(Received for publication, November 20, 1996, and in revised form, February 21, 1997)
From the Department of Medical Biochemistry, Semmelweis University of Medicine, H-1088 Budapest, Hungary
A new model has been introduced to characterize
the action of a fluid phase enzyme on a solid phase substrate. This
approach is applied to evaluate the kinetics of fibrin dissolution with several proteases. The model predicts the rate constants for the formation and dissociation of the protease-fibrin complex, the apparent
order of the association reaction between the enzyme and the substrate,
as well as a global catalytic constant (kcat) for the dissolution process. These kinetic parameters show a strong dependence on the nature of the applied protease and on the structure of the polymerized substrate. The kinetic data for trypsin,
PMN-elastase, and three plasminogen-derived proteases with identical
catalytic domain, but with a varied N-terminal structure, are compared. The absence of kringle5 in
des-kringle1-5-plasmin (microplasmin) is related to a
markedly lower kcat (0.008 s1) compared with plasmin and
des-kringle1-4plasmin (miniplasmin) (0.039 s
1). The essentially identical kinetic parameters for
miniplasmin and plasmin with the exception of
kdiss, which is higher for miniplasmin (81.8 s
1 versus 57.6 s
1), suggest
that the first four kringle domains are needed to retain the enzyme in
the enzyme-fibrin complex. Trypsin, a protease of similar primary
specificity to plasmin, but with a different catalytic domain, shows
basically the same kcat as plasmin, but its
affinity to fibrin is markedly lower compared with plasmin and even
microplasmin. The latter suggests that in addition to the kringle
domains, the structure of the catalytic domain in plasmin also
contributes to its specificity for fibrin. The thinner and extensively
branched fibers of fibrin are more efficiently dissolved than the
fibers with greater diameter and lower number of branching points. When the polymer is stabilized through covalent cross-linking, the kcat for plasmin and miniplasmin is 2-4-fold
higher than on non-cross-linked fibrin, but the decrease in the
association rate constant for the formation of enzyme-substrate complex
explains the relative proteolytic resistance of the cross-linked
fibrin. Thus, the functional evaluation of the discrete steps of the
fibrinolytic process reveals new aspects of the interactions between
proteases and their polymer substrate.
The complex network of blood coagulation culminates in the conversion of fibrinogen to the rod-shaped fibrin monomers (for review, see Ref. 1). The non-covalent interactions among the fibrin monomers result in the formation of long double-stranded polymers in which the protomeric units are positioned in a half-staggered manner. Factors that influence the rate of the conversion of fibrinogen to fibrin and the strength of the polymerization interactions determine the final fiber diameter and clot structure. Increase in the NaCl concentration or in the pH reduces the number of protofibrils in a single fiber and the porosity of the fibrin network (2, 3); e.g. as the NaCl concentration is changed from 50 to 400 mM, the fiber diameter decreases from 180 to 39 nm, and the average distance between two branching points of the network falls from 1690 to 93 nm. Lowering the concentration of fibrinogen, the rate of its conversion to fibrin, and the presence of Ca2+ increases the fiber diameter and the porosity of the gel (4, 5). The structural features of the fibrin gel affect the physicochemical properties of fibrin (permeability and light scattering) (4, 5), but their influence on the susceptibility of fibrin to proteolytic enzymes has not been quantitatively characterized. The structure of fibrin is covalently modified by the activated factor XIII that inserts isopeptide bonds between fibrin monomers (for review, see Ref. 6). The functional aspects of this modification with respect to proteolytic susceptibility are documented. The cross-linking renders fibrin relatively resistant to degradation with plasmin (7-10) and leukocyte proteases (11, 12), but exact quantitative evaluation of this relative resistance has not been performed.
The serine protease plasmin is generally considered to be the enzyme responsible for the dissolution of fibrin clots under physiological and pathological conditions (for review, see Ref. 13). Morphological data for the presence of polymorphonuclear neutrophils in thrombi (14, 15), the immunochemical detection of specific fibrinogen degradation products in plasma samples in vivo (16), and the identification of PMN-elastase1 and cathepsin-G as the major fibrinolytic enzymes of polymorphonuclear leukocytes (17) suggested a concept for an alternative fibrinolytic pathway (18, and for review, see Ref. 19). According to this concept, in addition to plasmin and PMN-proteases, a fibrinolytic role is attributed to miniplasmin,2 which is generated after activation of the elastase-degraded form of plasminogen (Mr 38,000) lacking four of the five kringle domains that represent the N-terminal Glu1-Val441 sequence of plasminogen (20). The catalytic properties of some of these proteases have been evaluated in homogeneous systems for fibrinogen as a substrate (21, 22) or for fibrin with enzymes dispersed within the clot (23). According to these kinetic data plasmin, miniplasmin and PMN-elastase are the most efficient fibrinolytic enzymes, and a conclusion is drawn that the high affinity lysine binding site in the N-terminal kringle domains of plasmin is involved in the interactions with the native polymerized fibrin, whereas the fifth kringle found in both enzymes participates in binding to lysine residues newly exposed in the course of fibrin degradation (23).
