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Blood coagulation is a vital physiological process involving a complex network of biochemical reactions, which converge to form a blood clot that repairs vascular injury. This process unfolds in three phases: initiation, amplification, and propagation, ultimately leading to thrombin formation. Coagulation begins when tissue factor (TF) is exposed on an injured vessel's wall. The first step is when activated factor VII (VIIa) in the plasma binds to TF, forming complex TF:VIIa, which activates factor X. Activated factor X (Xa) is necessary for coagulation, so the regulation of its activation is crucial. Tissue Factor Pathway Inhibitor (TFPI) is a critical regulator of the initiation phase as it inhibits the activation of factor X. While previous studies have proposed two pathways-direct and indirect binding-for TFPI's inhibitory role, the specific biochemical reactions and their rates remain ambiguous. Many existing mathematical models only assume an indirect pathway, which may be less effective under physiological flow conditions. In this study, we revisit datasets from two experiments focused on activated factor X formation in the presence of TFPI. We employ an adaptive Metropolis method for parameter estimation to reinvestigate a previously proposed biochemical scheme and corresponding rates for both inhibition pathways. Our findings show that both pathways are essential to replicate the static experimental results. Previous studies have suggested that flow itself makes a significant contribution to the inhibition of factor X activation. We added flow to this model with our estimated parameters to determine the contribution of the two inhibition pathways under these conditions. We found that direct binding of TFPI is necessary for inhibition under flow. The indirect pathway has a weaker inhibitory effect due to removal of solution phase inhibitory complexes by flow., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2024 Santiago et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

Background: Tissue factor (TF) pathway inhibitor (TFPI) is an anticoagulant protein that inhibits factor (F)Xa, the TF-FVIIa-FXa complex, and early forms of the prothrombinase complex. Concizumab is a monoclonal antibody that blocks FXa inhibition by TFPI and reduces bleeding in hemophilia., Objectives: To examine how concizumab impacts various reactions of TFPI to restore thrombin generation in hemophilia A using mathematical models., Methods: A compartment model was used to estimate plasma concentrations of free concizumab and its complexes with TFPIα and TFPIβ. Concizumab was integrated into a flow-mediated mathematical model of coagulation, and a small injury was simulated under hemophilia A conditions. Simulations were then analyzed to determine how concizumab's blockade of TFPI anticoagulant activities, specifically the inhibition of FXa in plasma and on platelets, inhibition of TF:FVIIa at the subendothelium, and prior sequestration of plasma TFPIα to the endothelium via TFPIβ, altered thrombin generation., Results: Concizumab improved simulated thrombin generation in hemophilia A by simultaneously altering all 3 mechanisms of the TFPI anticoagulant blockade examined. Concizumab sequestered ∼75% of plasma TFPIα through the formation of ternary TFPIα-concizumab-TFPIβ-complexes. For all TF levels, reducing the TFPIα plasma concentration had the largest impact on the lag time, followed by blocking TFPIα inhibition of TF:FVIIa:FXa and subsequently by blocking TFPIα inhibition of FXa in plasma and on the platelet surface., Conclusion: The effectiveness of concizumab is mediated through the blockade of TFPI anticoagulant activities in plasma and on multiple physiological surfaces. An important and previously unrecognized function of concizumab was the sequestration of plasma TFPIα to the endothelium., Competing Interests: Declaration of competing interests A.E.M. receives research funding from Novo Nordisk and Pharmacosmos and has received honoraria for serving on Novo Nordisk Advisory Boards. A.R.W., M.D., and M.K. are employees of Novo Nordisk. M.K. is a minor shareholder of Novo Nordisk. K.M. and K.L. received research funding from Novo Nordisk., (Copyright © 2024 The Authors. Published by Elsevier Inc. All rights reserved.)

