Self-assembly of soluble unlinked and cross-linked fibrin oligomers

Self-assembly of soluble unlinked and cross-linked fibrin oligomers formed from desA-fibrin monomer under the influence of factor XIIIa was studied in the presence of non-denaturing urea concentrations. By methods of elastic and dynamic light scattering combined with analytical ultracentrifugation, desA-fibrin oligomers formed in both the presence and absence of the factor XIIIa were shown to be ensembles consisting of soluble rod-like double-stranded protofibrils with diverse weight and size. Unlinked and cross-linked soluble double-stranded protofibrils can reach the length of 350–450 nm. The structure of soluble covalently-linked protofibrils is stabilized by isopeptide γ-dimers. Electrophoretic data indicate a complete absence of isopeptide bonds between α-chains of desA-fibrin molecules. The molecular mechanism of formation of soluble rod-like fibrin structures and specific features of its covalent stabilization under the influence of factor XIIIa are discussed.     

The strengthening role of <Emphasis Type="Italic">D</Emphasis>:<Emphasis Type="Italic">D</Emphasis> interactions in fibrin self-assembly under oxidation

The effect on ozone-induced oxidation on the self-assembly of fibrin in the presence of fibrin-stabilizing factor FXIIIa of soluble cross-linked fibrin oligomers was studied in a medium containing moderate urea concentrations. It is established that fibrin oligomers were formed by the protofibrils cross-linked through γ-γ dimers and the fibrils additionally cross-linked by through α-polymers. The oxidation promoted both the accumulation of greater amounts of γ-γ dimers and the formation of protofibrils, fibrils, and their dissociation products emerging with increasing urea concentrations, which have a high molecular weight. It is concluded that the oxidation enhances the axial interactions between D-regions of fibrin molecules.     

Characterization of soluble polymerized fibrin formed in the presence of excess fibrinogen fragment D

Polymerization of fibrin is inhibited in the presence of excess fibrinogen fragment D. This study was performed in order to test the proposal that these inhibited solutions contain short linear polymers of fibrin (protofibrils) whose further polymerization is prevented as a result of attachment of a molecule of fragment D at each end. Negative-stain electron micrographs, intrinsic viscosities, angular dependence of light scattering intensity, and kinetics of the increase of the scattered intensity with polymerization all were found to support the above model of the inhibited polymer and to reflect the presence of a broad distribution of the lengths of the inhibited fibrin polymers. Furthermore, sodium dodecyl sulfate-polyacrylamide gel electrophoresis of polymers stabilized with gamma-dimer cross-links introduced by factor XIIIa demonstrates cross-linking of fragment D to fibrin oligomers. Cross-linked polymers have been separated from excess fragment D by gel exclusion chromatography in 1 M urea. (In the absence of urea, the purified polymers very slowly associate to fibers.) The observation of the relative stability of short isolated inhibited protofibrils and the decrease or absence of inhibition of fibrin gelation when fragment D was added to solutions in which fibrin had been given time to polymerize to long protofibrils demonstrate that the inhibitory effect of fragment D occurs as a result of inhibition of the first fibrin polymerization step.     

Mass–length ratio of fibrin fibers from gel permeation and light scattering

Mass–length ratios of fibers in fibrin gels were obtained from measurements of the angular dependence of the intensity of light scattered by dilute gels and from the permeability of more concentrated gels. The permeability was determined by measuring the forced flow of buffer through a short column of gel held in a glass tube. The results obtained with the two methods are consistent. At high pH and high ionic strength the mass–length ratio is found to be that calculated for a protofibril, i.e., a double strand of fibrin molecules laid end to end at a separation of 450 Å. This same value is found under conditions where the polymerized fibrin is not gelled (pH 10.25) and where it is just gelled (pH 10.0). At pH 7.4, ionic strength 0.35, the fibers are found to have a higher mass–length ratio, the average fiber consisting of some three protofibrils associated laterally. At pH 7.4 and low ionic strength (0.10 M) the fibers are up to a hundred times more massive.     