Two possible access routes to the fibrin substrate exist for an enzyme (24), an intrinsic one (when the protease is entrapped within the fibrin network) and an extrinsic one (when the fluid phase-borne enzyme attacks the surface of the clot). Since even contracted blood clots do not hinder the free diffusion of particles with Mr up to 470,000 (25) in the former case, classical approaches can be used for the kinetic evaluation of the fibrinolytic proteases if the enzymes are uniformly dispersed within the clot (23). In the second case, the catalysis at a phase interface introduces major complications into enzyme kinetics. The extrinsic access route for fibrin, however, is of primary physiological importance since the blood-borne plasminogen activators and plasminogen assemble on the surface of the clot or propagate into it (for review, see Refs. 13 and 26), and the neutrophil leukocytes adhere to this surface (27, 28) where the generated or released proteases are protected against the plasma-derived protease inhibitors (29). Consequently, it is an intriguing task to characterize with exact kinetic data the proteolytic events at this interface. The complex character of the problem comes from the high heterogeneity of the system. The interface surface contains a relatively limited number of enzyme-susceptible cleavage sites. Thus, the fluid phase-borne protease binds to nonspecific sites of the substrate surface (absorption) or, less frequently, directly attacks the cleavage-sites. After the absorption, the enzyme can approach the susceptible site through lateral movement along the fiber (first change in the dimensionality of the reaction), or, alternatively, it can move into the fiber (a second change in the dimensionality), desorb into the fluid phase, or move into the inner pores of the gel. Consequently, the kinetic evaluation of the overall rate of fibrin dissolution should consider the relative rates of these movements, the changes in the dimensionality, and the rate of the hydrolytic reactions. In similar systems where degradation of different biopolymers or phospholipids occurs on phase interface, special mathematical approaches have been introduced for kinetic characterization of the hydrolytic events (30-36). Recently, elaborate models have been developed for the evaluation of fibrin dissolution, which take into account the details of the transport phenomena (37-39). According to these data, the diffusion of fibrin-binding proteins into the clot is severely impaired, which results in accumulation of the extrinsically applied enzyme in a thin superficial layer of the gel. Morphological data confirm that the lytic process is restricted to a several micrometer deep zone at the surface of the clot (40, 41). In the present study, we introduce a new model for the kinetics of fibrin surface degradation with proteases and apply it for the comparison of several fibrinolytic enzymes and for the evaluation of the impact of fibrin and enzyme structure on the lytic process. Further insights into the structure-function relationship of the fibrinolytic proteases can be gained from the kinetic behavior of trypsin (a protease of similar primary specificity to plasmin) and of a lower molecular weight derivative of plasmin, microplasmin (Mr 29,000) lacking all kringle domains (42, 43).
Human plasma was collected from healthy volunteers.
Streptokinase, aprotinin, porcine pancreatic elastase, Factor XIIIa
(from human placenta), and the chromogenic elastase substrate
(methoxysuccinyl-L-alanyl-L-alanyl-L-prolyl-L-valine-p-nitroanilide) were from Calbiochem (La Jolla, CA). The chromogenic plasmin substrate Spectrozyme-PL
(H-D-norleucyl-hexahydrotyrosyl-lysine-p-nitroanilide) was a product of American Diagnostica (Hartford, CT). Human fibrinogen was purchased from Chromogenix AB (Mölndal, Sweden).
Lysine-Sepharose 4B, Sephadex G-25, and Sephacryl S-400 HR were from
Pharmacia Biotech Inc. (Uppsala, Sweden). PMN-elastase, human thrombin
(1000 NIH units/mg), p-nitrophenyl
p-guanidinobenzoate and phenylmethylsulfonyl fluoride were
the products of Sigma. Bovine serum albumin and bovine pancreas trypsin
were from Serva (Heidelberg, Germany), lactoperoxidase and
4-(2-aminoethyl)-benzenesulfonyl fluoride (Pefabloc®) were
from Boehringer Mannheim (Germany). Na125I (carrier-free)
was purchased from Izinta Ltd. (Budapest, Hungary). All other reagents
were purchased from Reanal (Budapest, Hungary).
