $r$ and ${\tilde{c}}_{020}$ using Eq. (E2) for $\gamma ={10}^6$ and $\gamma ={10}^8$, respectively. Note the different $x$ axis scales.

Figure 16

End time concentration of (a) free reaction sites and (b) branch points for Case 2 (red) and Case 3 (blue) with $\gamma =1,{\eta }_m=1$.

Figure 17

(Case 2) Each concentration or gel time is a function of monomeric conversion rate ${\eta }_m$ and varying $\gamma$, fixed $\kappa =10$. (a) Gel time ${\tau }_{\text{gel}}$ and concentrations of (b) branches $b$, (c) fibrinogen monomer ${\tilde{c}}_{020}$, and (d) fibrinogen in oligomer $\widehat{o}$ at end of simulation. Black curves show the gel/no-gel boundary.

Figure 18

(Case 2) Heatmap of ${\tau }_{\text{end}}$ for various ${\eta }_m$ and $\kappa$ values with $\gamma =1$. The base case considered here is point “2” and has ${\eta }_m=1, \kappa =10$, and $\gamma =1$.

Figure 19

(Case 2) Variations in ${\eta }_m$. Time course of the integral of the individual terms in that contribute to $r\left(\tau \right)$ (top row) and time course of the concentrations of fibrinogen monomers (${\tilde{c}}_{020}$), fibrin monomers (${\tilde{c}}_{102}$), fibrinogen-fibrin dimers (${\tilde{c}}_{111}$), inert trimers (${\tilde{c}}_{120}$), free fibrin reaction sites ($r$), branch points ($b$), and gel indicator function $v$ (bottom row) for three ${\eta }_m=0.1,1.0,10.0$ values with $\kappa =10,\gamma =1$. The gray dotted lines refer to upper and lower bounds on $r$ in Eq. (36).

Figure 20

(Case 2) Variations in $\kappa$. Time course of the integral of the individual terms in that contribute to $r\left(\tau \right)$ (top row) and time course of the concentrations of fibrinogen monomers (${\tilde{c}}_{020}$), fibrin monomers (${\tilde{c}}_{102}$), fibrinogen-fibrin dimers (${\tilde{c}}_{111}$), inert trimers (${\tilde{c}}_{120}$), free fibrin reaction sites ($r$), branch points ($b$), and gel indicator function $v$ (bottom row) for three $\kappa =1,10,100$ values with ${\eta }_m=1,\gamma =1$. The gray dotted lines refer to upper and lower bounds on $r$ in Eq. (36).

Figure 21

(Case 2) Variations in $\gamma$. Time course of the integral of the individual terms in that contribute to $r\left(\tau \right)$ (top row) and time course of the concentrations of fibrinogen monomers (${\tilde{c}}_{020}$), fibrin monomers (${\tilde{c}}_{102}$), fibrinogen-fibrin dimers (${\tilde{c}}_{111}$), inert trimers (${\tilde{c}}_{120}$), free fibrin reaction sites ($r$), branch points ($b$), and gel indicator function $v$ (bottom row) for three $\gamma =0.1,1,10$ values with ${\eta }_m=1,\kappa =10$. The gray dotted lines refer to upper and lower bounds on $r$ in Eq. (36).

Figure 22

(Case 2) Nongelling simulations: (a) ${\eta }_m=0.1, \kappa =10, \gamma =1$, (b) ${\eta }_m=1, \kappa =1, \gamma =1$, (c) ${\eta }_m=1, \kappa =10, \gamma =10$. Timecourse of concentrations of reactive sites $r$, fibrinogen monomers ${\tilde{c}}_{020}$, and inert trimers ${\tilde{c}}_{120}$. Plots of reaction rate boundary curves: $r<\frac{{\eta }_m}{\gamma }$ if the rate of reaction site production by fibrinogen conversion exceeds the rate of reaction site use by fibrinogen binding, $r>\sqrt{4\gamma {\tilde{c}}_{020}/\kappa }$ if the rate that reaction sites are used for branch formation exceeds their use by fibrinogen binding, $r>\frac{2}{\kappa }$ if the rate of reactive site use for branch formation exceeds that by link formation, and $r>2\gamma {\tilde{c}}_{020}$ if the rate of reaction site use for link formation exceeds that by fibrinogen binding.