Molecular mechanisms of the polymerization of fibrin and the formation of its three-dimensional network

The results of biochemical, immunochemical, and X-ray studies of the structures of fibrinogen and fibrin molecules were analyzed. The mechanisms of the successive formation of the fibrin three-dimensional network were described: the polymerization of monomeric molecules with the formation of bifilar protofibrils, the lateral association of protofibrils, and the embranchment of the forming fibrils. Data on the electron and confocal microscopy of the polymeric fibrin were considered. The role of the known polymerization centers of fibrin which participated in the formation of protofibrils and their lateral association was discussed. Data on the existence of the previously unknown polymerization centers were given. In particular, the experimental results demonstrated that one of such centers which participated in the formation of protofibrils was located in the Bβ12–46 fragment, and did not require the cleavage of fibrinopeptide B for its functioning. The results of the computer modeling of the spatial structure of the fibrin(ogen) molecule and the intermolecular interactions in the course of the fibrin polymerization were presented. The location of the αC domains in the fibrin(ogen) molecule and their role in the polymerization process were discussed. Information on the structure of the calciumbinding sites of fibrin(ogen) and the functional role of Ca²⁺ in fibrin polymerization was published. The structure of factor XIII(a) and the mechanisms of fibrin stabilization by this factor were briefly described.     

Mechanism of fibrin coagulation based on selective, cation-driven, protofibril association

The cation sensitivity of linear and lateral assembly processes of thrombin- and reptilase-activated fibrinogen was examined. Analytic ultracentrifugation shows that the linear assembly of fibrin oligomers (protofibrils) is neither cation dependent nor sensitive to chelating agents. Protofibrils generated with thrombin–hirudin gelate with either 1–2 mM Ca(II) or 15–100 μM Zn(II). By contrast, protofibril B, generated with reptilase–diisopropylfluorophosphonate, gelates only with Ca(II) but is insensitive to Zn(II). These results indicate that the release of fibrinopeptides A and B (FPA and FPB) expose two types of lateral binding sites that are sensitive to Ca(II) and Zn(II) respectively. Transmission electron (TEM) micrographs of negatively stained gels indicate that the linear packing of the monomers within the fibrin- and cation-induced protofibrin fibers is essentially identical. Scanning electron (SEM) micrographs show that the Ca(II)-induced protofibrin B gel is similar to fibrin. In all, it seems that branching and gelation derive from two types of cation-sensitive, lateral associative processes. Based on these findings, a new paradigm for fibrin coagulation is proposed.     

Protofibrin clots induced by calcium and zinc

During the course of studies with fibrin protofibrils, produced by adding hirudin to thrombin-activated fibrinogen prior to the onset of gelation, turbid clots were observed to be generated merely by adding Ca(II) or Zn(II) to protofibrils. The rate of gelation (CT) and turbidity of the “protofibrin” clots increases with cation levels in a concentration-dependent manner, with Zn(II) much more potent than Ca(II). For example, 50 μM Zn(II) generated a more turbid protofibrin clot than 0.5 mM Ca(II). In combination, levels of Zn(II) and Ca(II), which individually have no effect, induce protofibril gelation. The generation of protofibrin clots by Zn(II) is decreased at increasing ionic strength. Apparently, the underlying electrostatic forces that bind the monomers in fibrin and protofibrin gels are similar. SEM micrographs show that Ca(II)- or Zn(II)-induced protofibrin clots (600–1500Å thick) are essentially indistinguishable from those formed directly from fibrinogen and thrombin with divalent cation. The protofibrin fibers induced by the cations are thicker than the fibers formed directly from fibrinogen and thrombin in the absence of divalent cation. Branching appears brought about the the divalent cation-sensitive lateral association of different protofibril strands. These findings describe simple experimental methods for separately studying the early and late stages of fibrin gelation.     

Relations between enzymatic and association reactions in the development of bovine fibrin clot structure