Background: In healthy individuals, plasma levels of clotting proteins naturally vary within a range of 50% to 150% of their mean values. We do not know how these variations modify thrombin generation., Objectives: To assess the impact of protein level variations on simulated thrombin generation in normal and factor (F)VIII-, FIX-, or FXI-deficient blood., Methods: We used a mathematical model of flow-mediated coagulation to simulate thrombin generation with all possible combinations of clotting protein variations within the normal range and for various tissue factor levels. We selected, analyzed, and ranked combinations that enhanced thrombin generation compared with baseline., Results: Protein variations most strongly affected thrombin generation at intermediate tissue factor levels. Low tissue factor levels prevented coagulation initiation, while high tissue factor levels always triggered thrombin generation. At intermediate levels, we identified protein variations that substantially modified thrombin generation. Low-normal FV shortened lag times and increased thrombin generation, whereas high-normal FV lengthened lag times and reduced thrombin generation. With severe FVIII and FIX deficiencies, low-normal tissue factor pathway inhibitor α and antithrombin amplified the effect of low-normal FV. For moderate FVIII and FIX deficiencies, high-normal tissue factor pathway inhibitor α and antithrombin enhanced the impact of high-normal FV in reducing thrombin production. In normal and FXI-deficient blood, high-normal FVIII and FIX significantly boosted thrombin generation., Conclusion: Our mathematical model predicted how variations in clotting protein levels, within the normal range, could contribute to the variability of bleeding phenotypes observed with clotting factor deficiencies. Our study generated experimentally testable hypotheses that could aid in developing new therapies toward normal hemostasis., (© 2024 The Author(s).)

Blood coagulation is a network of biochemical reactions wherein dozens of proteins act collectively to initiate a rapid clotting response. Coagulation reactions are lipid-surface dependent, and this dependence is thought to help localize coagulation to the site of injury and enhance the association between reactants. Current mathematical models of coagulation either do not consider lipid as a variable or do not agree with experiments where lipid concentrations were varied. Since there is no analytic rate law that depends on lipid, only apparent rate constants can be derived from enzyme kinetic experiments. We developed a new mathematical framework for modeling enzymes reactions in the presence of lipid vesicles. Here the concentrations are such that only a fraction of the vesicles harbor bound enzymes and the rest remain empty. We call the lipid vesicles with and without enzyme TF:VIIa + and TF:VIIa - lipid, respectively. Since substrate binds to both TF:VIIa + and TF:VIIa - lipid, our model shows that excess empty lipid acts as a strong sink for substrate. We used our framework to derive an analytic rate equation and performed constrained optimization to estimate a single, global set of intrinsic rates for the enzyme-substrate pair. Results agree with experiments and reveal a critical lipid concentration where the conversion rate of the substrate is maximized, a phenomenon known as the template effect. Next, we included product inhibition of the enzyme and derived the corresponding rate equations, which enables kinetic studies of more complex reactions. Our combined experimental and mathematical study provides a general framework for uncovering the mechanisms by which lipid mediated reactions impact coagulation processes., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2024 The Authors. Published by Elsevier Inc. All rights reserved.)

Human immunodeficiency virus type 1 typically requires a high density of CD4 for efficient entry as a mechanism to target CD4+ T cells (T-tropic), with CCR5 being used most often as the coreceptor. When target T cells are limiting, the virus can evolve to infect cells with a low density of CD4 such as macrophages (M-tropic). The entry phenotype is known to be encoded in the viral Env protein on the surface of the virus particle. Using data showing a dose response for infectivity based on CD4 surface density, we built a model consistent with T-tropic viruses requiring multiple CD4 molecules to mediate infection, whereas M-tropic viruses can infect cells using a single CD4 receptor molecule interaction. We also found that T-tropic viruses bound to the surface of cells with a low density of CD4 are released more slowly than M-tropic viruses which we modeled to be due to multiple interactions of the T-tropic virus with multiple CD4 molecules to allow the initial stable binding. Finally, we found that some M-tropic Env proteins, as the gp120 subunit, possess an enhanced affinity for CD4 compared with their T-tropic pair, indicating that the evolution of macrophage tropism can be reflected both in the closed Env trimer conformation on the virion surface and, in some cases, also in the open confirmation of gp120 Env. Collectively, these studies reveal differences in the stoichiometry of interaction of T-tropic and M-tropic viruses with CD4 and start to identify the basis of binding differences at the biochemical level., Importance: Human immunodeficiency virus type 1 normally targets CD4+ T cells for viral replication. When T cells are limiting, the virus can evolve to infect myeloid cells. The evolutionary step involves a change from requiring a high surface density of CD4 for entry to being able to infect cells with a low density of CD4, as is found on myeloid lineage cells such as macrophage and microglia. Viruses able to infect macrophages efficiently are most often found in the CNS late in the disease course, and such viruses may contribute to neurocognitive impairment. Here, we examine the CD4 binding properties of the viral Env protein to explore these two different entry phenotypes., Competing Interests: The authors declare no conflict of interest.