Published procedures were used for their isolation from citrated human plasma and further purification (20, 42, 44).
Plasmin and Miniplasmin Generation, Determination of Active Enzyme ConcentrationThese were performed as described previously for plasmin, miniplasmin, and PMN-elastase (29). The concentration of microplasmin and trypsin was determined with active-site titration (45).
Fibrinogen Purification and Fibrin FormationPlasminogen-free fibrinogen with no contaminant factor
XIII activity was prepared from freeze-dried human fibrinogen as
described previously (23). In a microplate well, 200 µl of the
purified fibrinogen (2 g/l) in 10 mM imidazole buffer, pH
7.4, containing 3 mM Ca2+ and varying amounts
of NaCl was clotted with 1 NIH unit/ml thrombin to gain
non-cross-linked fibrin. It has recently been shown (46) that the
Cl concentration determines the structure of the
polymerizing fibrin and not the ionic strength of the solution as it
was previously believed (2). For the preparation of cross-linked
fibrin, the inactivation of factor XIII was omitted from the
purification procedure of fibrinogen, and, after clotting as above,
fibrin was completely cross-linked in 16 h by the contaminant
factor XIII as evidenced by gel electrophoresis.
A fibrinolytic enzyme in a homogeneous solution is
applied to the surface of the preformed fibrin gel in a microplate
well, and the turbidity of the clot that reflects the mass/length ratio of the fibrin fibers (23) is followed by measuring
A340 with a Dynatech MR 5000 microplate reader.
In the course of clot dissolution, the turbidity decreases due to
reduction in the mass/length ratio of the fibrin fibers, resulting in
solubilization in the interface layer and consumption of the gel phase.
When a fluid phase-borne protease reaches the surface of the
fibrin-gel, the enzyme molecules diffuse through the interface of the
two phases and bind to the surface of the fibrin fibers either at
proteolysis-susceptible sites or at nonspecific sites. In the former
case, the protease molecules catalyze hydrolysis of peptide bonds in
the fibrin, whereas, in the latter case, they migrate along the fibers
to the cleavable bond or dissociate without hydrolytic effect
(ineffective binding) and further diffuse among the fibrin fibers or
into the fibers (internal diffusion). According to our measurements
(see "Results") and to the experimental data of others (37-41), no
significant diffusion is expected for molecules that specifically bind
to fibrin. Consequently, it is reasonable to assume the existence of a
reactive boundary layer to which the dissolution process is restricted
(see Fig. 1). The depth of this layer does not depend on the
concentration of the applied enzyme, and the enzyme concentration in it
is approximately 10-fold higher than in the bulk phase (40, 41). For
the sake of generality (to eliminate the dependence of the parameters
on the actual size of the examined surface), we restrict the evaluation
of the dissolution reaction to a subvolume Vx of the
reactive boundary layer with surface area S = 1 []
(see Fig. 1). The basic process in this layer can be described,
![]() |
(Eq. 1) |
![]() |
(Eq. 2) |
![]() |
The exponent of the enzyme concentration is introduced to
express the changes in the dimensionality of the distribution of the
kinetic energy of the enzyme molecules. 1) From the three-dimensional free movement in the bulk fluid phase, they bind to the surface of the
fibrin fibers, suggesting two-dimensional movement (34, 35); and 2) on
the other hand, internal diffusion or dissociation from the surface of
the fiber increases the dimensionality of the reaction. These changes
finally yield a non-integer value for the exponent, as evidenced by our
measurements (see "Results"). The equation E() = ((E0
EF(
))/(h1(
)))·r,
where E0 [nmol/m2] is the initial
number of enzyme molecules per unit surface (calculated using the
10-fold concentration factor reported in Ref. 40) and
h1(
) [m] is the height of the fluid phase
at time
, reflects the fact that the number of enzyme molecules is
distributed between a free enzyme fraction and a fibrin-bound fraction.
Applying the equation for area S = 1 and introducing
K1
=k1·Fl
[nmol1
p·m2
3p/s], the following equation
is derived.