Development of fibrin clot structure was examined at pH 7.0, Γ/2 0.15, and 29 °C as a function of thrombin and fibrinogen concentrations. Parameters for the release of Apeptides, to give $\text{?}{}_{\mathrm{<mtext>A</mtext>}}$ were evaluated. Characteristics of time dependencies of development of turbidity, 90 ° light scattering, network, and compactible network were established. Mean mass/length ratios of fibrin in developing and mature networks were determined. Relationships between results combined with an inferred dependence of lateral interaction on release of B-peptides are used to disclose a model in which a protofibril network is formed first and the intrinsic length of this network (i.e., length exclusive of overlap or loose ends) determines network length, thus mean mass/length ratio, at maturity. Statements regarding initial protofibril network are: (i) A dominant group of $\text{?}{}_{\mathrm{<mtext>A</mtext>}}$-protofibrils appears first. With decreasing rate of production of $\text{?}{}_{\mathrm{<mtext>A</mtext>}}$ their average length increases and number decreases. (ii) Slower release of B-peptides produces $\text{?}{}_{\mathrm{<mtext>AB</mtext>}}$ whose fraction $\text{θ}{}_{\mathrm{<mtext>AB</mtext>}}\text{=}\text{?}{}_{\mathrm{<mtext>AB</mtext>}}\text{(?}{}_{\mathrm{A}}\text{+?}{}_{\mathrm{AB}}$) determines the occurrence of protofibril regions capable of contributing to a lateral interaction sufficiently stable for the formation of network. (iii) When dominant protofibrils attain a minimum combination of average length, number concentration, and frequency of occurrence of capable regions, an initial protofibril network is rapidly generated. (iv) Capable regions near protofibril ends are preferentially involved in initial network formation. (v) The initial network mesh size is large compared to average concomitant free protofibril length. (vi) With B-peptide release dependent on prior A-peptide release, protofibrils in the initial network have the highest capable region frequency, and this is maintained as lateral interaction progresses. Then, fibrin which is free at initial network formation and fibrin which is produced subsequently interact mainly to increase the mean mass/length ratio of initial network elements.     

Electron microscope structural study of modified fibrin and a related modified fibrinogen aggregate

The structure of proteolytically modified fibrin and a closely related modified fibrinogen aggregate have been studied by analysis of electron microscope images. For both structures, we propose a model that consists of double-stranded, 2-fold helical protofibrils, which are associated laterally to form ordered fibrils, with a C222 space group: a = 44.0 nm, b = c = 9.4 nm. Each fibril is 80 nm or less in diameter, and twists along its length in a right-handed sense, with a pitch from 7 to 12 times the molecular length. The fibrils associate laterally to form bundles, which tend to twist in a left-handed sense, with a pitch of the order of 40 times the molecular length. The specific volume of modified fibrin calculated from this model is 3.9 A3 per dalton, which is comparable to the specific volume of 3.6 A3 per dalton for modified fibrinogen crystals but is lower than the 6 A3 per dalton determined for fibrin from light-scattering experiments. Comparison of our electron microscope results with X-ray and neutron diffraction data suggest a similar, but less well-ordered, structure for native fibrin, with a smaller fibril, approximately 18.4 nm wide, consisting of eight protofibrils.     

α-Synuclein Amyloid Fibrils with Two Entwined,Asymmetrically Associated Protofibrils

Parkinson disease and other progressive neurodegenerative conditions are characterized by the intracerebral presence of Lewy bodies, containing amyloid fibrils of α-synuclein. We used cryo-electron microscopy and scanning transmission electron microscopy (STEM) to study in vitro-assembled fibrils. These fibrils are highly polymorphic. Focusing on twisting fibrils with an inter-crossover spacing of 77 nm, our reconstructions showed them to consist of paired protofibrils. STEM mass per length data gave one subunit per 0.47 nm axial rise per protofibril, consistent with a superpleated β-structure. The STEM images show two thread-like densities running along each of these fibrils, which we interpret as ladders of metal ions. These threads confirmed the two-protofibril architecture of the 77-nm twisting fibrils and allowed us to identify this morphotype in STEM micrographs. Some other, but not all, fibril morphotypes also exhibit dense threads, implying that they also present a putative metal binding site. We propose a molecular model for the protofibril and suggest that polymorphic variant fibrils have different numbers of protofibrils that are associated differently.     

Ligation of fibrinogen by factor XIIIa with dithiothreitol: mechanical properties of ligated fibrinogen gels

Human fibrinogen (concentration 8.4 mg/mL) was ligated (cross-linked) with factor XIIIa and dithiothreitol (DTT) at pH 8.5, ionic strength 0.45. With 7.5 μg/mL of factor XIIIa alone, there was almost no γ-γ ligation, but with 2 mM DTT added, oligomers appeared, and γ-γ and Aα-Aα ligation was nearly complete after 3 days. At 38 μg/mL of factor XIIIa, some γ-γ and Aα-Aα ligation occurred even without DTT. For fibrinogen concentrations of 4.0 and 8.4 mg/mL, 38 μ/mL factor XIIIa, 2.0 mM DTT, clot-like gels formed and the shear modulus of elasticity increased slowly over several days to a constant value. The final modulus was similar in magnitude to those of ligated clots of α-fibrin (clotted by thrombin) and α-fibrin (clotted by batroxobin) under the same conditions. However, the opacity was somewhat higher; whereas in fine fibrin clots there is minimal lateral association of the protofibrils, in fibrinogen gels at the same pH and ionic strength the protofibrils (which are presumably single chains of fibrinogen monomers joined end to end at their D domains) are evidently associated in bundles (although not to the degree seen in coarse fibrin clots). Creep and creep recovery measurements showed almost perfect elastic behavior, with essentially no creep under stress and complete recovery after removal of stress. The modulus was scarcely affected by introduction of lithium bromide by diffusion to a concentration of 0.6M, which in unligated fibrin clots causes substantial softening. Whereas in fine fibrin clots (both αβ-fibrin and α-fibrin) factor XIIIa causes only γ-γ ligation, addition of 2 mM DTT produced some α-α ligation in these also.     