Blood clotting involves the coupled processes of platelet aggregation and coagulation. Simulating clotting under flow in complex geometries is challenging due to multiple temporal and spatial scales and high computational cost. clotFoam is an open-source software developed in OpenFOAM that employs a continuum model of platelet advection, diffusion, and aggregation in a dynamic fluid environment and a simplified coagulation model with proteins that advect, diffuse, and react within the fluid and with wall-bound species through reactive boundary conditions. Our framework provides the foundation on which one can build more complex models and perform reliable simulations in almost any computational domain., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Blood coagulation is a self-repair process regulated by activated platelet surfaces, clotting factors, and inhibitors. Tissue factor pathway inhibitor (TFPI) is one such inhibitor, well known for its inhibitory action on the active enzyme complex comprising tissue factor (TF) and activated clotting factor VII. This complex forms when TF embedded in the blood vessel wall is exposed by injury and initiates coagulation. A different role for TFPI, independent of TF:VIIa, has recently been discovered whereby TFPI binds a partially cleaved form of clotting factor V (FV-h) and impedes thrombin generation on activated platelet surfaces. We hypothesized that this TF-independent inhibitory mechanism on platelet surfaces would be a more effective platform for TFPI than the TF-dependent one. We examined the effects of this mechanism on thrombin generation by including the relevant biochemical reactions into our previously validated mathematical model. Additionally, we included the ability of TFPI to bind directly to and inhibit platelet-bound FXa. The new model was sensitive to TFPI levels and, under some conditions, TFPI could completely shut down thrombin generation. This sensitivity was due entirely to the surface-mediated inhibitory reactions. The addition of the new TFPI reactions increased the threshold level of TF needed to elicit a strong thrombin response under flow, but the concentration of thrombin achieved, if there was a response, was unchanged. Interestingly, we found that direct binding of TFPI to platelet-bound FXa had a greater anticoagulant effect than did TFPI binding to FV-h alone, but that the greatest effects occurred if both reactions were at play. The model includes activated platelets' release of FV species, and we explored the impact of varying the FV/FV-h composition of the releasate. We found that reducing the zymogen FV fraction of this pool, and thus increasing the fraction that is FV-h, led to acceleration of thrombin generation., Competing Interests: Declaration of interests K.L. received research support from Novo Nordisk., (Copyright © 2022 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

Blood coagulation is a self-repair process regulated by activated platelet surfaces, clotting factors, and inhibitors. Antithrombin (AT) is one such inhibitor that impedes coagulation by targeting and inactivating several key coagulation enzymes. The effect of AT is greatly enhanced in the presence of heparin, a common anticoagulant drug. When heparin binds to AT, it either bridges with the target enzyme or induces allosteric changes in AT leading to more favorable binding with the target enzyme. AT inhibition of fluid-phase enzymes caused little suppression of thrombin generation in our previous mathematical models of blood coagulation under flow. This is because in that model, flow itself was a greater inhibitor of the fluid-phase enzymes than AT. From clinical observations, it is clear that AT and heparin should have strong inhibitory effects on thrombin generation, and thus we hypothesized that AT could be inhibiting enzymes bound to activated platelet surfaces that are not subject to being washed away by flow. We extended our mathematical model to include the relevant reactions of AT inhibition at the activated platelet surfaces as well as those for unfractionated heparin and a low molecular weight heparin. Our results show that AT alone is only an effective inhibitor at low tissue factor densities, but in the presence of heparin, it can greatly alter, and in some cases shut down, thrombin generation. Additionally, we studied each target enzyme separately and found that inactivation of no single enzyme could substantially suppress thrombin generation., Competing Interests: Declaration of interests K.L. received research support from Novo Nordisk., (Copyright © 2022 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