![]() |
(Eq. 3) |
The rate of EF dissociation to E and F (dissociation 1) is as follows.
![]() |
(Eq. 4) |
![]() |
The rate of EF degradation to E and P (dissociation 2) is as follows.
![]() |
(Eq. 5) |
![]() |
In this case, the exponent of the EF was introduced
in analogy to p as a mathematical experiment despite our
expectation to gain a value of 1 (if the model assigns any value
different from 1 to q, this will question the validity and
the physical meaning of p in Equation 3). Considering that
Vx is constant, K2 is
defined as follows.
![]() |
(Eq. 6) |
![]() |
(Eq. 7) |
Thus, the overall rate of change of EF is as follows.
![]() |
(Eq. 8) |
Following an initial phase when the homogeneity of the reactive
layer is established, the number of intact fibrin molecules per unit
surface at time F(
) is proportional to the height of
the solid phase h2(
), thus there is a
geometrical dependence,
![]() |
(Eq. 9) |
Combining Equations 8 and 9 results in the following.
![]() |
(Eq. 10) |
The overall rate of substrate consumption is as follows.
![]() |
(Eq. 11) |
![]() |
In the experiments, we measured the time needed for the turbidity of the clot to decrease to a certain value, at which we have independent data for the amount of undegraded fibrin (the residual non-cross-linked fibrin was determined with gel filtration as in Kolev et al. (23); the residual cross-linked fibrin was calculated from the data for protein concentration in the fluid phase determined with the method of Lowry et al. (47) and turbidimetric estimation of the volume of the fluid phase). Thus, the time represents the dependent variable that contains the experimental error. Consequently, it is justified to consider the momentary number of substrate molecules as an independent variable designated F [nmol/m2].
![]() |
(Eq. 12) |
![]() |
(Eq. 13) |
Using Equations 10 and 11, Equation 12 is modified as follows:
![]() |
(Eq. 14) |
Summarizing the basic equations of the model follows.
![]() |
(Eq. 15) |
From Equation 11, the exchange of the independent variable results in the following.
![]() |
(Eq. 16) |
At time = 0, the following initial conditions are valid.
![]() |
(Eq. 17) |
![]() |
(Eq. 18) |
For the system described so far, we formulate two mathematical
problems: 1) to predict the time required for
F0 to decrease to a pre-determined value
Fkp in a single experiment, if the parameters of the
model (p, q, K1
,
k
1, K2
) are given; and
2) to determine the parameters of the model yielding minimal
discrepancy between the values
kp predicted as in problem 1 and the values measured in a set of experiments for a group of enzymes
with identical p and q exponents as introduced in
Equations 3 and 5). Thus, we can confirm or rule out the necessity for
such exponents. These problems are solved as described under "Appendix" with an original software written under Matlab®
4.0 with SimulinkTM 1.2c (The MathWorks Inc.,
Natick, MA).
Fibrinogen, plasmin, miniplasmin, and PMN-elastase were labeled by lactoperoxidase-catalyzed iodination, and 125I-fibrin-coated tubes were prepared as described previously (29). The degradation of the fibrin monomer surface in the 125I-fibrin-coated tubes was followed by measuring the radioactivity released into the fluid phase and by autoradiography after gel-electrophoresis of fluid phase samples. Penetration of 125I-labeled and active site-blocked proteases into fibrin clots pre-formed in microplate wells was followed by measuring the residual radioactivity in the clot after removal of the fluid phase and rapid (5 s) washing of the fibrin surface.