Oxidation-induced destabilization of the fibrinogen αC-domain dimer investigated by molecular dynamics simulations

Upon activation, fibrinogen is converted to insoluble fibrin, which assembles into long strings called protofibrils. These aggregate laterally to form a fibrin matrix that stabilizes a blood clot. Lateral aggregation of protofibrils is mediated by the αC domain, a partially structured fragment located in a disordered region of fibrinogen. Polymerization of αC domains links multiple fibrin molecules with each other enabling the formation of thick fibrin fibers and a fibrin matrix that is stable but can also be digested by enzymes. However, oxidizing agents produced during the inflammatory response have been shown to cause thinner fibrin fibers resulting in denser clots, which are harder to proteolyze and pose the risk of deep vein thrombosis and lung embolism. Oxidation of Met⁴⁷⁶ located within the αC domain is thought to hinder its ability to polymerize disrupting the lateral aggregation of protofibrils and leading to the observed thinner fibers. How αC domains assemble into polymers is still unclear and yet this knowledge would shed light on the mechanism through which oxidation weakens the lateral aggregation of protofibrils. This study used temperature replica exchange molecular dynamics simulations to investigate the αC-domain dimer and how this is affected by oxidation of Met⁴⁷⁶. Analysis of the trajectories revealed that multiple stable binding modes were sampled between two αC domains while oxidation decreased the likelihood of dimer formation. Furthermore, the side chain of Met⁴⁷⁶ was observed to act as a docking spot for the binding and this function was impaired by its conversion to methionine sulfoxide.     

Fibrin protofibril and fibrinogen binding to ADP-stimulated platelets: evidence for a common mechanism

The molecular basis of platelet-fibrin binding has been elucidated by studying interactions between platelets and protofibrils, soluble two-stranded polymers of fibrin which are intermediates on the fibrin assembly pathway. The fibrinogen degradation product, fragment D, has been used to block fibrin assembly, thus enabling the preparation of stable solutions of short protofibrils, composed of fewer than twenty fibrin monomer molecules per polymer. Fibrin protofibrils bound to ADP-activated platelets in a time- and concentration-dependent process which was effectively blocked by excess unlabelled fibrinogen, i.e., the binding was specific and appeared to involve a common receptor. ADP-stimulated cells bound approx. 3 micrograms of fibrin protofibrils/10(8) platelets, compared to 4 micrograms of fibrinogen/10(8) cells, following a 30-min incubation period at room temperature. Binding of both ligands was inhibited by high concentrations of fragment D, further indicating a similar mechanism. The kinetic data obtained were well described by an apparent first-order mechanism in which the rate constant for fibrin protofibril binding was found to be 5-fold slower than that measured for fibrinogen. Two monoclonal antibodies, each directed against the platelet glycoprotein IIb-IIIa complex, inhibited the binding of fibrin protofibrils and fibrinogen in a similar, concentration-dependent manner, providing strong evidence for a common receptor. Binding of GPRP-fibrin (soluble fibrin oligomers formed in the presence of 1 mM Gly-Pro-Arg-Pro) to ADP-stimulated platelets was also inhibited by a monoclonal antibody directed against the GPIIb-IIIa complex. Neither fibrin protofibrils nor fibrinogen bound to Glanzmann's thrombasthenic platelets, which lack normal quantities of functional glycoprotein IIb-IIIa complex, further supporting the hypothesis that fibrinogen and fibrin bind to a common platelet receptor present on the glycoprotein IIb-IIIa complex.     