Thrombin is an enzyme produced during blood coagulation that is crucial to the formation of a stable clot. Thrombin cleaves soluble fibrinogen into fibrin, which polymerizes and forms an insoluble, stabilizing gel around the growing clot. A small fraction of circulating fibrinogen is the variant γA/γ', which has been associated with high-affinity thrombin binding and implicated as a risk factor for myocardial infarctions, deep vein thrombosis, and coronary artery disease. Thrombin is also known to be strongly sequestered by polymerized fibrin for extended periods of time in a way that is partially regulated by γA/γ'. However, the role of γA/γ'-thrombin interactions during fibrin polymerization is not fully understood. Here, we present a mathematical model of fibrin polymerization that considered the interactions between thrombin, fibrinogen, and fibrin, including those with γA/γ'. In our model, bivalent thrombin-fibrin binding greatly increased thrombin residency times and allowed for thrombin-trapping during fibrin polymerization. Results from the model showed that early in fibrin polymerization, γ' binding to thrombin served to localize the thrombin to the fibrin(ogen), which effectively enhanced the enzymatic conversion of fibrinogen to fibrin. When all the fibrin was fully generated, however, the fibrin-thrombin binding persisted but the effect of fibrin on thrombin switched quickly to serve as a sink, essentially removing all free thrombin from the system. This dual role for γ'-thrombin binding during polymerization led to a paradoxical decrease in trapped thrombin as the amount of γ' was increased. The model highlighted biochemical and biophysical roles for fibrin-thrombin interactions during polymerization and agreed well with experimental observations., Competing Interests: The authors have declared that no competing interests exist.

The ISTH London 2022 Congress is the first held (mostly) face-to-face again since the COVID-19 pandemic took the world by surprise in 2020. For 2 years we met virtually, but this year's in-person format will allow the ever-so-important and quintessential creativity and networking to flow again. What a pleasure and joy to be able to see everyone! Importantly, all conference proceedings are also streamed (and available recorded) online for those unable to travel on this occasion. This ensures no one misses out. The 2022 scientific program highlights new developments in hemophilia and its treatment, acquired and other inherited bleeding disorders, thromboinflammation, platelets and coagulation, clot structure and composition, fibrinolysis, vascular biology, venous thromboembolism, women's health, arterial thrombosis, pediatrics, COVID-related thrombosis, vaccine-induced thrombocytopenia with thrombosis, and omics and diagnostics. These areas are elegantly reviewed in this Illustrated Review article. The Illustrated Review is a highlight of the ISTH Congress. The format lends itself very well to explaining the science, and the collection of beautiful graphical summaries of recent developments in the field are stunning and self-explanatory. This clever and effective way to communicate research is revolutionary and different from traditional formats. We hope you enjoy this article and will be inspired by its content to generate new research ideas., (© 2022 The Authors. Research and Practice in Thrombosis and Haemostasis published by Wiley Periodicals LLC on behalf of International Society on Thrombosis and Haemostasis (ISTH).)

Computational models of various facets of hemostasis and thrombosis have increased substantially in the last decade. These models have the potential to make predictions that can uncover new mechanisms within the complex dynamics of thrombus formation. However, these predictions are only as good as the data and assumptions they are built upon, and therefore model building requires intimate coupling with experiments. The objective of this article is to guide the reader through how a computational model is built and how it can inform and be refined by experiments. This is accomplished by answering six questions facing the model builder: (1) Why make a model? (2) What kind of model should be built? (3) How is the model built? (4) Is the model a "good" model? (5) Do we believe the model? (6) Is the model useful? These questions are answered in the context of a model of thrombus formation that has been successfully applied to understanding the interplay between blood flow, platelet deposition, and coagulation and in identifying potential modifiers of thrombin generation in hemophilia A., Competing Interests: K.B.N. reports grants from National Institutes of Health, during the conduct of the study. D.M.M. reports grants from Army Research Office, during the conduct of the study., (The Author(s). This is an open access article published by Thieme under the terms of the Creative Commons Attribution-NonDerivative-NonCommercial License, permitting copying and reproduction so long as the original work is given appropriate credit. Contents may not be used for commercial purposes, or adapted, remixed, transformed or built upon. (https://creativecommons.org/licenses/by-nc-nd/4.0/).)