For determination of each set of kinetic parameters, at
least 8 different concentrations of the protease in the range 0.02-2.4 µM are applied to the surface of preformed fibrin clots
(cross-linked and non-cross-linked). All measurements are performed at
least in triplicate, and the mean value and its standard error are used for the calculations as described in the mathematical procedures (see
"Appendix"). The experimental approach is illustrated in
Fig. 2 for the dissolution of cross-linked fibrin with
plasmin. Identical experiments are carried out with all fibrinolytic
enzymes on both cross-linked and non-cross-linked fibrin. Essentially
identical parameter values are gained independently of the criterion
used in the optimization procedure (absolute, relative, or general). For the sake of uniformity, only data from procedures with the relative
criterion are reported in Tables I and II
(in this case the relative square error of the function can be used to
evaluate the reliability of the parameter values). The predictive value of the model can be estimated on the basis of the comparison of the
empirical data with the corresponding calculated values. An example for
the good agreement of theoretical and empirical data is provided in
Fig. 3 for the dissolution of non-cross-linked fibrin
with miniplasmin. In addition, the model predicts the absolute surface
concentration of the enzyme-fibrin complexes. Its value does not exceed
10% of the respective initial enzyme concentration. Even when the
optimization procedure does not use the steady-state assumption (first
and second method under "Appendix"), this value does not vary by
more than 5% in the course of the studied reactions following the
initial negligibly short pre-steady-state period. The identity of the
results gained with the three computation methods supports the validity
of the steady-state assumption for the examined system and justifies
its usage in the computation procedure (computation time is at least
100-fold shorter when working with this assumption). If the
optimization procedure is allowed to freely vary, the values of the
exponents p and q in Equations 3 and 5, it
consistently assigns a value of 1 to q and a non-integer
value to p. Thus, the model does not tolerate arbitrarily introduced parameters (there is no reason to suggest a value different from 1 for the order of the EFE + P
reaction, whereas sound arguments can be brought in favor of the
non-integer value of p). For q = 1 K2
is equal to k2
(Equation 6). Since we interpret the values of p as a result
of changes in the dimensionality of the E + Fl
EF reaction, in separate
experiments, we evaluate indirectly the relative contribution of the
attacks from the bulk phase (three-dimensional reaction) and from
lateral migration (two-dimensional reaction) to the rate of
EF complex formation. In the initial period after the
application of the enzyme solution over the fibrin surface, obviously,
the three-dimensional events dominate. In accordance with this, if the
enzyme solution is replaced with buffer within 2 min after the
application and subsequently the clot is washed 3 times with an
enzyme-free buffer (each volume of the washing buffer is left for 2 min
over the fibrin), the dissolution process is completely stopped. If the
fibrin surface is washed 3 or 5 times in the same way with a
protease-free buffer later than 2 min following the application of the
enzyme, the rate of A340-change decreases
3-fold, but no complete inhibition of fibrinolysis can be achieved.
This residual fibrinolytic activity can be attributed to the migration
of the enzyme molecules along the fibrin fibers. The data of this
functional assay are confirmed by the rate of penetration of
125I-labeled and active-site blocked enzymes into fibrin
clots. When the incorporation of radioactivity from solutions of
125I-plasmin-PMSF, 125I-miniplasmin-PMSF or
125I-PMN-elastase-PMSF into fibrin is measured, in the
first minute after the application of the enzyme over the fibrin
surface, a burst in the radioactivity of the clot is observed, whereas
in the next 5 h, a linear, but 100-fold slower, increase in the
clot-bound radioactivity can be detected (not shown). The dissociation
of the clot-bound proteases is relatively slow compared with the binding. If, following 1-min binding, the clot is rapidly (10 s) washed
three times with enzyme-free buffer and the last volume of washing
buffer is left over the clot, 70% of the enzyme-related radioactivity
is retained in the clot 1 h later. If plasmin is applied to a
fibrin monomer surface, 6AH (which interferes with the interaction
between the kringle domains of plasmin and lysine residues in fibrin)
inhibits its fibrinolytic activity at concentrations in the micromolar
range when it has been pre-incubated with the enzyme, whereas an order
of magnitude higher concentration of 6AH is needed for a similar effect
when the 6AH is added following the enzyme binding to the fibrin
(Fig. 4, A and C). This, together with the time-course of the inhibition in the second case (30 min are
needed to reach the full inhibiting effect, Fig. 4C,
inset) suggests high processivity for the plasmin molecule.
Once bound to the substrate, the enzyme dissociates relatively rarely
into the bulk phase where 6AH blocks its lysine-binding sites. Blocking of the single lysine-binding site of miniplasmin with 6AH (Fig. 4B) has a less expressed effect on the fibrinolytic activity
of this protease compared with plasmin, and 60 min are needed for the
maximal inhibition when 6AH is added after the miniplasmin binding to
fibrin (Fig. 4D, inset). The latter fact
emphasizes the importance of the fifth kringle in the processive
properties of the enzyme.