Differences between the complexes formed by monomeric fibrin with fragment D and dimer D

The paper is concerned with studies in formation of monomeric fibrin (fm) complexes with fragment D (D) of fibrinogen and dimer D (DD) of stabilized fibrin. The complexes are shown to be essentially different. The fm-D complexes are unstable, their composition is a function of D concentration in the mixture, the ultimate molar D/fm ratio is equal to 3. The fm-DD complexes are quite stable, their composition is constant: the molar DD/fm ratio is equal to 1. In mixtures containing fm, DD and different amounts of D complexes of different composition are formed but the total number of D-units in them approaches 3. A model is suggested showing interaction of fm molecules in protofibril formation with allowance for the retention of binding centres which provide the lateral link between protofibrils.     

Physiochemical characterization of the Alzheimer's disease-related peptides A beta 1-42Arctic and A beta 1-42wt

The amyloid beta peptide (A beta) is crucial for the pathogenesis of Alzheimer's disease. Aggregation of monomeric A beta into insoluble amyloid fibrils proceeds through several soluble A beta intermediates, including protofibrils, which are believed to be central in the disease process. The main reason for this is their implication in familial Alzheimer's disease with the Arctic amyloid precursor protein mutation (E693G). This mutation gives rise to early onset Alzheimer's disease, and synthetic A beta 1-40Arctic displays an enhanced rate of protofibril formation in vitro[Nilsberth C, Westlind-Danielsson A, Eckman CB, Condron MM, Axelman K, Forsell C, Stenh C, Luthman J, Teplow DB, Younkin SG, Naslund J & Lannfelt L. (2001) Nat Neurosci4, 887-893]. To increase our understanding of the mechanisms involved in A beta aggregation, especially A beta monomer oligomerization into protofibrils and protofibril fibrillization into fibrils, the kinetics of A beta 1-42wt and A beta 1-42Arctic aggregation were examined under different physiochemical conditions, such as concentration, temperature, ionic strength and pH. We used size exclusion chromatography for this purpose, where monomers are separated from protofibrils, and fibrils are separated from protofibrils in a centrifugation step. The Arctic mutation significantly accelerated both A beta 1-42wt protofibril formation and protofibril fibrillization. In addition, we demonstrated that two distinct chemical processes - monomer oligomerization and protofibril fibrillization - were affected differently by changes in the micro-environment and that the Arctic mutation alters the peptide response to such changes.     

Polymorphism of fibrin equilibrium oligomers in the presence of fragment D

The methods of viscosimetry, the Rayleigh light-scattering and analytical ultracentrifugation were applied to study the physicochemical mechanism of the effect of fragment D on the structure of fibrin equilibrium oligomers. Using the values of intrinsic viscosity, weight average molecular masses and mass/length ratio it was shown that when producing an antipolymerization effect the fragment D retains the three-dimensional organization of fibrin polymers, i.e. rigid rod-like single- and double-stranded protofibrillas. The paper has proved that along with the traditional mechanism of inhibiting self-assembly of of the double-stranded structure due to the competition of fragment D with fibrin monomer for central domain E there is an alternative attributed to its attachment to a peripheral region of the fibrin monomer. The second mechanism is the only one which occurs in the region of single-stranded pseudoprotofibrillas existence. The role of alpha C-domains in protein-protein interactions is also discussed.     

The extraction effect of ultrasound during plasma clot disruption

The products of the plasma clot destruction by the low-frequency ultrasound (US) were analyzed using the combination of SDS gel-electrophoresis, gel filtration chromatography and scanning electron microscopy. It was found that US (27 kHz) did not cause activation of the plasmin system or covalent bonds cleavage in the fibrin molecules. At US intensities less than 21.6 W/cm² there was extraction of blood serum proteins, which are located in the pores of the fibrin network. The increase in intensity of ultrasonic action resulted in protofibril dissociation, which was accompanied by further release into the solution of the blood serum proteins, located inside fibrin fibers. After US cavitation protein extracted from the plasma clot underwent aggregation. Interaction between free protofibrils resulted in formation of insoluble fibrin particles.     