Bleeding frequency and severity within clinical categories of hemophilia A are highly variable and the origin of this variation is unknown. Solving this mystery in coagulation requires the generation and analysis of large data sets comprised of experimental outputs or patient samples, both of which are subject to limited availability. In this review, we describe how a computationally driven approach bypasses such limitations by generating large synthetic patient data sets. These data sets were created with a mechanistic mathematical model, by varying the model inputs, clotting factor, and inhibitor concentrations, within normal physiological ranges. Specific mathematical metrics were chosen from the model output, used as a surrogate measure for bleeding severity, and statistically analyzed for further exploration and hypothesis generation. We highlight results from our recent study that employed this computationally driven approach to identify FV (factor V) as a key modifier of thrombin generation in mild to moderate hemophilia A, which was confirmed with complementary experimental assays. The mathematical model was used further to propose a potential mechanism for these observations whereby thrombin generation is rescued in FVIII-deficient plasma due to reduced substrate competition between FV and FVIII for FXa (activated factor X).

Flagella are hairlike appendages attached to microorganisms that allow the organisms to traverse their fluid environment. The algae Volvox are spherical swimmers with thousands of individual flagella on their surface, and their coordination is not fully understood. In this work, a previously developed minimal model of flagella synchronization is extended to the outer surface of a sphere submerged in a fluid. Each beating flagellum tip is modeled as a small sphere, elastically bound to a circular orbit just above the spherical surface and a regularized image system for Stokes flow outside of a sphere is used to enforce the no-slip condition. Biologically relevant distributions of rotors results in a rapidly developing and robust symplectic metachronal wave traveling from the anterior to the posterior of the spherical Volvox body.

Background: The variability in bleeding patterns among individuals with hemophilia A, who have similar factor VIII (FVIII) levels, is significant and the origins are unknown., Objective: To use a previously validated mathematical model of flow-mediated coagulation as a screening tool to identify parameters that are most likely to enhance thrombin generation in the context of FVIII deficiency., Methods: We performed a global sensitivity analysis (GSA) on our mathematical model to identify potential modifiers of thrombin generation. Candidates from the GSA were confirmed by calibrated automated thrombography (CAT) and flow assays on collagen-tissue factor (TF) surfaces at a shear rate of 100 per second., Results: Simulations identified low-normal factor V (FV) (50%) as the strongest modifier, with additional thrombin enhancement when combined with high-normal prothrombin (150%). Low-normal FV levels or partial FV inhibition (60% activity) augmented thrombin generation in FVIII-inhibited or FVIII-deficient plasma in CAT. Partial FV inhibition (60%) boosted fibrin deposition in flow assays performed with whole blood from individuals with mild and moderate FVIII deficiencies. These effects were amplified by high-normal prothrombin levels in both experimental models., Conclusions: These results show that low-normal FV levels can enhance thrombin generation in hemophilia A. Further explorations with the mathematical model suggest a potential mechanism: lowering FV reduces competition between FV and FVIII for factor Xa (FXa) on activated platelet surfaces (APS), which enhances FVIII activation and rescues thrombin generation in FVIII-deficient blood., (© 2019 International Society on Thrombosis and Haemostasis.)

We present the first mathematical model of flow-mediated primary hemostasis in an extravascular injury which can track the process from initial deposition to occlusion. The model consists of a system of ordinary differential equations (ODEs) that describe platelet aggregation (adhesion and cohesion), soluble-agonist-dependent platelet activation, and the flow of blood through the injury. The formation of platelet aggregates increases resistance to flow through the injury, which is modeled using the Stokes-Brinkman equations. Data from analogous experimental (microfluidic flow) and partial differential equation models informed parameter values used in the ODE model description of platelet adhesion, cohesion, and activation. This model predicts injury occlusion under a range of flow and platelet activation conditions. Simulations testing the effects of shear and activation rates resulted in delayed occlusion and aggregate heterogeneity. These results validate our hypothesis that flow-mediated dilution of activating chemical adenosine diphosphate hinders aggregate development. This novel modeling framework can be extended to include more mechanisms of platelet activation as well as the addition of the biochemical reactions of coagulation, resulting in a computationally efficient high throughput screening tool of primary and secondary hemostasis.