|
|
Effect of Fibrin Network Structure on the Catalytic Properties of Plasmin
Fibrin gel with varying fiber diameter and porosity was
prepared by clotting fibrinogen in solutions containing different concentrations of Cl, which determines the polymerization
pattern (46). For each type of fibrin gel, plasmin (at eight different
concentrations in the range 0.05-1.2 µM) is applied to
the surface of the clot in a solution yielding a final concentration of
150 mM NaCl in the total volume of the clot and fluid
phase. Using the approach described under "Experimental
Procedures," the kinetic parameters of Equations 3-5 and 7. can be
determined for the dissolution of the different fibrin clots by
plasmin. In parallel with the increase of the Cl
concentration at polymerization, the apparent k2
increases, whereas the exponent of the enzyme concentration
p decreases (Fig. 5). The parameters for the
formation (K1
=
k1·Fl) and dissociation (k
1) of the enzyme-fibrin complex do not
change significantly (K1
= 105.2 ± 8.8 m3p-2·nmol1-p/s, k
1 = 54.1 ± 4.1 s
1). Thus, the catalytic efficiency
ratio
(k2·K1
/(k2 + k-1)) follows the tendency of
k2 (Fig. 5.).
Quantitative Comparison of Fibrin Surface Degradation with Different Proteases
Each of the studied proteases is applied to
the surface of a pre-formed fibrin clot at eight different
concentrations and from the turbidimetric data the kinetic parameters
(p, k2, K1, k-1) introduced in Equations 3-5 and 7 are
calculated for each separate enzyme regarded as a "group" in the
sense of problem 2. The results for non-cross-linked fibrin are
summarized in Table I and for cross-linked fibrin in Table II. For
estimation of the role of polymerization in the dissolution process,
the degradation of 125I-labeled fibrin monomer surface is
also followed (Fig. 6). The inset shows that
digestion of the labeled fibrin yields products that are essentially
identical with those of native, unlabeled polymerized fibrin (23),
suggesting that the labeling technique does not modify the region of
the fibrin molecule degraded by the proteases.
With the introduced approach, several discrete steps in the
process of fibrin dissolution with extrinsic proteases can be characterized. The detachment of water-soluble degradation products from the fibrin fibers that is detected as dissolution of the clot or
change in its turbidity is rate-limited by the proteolytic reaction
that is characterized by the apparent k2
constant. This constant is apparent because more than one peptide bond
is cleaved for the release of a single soluble product (thus this value
is a global integrated constant), and its value depends on the actual accumulation of enzyme in the reactive boundary layer (in all calculations, we assume an average 10-fold accumulation compared with
the fluid phase as reported in Sakharov and Rijken (40)). We do not
correct the value of k2 for the number of
cleaved bonds because, in this way, the constant is more suitable for
comparison of different proteases in their efficiency in the global
process of fibrin dissolution. The soundness of the approach is
supported by the data gained for clot-embedded plasmin. We earlier
found a k2 of 6 × 102
s
1 for 2.5 g/l fibrin at 150 mM NaCl and 3 mM CaCl2 (23), whereas our present result shown
in Fig. 5 is 8.6 × 10
2 s
1. Our
findings are in good agreement with the single available report in the
literature (39). Under conditions similar to ours (for 3 g/l fibrin
prepared at 100 mM NaCl and 5 mM
CaCl2) a k2 of 5 s
1 is
found for plasmin if a solubilization factor (number of lyzed bonds per
solubilized monomer) of 10 is used (with the same assumption and
ignoring the accumulation of enzyme in the boundary layer, our result
from Fig. 5 should be 5.61 s
1). Our data for the
parameters of enzyme-fibrin complex formation (k1·Fl and
k-1) are remarkably constant, independent of the
changes in the structure of the fibrin substrate as the
Cl
concentration at polymerization is varied. This
justifies the application of a constant accumulation ratio for the
enzyme in the boundary reactive layer, which was mentioned as a second
factor for the ostensibility of k2. Thus, the
increase of k2, observed as the substrate
polymer fibers become thinner and their branching points more frequent
at higher Cl
concentration (Fig. 5), can be interpreted
as a consequence either of a more suitable exposure of the
proteolysis-susceptible bonds or of a reduction in the number of bonds
that should be broken for the solubilization of a single monomer.