Localization of the domains of fibrin involved in binding to platelets

The molecular basis of platelet-fibrin interactions has been investigated by using synthetic peptides as potential inhibitors of fibrin protofibril and fibrinogen binding to ADP-stimulated platelets, adhesion of fibrin fibers to the platelet surface, and platelet-mediated clot retraction. Synthetic peptides of sequence RGDS and HHLGGAKQAGDV, corresponding to regions of the fibrinogen alpha- and gamma-chains previously identified as platelet recognition sites, inhibited the binding of radiolabelled soluble fibrin oligomers to ADP-stimulated platelets with IC50 values of 10 and 40 microM, respectively. Synthetic GPRP and GHRP, corresponding to the N-terminal tripeptide sequence of the fibrin alpha-chains and the tetrapeptide sequence of the beta-chains, respectively, were minimally effective in blocking soluble fibrin polymer binding to ADP-stimulated platelets. Platelet functions which are unique to the three-dimensional fibrin network were examined by measurements of the extent of adhesion of fluorophore-labelled fibrin to platelets with a microfluorimetric technique and by light scattering measurements of the time course of clot retraction. Inhibition of fibrin-platelet adhesion by RGDS, HHLGGAKQAGDV and GHRP exhibited a similar, linear dependence reaching 1/2 maximum at about 200 microM, suggesting nonspecific effects. GPRP inhibited fibrin assembly but did not appear to have specific effects on fibrin-platelet adhesion. Only RGDS effected clot retraction, causing a 4-6-fold decrease in rate at 230 microM. These results indicate that fibrinogen and fibrin protofibrils, which are obligatory intermediates on the fibrin assembly pathway, share a set of common platelet recognition sites located at specific regions of the alpha- and gamma-chains of the multinodular fibrin(ogen) molecules. The RGDS site is also involved in mediating interactions between the three-dimensional fibrin network and stimulated platelets.     

Structural Characterization of Heparin-induced Glyceraldehyde-3-phosphate Dehydrogenase Protofibrils Preventing α-Synuclein Oligomeric Species Toxicity

Glyceraldehyde-3-phosphate dehydrogenase (GAPDH) is a multifunctional enzyme that has been associated with neurodegenerative diseases. GAPDH colocalizes with α-synuclein in amyloid aggregates in post-mortem tissue of patients with sporadic Parkinson disease and promotes the formation of Lewy body-like inclusions in cell culture. In a previous work, we showed that glycosaminoglycan-induced GAPDH prefibrillar species accelerate the conversion of α-synuclein to fibrils. However, it remains to be determined whether the interplay among glycosaminoglycans, GAPDH, and α-synuclein has a role in pathological states. Here, we demonstrate that the toxic effect exerted by α-synuclein oligomers in dopaminergic cell culture is abolished in the presence of GAPDH prefibrillar species. Structural analysis of prefibrillar GAPDH performed by small angle x-ray scattering showed a particle compatible with a protofibril. This protofibril is shaped as a cylinder 22 nm long and a cross-section diameter of 12 nm. Using biocomputational techniques, we obtained the first all-atom model of the GAPDH protofibril, which was validated by cross-linking coupled to mass spectrometry experiments. Because GAPDH can be secreted outside the cell where glycosaminoglycans are present, it seems plausible that GAPDH protofibrils could be assembled in the extracellular space kidnapping α-synuclein toxic oligomers. Thus, the role of GAPDH protofibrils in neuronal proteostasis must be considered. The data reported here could open alternative ways in the development of therapeutic strategies against synucleinopathies like Parkinson disease.     

Kinetics of formation of fibrin oligomers. I. Theory

The course of formation of fibrin oligomers is treated theoretically for the condition that self-assembly of fibrin monomers is rapid compared with the loss of A peptides by the enzymatic action of thrombin. The rate constant for removal of the second A peptide is taken to be larger than that for the first by an arbitrary factor q; the association of activated A sites with their complementary a sites is assumed to be random and independent of oligomer size. Two types of oligomers are considered: noncovalently bonded protofibrils formed by the staggered overlap of thrombin-activated monomers and covalently bonded linear oligomers formed by factor XIIIa-mediated end-to-end ligation of adjacent monomers within protofibrils. Oligomers of the first type, if ligated, are dissociated to oligomers of the second type by solubilization in SDS–urea. Theoretical curves are presented for x _w and x ′_w (weight-average degree of polymerization of staggered overlap and linear ligated oligomers, respectively) and for the weight fractions of monomer, dimer, and decamer of both ligated and unligated species as functions of y, the fraction of A peptide removed; and also for w_x and w′_x, the weight fractions of x-mer of the respective oligomer types, as a function of x at y = 0.5. With increasing q, the maximum w_x or w′_x that a low oligomer will reach during the reaction decreases and the size distribution is broadened toward larger oligomers. Comparison with experiment is made in a companion paper.