Thrombin is an enzyme that plays many important roles in the blood clotting process; it activates platelets, cleaves coagulation proteins within feedback loops, and cleaves fibrinogen into fibrin, which polymerizes into fibers to form a stabilizing gel matrix in and around growing clots. Thrombin also binds to the formed fibrin matrix, but this interaction is not well understood. Thrombin-fibrin binding is often described as two independent, single-step binding events, one high-affinity and one low-affinity. However, kinetic schemes describing these single-step binding events do not explain experimentally-observed residency times of fibrin-bound thrombin. In this work, we study a bivalent, sequential-step binding scheme as an alternative to the high-affinity event and, in addition to the low-affinity one. We developed mathematical models for the single- and sequential-step schemes consisting of reaction-diffusion equations to compare to each other and to experimental data. We then used Bayesian inference, in the form of Markov chain Monte Carlo, to learn model parameter distributions from previously published experimental data. For the model to best fit the data, we made an additional assumption that thrombin was irreversibly sequestered; we hypothesized that this could be due to thrombin becoming physically trapped within fibrin fibers as they formed. We further estimated that ∼30% of thrombin in the experiments to which we compare our model output became physically trapped. The notion of physically trapped thrombin may provide new insights into conflicting observations regarding the speed of fibrinolysis. Finally, we show that our new model can be used to further probe scenarios dealing with thrombin allostery., (Copyright © 2019 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

Chromogenic substrates (CS) are synthetic substrates used to monitor the activity of a target enzyme. It has been reported that some CSs display competitive product inhibition with their target enzyme. Thus, in assays where enzyme activity is continuously monitored over long periods of time, the product inhibition may significantly interfere with the reactions being monitored. Despite this knowledge, it is rare for CSs to be directly incorporated into mathematical models that simulate these assays. This devalues the predictive power of the models. In this study, we examined the interactions between a single enzyme, coagulation factor Xa, and its chromogenic substrate. We developed, and experimentally validated, a mathematical model of a chromogenic assay for factor Xa that explicitly included product inhibition from the CS. We employed Bayesian inference, in the form of Markov-Chain Monte Carlo, to estimate the strength of the product inhibition and other sources of uncertainty such as pipetting error and kinetic rate constants. Our model, together with carefully calibrated biochemistry experiments, allowed for full characterization of the strength and impact of product inhibition in the assay. The effect of CS product inhibition in more complex reaction mixtures was further explored using mathematical models., (Copyright © 2019 Elsevier Inc. All rights reserved.)

Upon injury to a blood vessel, flowing platelets will aggregate at the injury site, forming a plug to prevent blood loss. Through a complex system of biochemical reactions, a stabilizing mesh forms around the platelet aggregate forming a blood clot that further seals the injury. Computational models of clot formation have been developed to a study intravascular thrombosis, where a vessel injury does not cause blood leakage outside the blood vessel but blocks blood flow. To model scenarios in which blood leaks from a main vessel out into the extravascular space, new computational tools need to be developed to handle the complex geometries that represent the injury. We have previously modeled intravascular clot formation under flow using a continuum approach wherein the transport of platelet densities into some spatial location is limited by the platelet fraction that already reside in that location, i.e., the densities satisfy a maximum packing constraint through the use of a hindered transport coefficient. To extend this notion to extravascular injury geometries, we have modified a finite element method flux-corrected transport (FEM-FCT) scheme by prelimiting antidiffusive nodal fluxes. We show that our modified scheme, under a variety of test problems, including mesh refinement, structured vs unstructured meshes, and for a range of reaction rates, produces numerical results that satisfy a maximum platelet-density packing constraint., (© 2019 John Wiley & Sons, Ltd.)

