Toward an understanding of fibrin branching structure

The blood clotting enzyme thrombin converts fibrinogen molecules into fibrin monomers which polymerize to form a fibrous three-dimensional gel. The concentration of thrombin affects the architecture of the resulting gel, in particular, a higher concentration of thrombin produces a gel with more branch points per unit volume and with shorter fiber segments between branch points. We propose a mechanism by which fibrin branching can occur and show that this mechanism can lead to dependence of the gel’s structure (at the time of gelation) on the rate at which monomer is supplied. A higher rate of monomer supply leads to a gel with a higher branch concentration and with shorter fiber segments between branch points. The origin of this dependence is explained.

Figure 1

A monomer $M$ consists of two reacting ends, denoted $S$. A linear chain consists of any number of links (L) and has two ends $S$.

Figure 2

Half-formed links $Z$ can “zipper” to form a link.

Figure 3

Half-formed links $Z$ can bind with an end $S$ to form a branch $B$.

Figure 4

Behavior in time of the nondimensional variables $u$, $r$, $\beta$, $c$, and $o$ for a constant supply rate $\sigma =0.25$ and branching rate $\kappa =10$.

Figure 5

Nondimensional gel time ${\tau }_g$ as a function of nondimensional supply rate $\sigma$. $\kappa =10$.

Figure 6

(a) Nondimensional concentrations of (A) free monomer, (B) branches, (C) monomer in oligomers, and (D) total monomer at the gel time ${\tau }_g$ as a function of the supply rate $\sigma$. (b) Percent composition of fibrin clot at gel time as a function of $\sigma$: (A) oligomer, (B) free monomer, and (C) branches. $\kappa =10$.

Figure 7

The structure parameter $o/\left(2\beta \right)$ gives an estimate of the mean number of monomers between branch points. It is plotted as a function of nondimensional supply rate $\sigma$. $\kappa =10$.

Figure 8

Variation in gel time ${\tau }_g$ and estimated mean segment length $p_s=o/\left(2\beta \right)$ with branching rate $\kappa$ and monomer supply rate $\sigma$. (a) Gel time as a function of $\sigma$ on a log-log scale, showing power-law behavior. (b) Structure parameter $p_s$ as a function of supply rate. From top to bottom, curves are for branching rates $\kappa =0.25$, 0.50, 1.0, 2.0, 4.0, 8.0, 16.0, 32.0, 64.0, and 128.0.

Figure 9

Formation of branches by different reactions as a function of nondimensional time for monomer supply rate $\sigma =0.25$ and branching rate $\kappa =10$. (a) Curves show percent of branches formed in reactions involving (A) one, (B) two, or (C) three free monomers, the total percent of branches formed in reactions involving at least one free monomer (D), and the percent of branches formed by reactions involving only oligomers (E). (b) Rate of branch formation by reactions that involve (A) one, (B) two, or (C) three monomers, or that involve oligomers only (D).

Figure 10

As in Fig. 9 with $\sigma =0.25$ and $\kappa =1.0$.

Figure 11

As in Fig. 9 with $\sigma =0.25$ and $\kappa =0.1$.

Figure 12

Behavior in time of the nondimensional variables, $\beta$, $c$, and $o$ for a (a) supply rate $\sigma =0.125$ and branching rate $\kappa =8$, and (b) supply rate $\sigma =0.5$ and branching rate $\kappa =8$.

Figure 13

A branch involving four bonds.