Another parameter that changes markedly with the alteration of the
substrate structure is the exponent p of the enzyme
concentration in the kinetic equation (Equation 14). Higher values of
this parameter suggest preservation of the kinetic energy of the enzyme
molecule for effective attacks, whereas low values reflect "energy
loss" for dissociation from the fiber surface and diffusion among the
fibers. If this interpretation is applied to the values of p
shown in Fig. 5., a conclusion can be drawn that the coarse fibrin
polymerized at low Cl
concentration channels the enzyme
within the thick fibers and reduces the dimensionality of its movement,
whereas in the thin fibrin formed at higher Cl
concentration, detachment from the fiber and migration among the fibers
occurs more frequently, resulting in increase in the dimensionality of
the enzyme movement and reduction of p. The 30 and 60 min
delays in the maximal effect of 6AH on the fibrinolytic activity of
plasmin and miniplasmin (Fig. 4.) also support the idea of enzyme
channeling along the fibrin surface. Due to the non-integer value of
p, the catalytic efficiency of plasmin on the different
fibrin substrates cannot be evaluated simply on the basis of the ratio
k2·K
1/(k2 + k
1), but the effect of p should
also be considered. From the simulated curves in Fig. 5,
inset, a definite conclusion can be drawn that the thinner and extensively branched fibers of fibrin formed at higher
Cl
concentrations are more efficiently dissolved by
plasmin at all enzyme concentrations. Similar changes of fibrin
structure can be induced by variations of fibrinogen and thrombin
concentrations (4, 5), which in vivo can yield fibrin of
different susceptibility to plasmin.
When the kinetic data for the three proteases with identical catalytic
domain (plasmin, miniplasmin, and microplasmin) are compared, important
conclusions about the structure-function relationship in the molecule
of plasmin can be drawn. Similarly to fibrinogen as a substrate (21,
22), the kinetic parameters of plasmin and miniplasmin on
non-cross-linked fibrin are essentially identical. The only exception
is k-1, which is markedly higher for miniplasmin (Table I; our model discriminates the association and dissociation constants for the formation of enzyme-substrate complex based on the
different order of the two reactions). This suggests that the first
four kringle domains are needed to retain the enzyme in the
enzyme-fibrin complex. The fifth kringle seems to contribute to the
maintenance of the appropriate conformation of the catalytic domain and
to the affinity of the enzyme to the proteolysis susceptible sites; the
absence of kringle5 in microplasmin is related to a markedly lower k2 and association constant
compared with plasmin and miniplasmin. Trypsin, a protease of similar
primary specificity to plasmin (it cleaves peptide bonds at basic amino
acids), shows the same k2 as plasmin, but its
affinity to fibrin estimated on the basis of the ratio of the
association and dissociation constants for the enzyme-substrate complex
is markedly lower compared with plasmin and even microplasmin. The
latter suggests that, in addition to the kringle domains, the structure
of the catalytic domain in plasmin also contributes to its specificity
for fibrin. PMN-elastase, another protease of potential relevance to
fibrinolysis in vivo, demonstrates similar affinity to
fibrin as plasmin, but its catalytic constant is almost an order of
magnitude lower. Due to the different values of the exponent
p for the enzyme concentration in Equation 14, comparison of
the catalytic efficiency of the separate proteases is possible only on
the basis of the simulated curves in Fig. 7A,
which indicates that among the physiologically relevant enzymes the
most efficient in the dissolution of non-cross-linked fibrin is plasmin
followed by miniplasmin and PMN-elastase. Since on fibrin monomers the
proteolytic activity of PMN-elastase is similar to that of plasmin and
better than that of miniplasmin (Fig. 6.), the estimated low efficiency
of PMN-elastase in polymerized fibrin dissolution can be explained by
the relatively higher number of bonds that should be broken for the
solubilization of a single monomer or alternatively by masking of the
elastase-susceptible peptide bonds in the polymer. The latter
possibility, however, is less probable if the identical degradation
products from fibrin monomers and clots are considered (Fig. 6,
inset and Ref. 23). The relative order of efficiency for
dissolution of non-cross-linked fibrin clots with extrinsic proteases
is the same as that established for the clot-embedded enzymes (23).
An important aspect of fibrinolysis in vivo is the influence
of cross-linking by factor XIIIa on the dissolution properties of
fibrin. Our results reveal new details of the well documented resistance of the cross-linked fibrin to proteolysis (7-12).
Surprisingly, the k2 on cross-linked fibrin for
plasmin and miniplasmin is 2-4-fold higher than on non-cross-linked
fibrin (this parameter is only slightly modified for the other three
studied enzymes) (Table II). The general tendency for significant
reduction of the association rate constant K1,
however, explains the relative proteolytic resistance of the
cross-linked fibrin. Cross-linking probably renders the fibrin monomers
a conformation to which the proteases show low affinity, but the rate
of the separate proteolytic steps is either not influenced
(microplasmin, PMN-elastase, trypsin) or even enhanced (plasmin,
miniplasmin). Using the ratio of the kinetic parameters and the value
of the exponent for the enzyme concentration p (Table II),
the proteolytic activity of the separate proteases on cross-linked
fibrin can be compared (Fig. 7B). The relative order of
catalytic efficiency of the physiologically relevant fibrinolytic
proteases is different on cross-linked fibrin surface; the most
effective is miniplasmin, followed by plasmin and PMN-elastase. Thus,
PMN-elastase, despite its relatively low direct fibrinolytic activity,
can contribute to the dissolution of fibrin through the generation of
miniplasmin, the most efficient protease on stabilized fibrin.