The hemostatic response involves blood coagulation and platelet aggregation to stop blood loss from an injured blood vessel. The complexity of these processes make it difficult to intuit the overall hemostatic response without quantitative methods. Mathematical models aim to address this challenge but are often accompanied by numerous parameters choices and thus need to be analyzed for sensitivity to such choices. Here we use local and global sensitivity analyses to study a model of coagulation and platelet deposition under flow. To relate with clinical assays, we measured the sensitivity of three specific thrombin metrics: lag time, maximum relative rate of generation, and final concentration after 20 minutes. In addition, we varied parameters of three different classes: plasma protein levels, kinetic rate constants, and platelet characteristics. In terms of an overall ranking of the model's sensitivities, we found that the local and global methods provided similar information. Our local analysis, in agreement with previous findings, shows that varying parameters within 50-150% of baseline values, in a one-at-a-time (OAT) fashion, always leads to significant thrombin generation in 20 minutes. Our global analysis gave a different and novel result highlighting groups of parameters, still varying within the normal 50-150%, that produced little or no thrombin in 20 minutes. Variations in either plasma levels or platelet characteristics, using either OAT or simultaneous variations, always led to strong thrombin production and overall, relatively low output variance. Simultaneous variation in kinetics rate constants or in a subset of all three parameter classes led to the highest overall output variance, incorporating instances with little to no thrombin production. The global analysis revealed multiple parameter interactions in the lag time and final concentration leading to relatively high variance; high variance was also observed in the thrombin generation rate, but parameters attributed to that variance acted independently and additively., Competing Interests: The authors have declared that no competing interests exist.

Here, we study the fluid dynamics of a pair of rigid helices rotating at a constant velocity, tethered at their bases, in a viscous fluid. Our computations use a regularized Stokeslet framework, both with and without a bounding plane, so we are able to discern precisely what flow features are unaccounted for in studies that ignore the surface from which the helices emanate. We examine how the spacing and phase difference between identical rotating helices affects their pumping ability, axial thrust, and power requirements. We also find that optimal mixing of the fluid around two helices is achieved when they rotate in opposite phase, and that the mixing is enhanced as the distance between the helices decreases.

Objective: Recent evidence suggests involvement of coagulation factor XIa (FXIa) in thrombotic event development. This study was conducted to explore possible synergies between tissue factor (TF) and exogenous FXIa (E-FXIa) in thrombin generation., Approach and Results: In thrombin generation assays, for increasing concentrations of E-FXIa with low, but not with high TF concentrations, peak thrombin significantly increased whereas lag time and time to peak significantly decreased. Similar dependencies of lag times and rates of thrombin generation were found in mathematical model simulations. In both in vitro and in silico experiments that included E-FXIa, thrombin bursts were seen for TF levels much lower than those required without E-FXIa. For in silico thrombin bursts initiated by the synergistic action of TF and E-FXIa, the mechanisms leading to the burst differed substantially from those for bursts initiated by high TF alone. For the synergistic case, sustained activation of platelet-bound FIX by E-FXIa, along with the feedback-enhanced activation of platelet-bound FVIIIa and FXa, was needed to elicit a thrombin burst. Furthermore, the initiation of thrombin bursts by high TF levels relied on different platelet FIX/FIXa binding sites than those involved in bursts initiated by low TF levels with E-FXIa., Conclusions: Low concentrations of TF and exogenous FXIa, each too low to elicit a burst in thrombin production alone, act synergistically when in combination to cause substantial thrombin production. The observation about FIX/FIXa binding sites may have therapeutic implications., (© 2016 American Heart Association, Inc.)

Many microorganisms swim in a highly heterogeneous environment with obstacles such as fibers or polymers. To better understand how this environment affects microorganism swimming, we study propulsion of a cylinder or filament in a fluid with a sparse, stationary network of obstructions modeled by the Brinkman equation. The mathematical analysis of swimming speeds is investigated by studying an infinite-length cylinder propagating lateral or spiral displacement waves. For fixed bending kinematics, we find that swimming speeds are enhanced due to the added resistance from the fibers. In addition, we examine the work and the torque exerted on the cylinder in relation to the resistance. The solutions for the torque, swimming speed, and work of an infinite-length cylinder in a Stokesian fluid are recovered as the resistance is reduced to zero. Finally, we compare the asymptotic solutions with numerical results for the Brinkman flow with regularized forces. The swimming speed of a finite-length filament decreases as its length decreases and planar bending induces an angular velocity that increases linearly with added resistance. The comparisons between the asymptotic analysis and computation give insight on the effect of the length of the filament, the permeability, and the thickness of the cylinder in terms of the overall performance of planar and helical swimmers.