We are grateful to Ida Horváth, Györgyi Oravecz, Antoaneta Krasteva, and Ágnes Himer for technical assistance.
The numerical solutions of the basic mathematical problems are as follows.
Problem No. 1
The parameters of each experiment are cons;
h0; h10;
E0; F0;
K1; K2
;
k
1; p; q; and
Fkp. Fkp [nmol/m2]
is the surface concentration of intact fibrin monomers at the examined
levels of clot turbidity determined by gel-filtration chromatography
for non-cross-linked fibrin or with protein determination for
cross-linked fibrin. The cons is a flag indicating the
availability of previously calculated parameters from the same
experiment for Fkp* > Fkp (cons = 1) or their absence
(cons = 0). The asterisk always designates calculated
parameters from a previous level, if cons = 1. The value of
kp should be calculated. Three different methods are applied
for the solution of the problem.
Equations 15 and 16 are solved as a system of two differential equations with initial conditions Equations 17 and 18 for the interval F = [F0;Fkp]. Equations 15 and 16 are integrated from F0 to Fkp at cons = 0 and from Fkp* to Fkp at cons = 1. At cons = 0, the real initial conditions (Equations 17 and 18) are singular and cannot be used as a start point for the solution. The modified boundary conditions,
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(Eq. 19) |
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(Eq. 20) |
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In this
procedure, we regard all the measurements with the same enzyme at a
given E0 from F0 to the
different Fkps as separate experiments belonging to
the same family. In each family, the experiment with the highest
Fkp has cons = 0, whereas all others
have cons = 1. In the system of differential Equations
15 and 16, Equation 15 can be solved separately because the right side
of the equation does not depend on . Solving Equation 15 in the
range
[F0;(Fkp)min]
(using the modified boundary condition for EF as in the
first method), a table for EF-F can be derived that
contains n + 1 pairs of values for EFi
Fi (i = 0, 1, 2, ...,
n). In each of the n intervals [Fi;Fi+1], the
function EF(
) can be replaced with the linear
function,
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(Eq. 21) |
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(Eq. 22) |
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(Eq. 23) |
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(Eq. 24) |
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(Eq. 25) |
With this method, we assume that the rate of formation of EF is equal to the rate of its degradation at each moment of time (this is not equivalent to the assumption that the amount of EF does not change in the course of the reaction). The range [F0;(Fkp)min] is divided by n + 1 points, F0,F1,F2,F3,...Fn = (Fkp)min.
For each Fi value, a value for
EFi is calculated from
dEF/dt = 0 using Equation 10. Thus, an
EFi Fi table is composed that
contains n + 1 pairs EFi
Fi. The last step is integration according to
Equation 25 as under "Second Method."
Problem No. 2
A group of "e" enzymes with common p and
q exponents (Equations 3 and 5) is considered
(i = 1, 2 ... e), where enzyme i is
characterized by parameters p, q,
K1i,
K
2i,
k
1i. For each
enzyme ni, different experiments are available (j = 1, 2 ... ni). The
experiment j with the enzyme i is repeated
mij > 1 times (3
mij
6), and the following experimental times are
measured
ij1,
ij2, ...
ijk, ...
ijmij.Let
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(Eq. 26) |
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The parameters
p,q,2,
1,
1
should be determined so that the calculated
optimally approaches
the experimental one. The mean
ijex and its standard
error
ij are determined as follows.
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(Eq. 27) |
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(Eq. 28) |
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(Eq. 29) |
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(Eq. 30) |
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(Eq. 31) |
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(Eq. 32) |
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(Eq. 33) |
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(Eq. 34) |
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(Eq. 35) |
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(Eq. 36) |
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(Eq. 37) |
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(Eq. 38) |
The optimization procedure uses Equations 29, 31, and 33, but the results are presented according to Equations 30, 32, and 34, which show the errors with their dimensions.