In the last decade numerous mathematical models have been formulated to investigate specific components of the clotting system such as the tissue factor pathway of coagulation. Sophisticated, multiscale models were developed to better understand the interplay of flow-mediated transport, platelet deposition, and coagulation kinetics, and their overall effect on thrombus formation. Promotion of thrombus growth is partially due to the well-known pro-coagulant roles of platelets like the surface-dependent coagulation reactions. Iterations of theoretical model predictions and experiment have helped to elucidate anticoagulant roles of platelets as well. These roles include paving over the subendothelium and hindering transport of substrates to and from enzyme complexes which can strongly affect thrombus formation. In this review, we give a brief overview of theoretical models of thrombus formation under flow and some of the experiments that motivated them and were motivated by them., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

Clinical evidence suggests that individuals with factor VIII (FVIII) deficiency (hemophilia A) are protected against venous thrombosis, but treatment with recombinant proteins can increase their risk for thrombosis. In this study we examined the dynamics of thrombus formation in individuals with hemophilia A and their response to replacement and bypass therapies under venous flow conditions. Fibrin and platelet accumulation were measured in microfluidic flow assays on a TF-rich surface at a shear rate of 100 s⁻¹. Thrombin generation was calculated with a computational spatial-temporal model of thrombus formation. Mild FVIII deficiencies (5-30% normal levels) could support fibrin fiber formation, while severe (<1%) and moderate (1-5%) deficiencies could not. Based on these experimental observations, computational calculations estimate an average thrombin concentration of ∼10 nM is necessary to support fibrin formation under flow. There was no difference in fibrin formation between severe and moderate deficiencies, but platelet aggregate size was significantly larger for moderate deficiencies. Computational calculations estimate that the local thrombin concentration in moderate deficiencies is high enough to induce platelet activation (>1 nM), but too low to support fibrin formation (<10 nM). In the absence of platelets, fibrin formation was not supported even at normal FVIII levels, suggesting platelet adhesion is necessary for fibrin formation. Individuals treated by replacement therapy, recombinant FVIII, showed normalized fibrin formation. Individuals treated with bypass therapy, recombinant FVIIa, had a reduced lag time in fibrin formation, as well as elevated fibrin accumulation compared to healthy controls. Treatment of rFVIIa, but not rFVIII, resulted in significant changes in fibrin dynamics that could lead to a prothrombotic state.

Vascular injury triggers two intertwined processes, platelet deposition and coagulation, and can lead to the formation of an intravascular clot (thrombus) that may grow to occlude the vessel. Formation of the thrombus involves complex biochemical, biophysical, and biomechanical interactions that are also dynamic and spatially-distributed, and occur on multiple spatial and temporal scales. We previously developed a spatial-temporal mathematical model of these interactions and looked at the interplay between physical factors (flow, transport to the clot, platelet distribution within the blood) and biochemical ones in determining the growth of the clot. Here, we extend this model to include reduction of the advection and diffusion of the coagulation proteins in regions of the clot with high platelet number density. The effect of this reduction, in conjunction with limitations on fluid and platelet transport through dense regions of the clot can be profound. We found that hindered transport leads to the formation of smaller and denser clots compared to the case with no protein hindrance. The limitation on protein transport confines the important activating complexes to small regions in the interior of the thrombus and greatly reduces the supply of substrates to these complexes. Ultimately, this decreases the rate and amount of thrombin production and leads to greatly slowed growth and smaller thrombus size. Our results suggest a possible physical mechanism for limiting thrombus growth.

Virchow's triad is traditionally invoked to explain pathophysiologic mechanisms leading to thrombosis, alleging concerted roles for abnormalities in blood composition, vessel wall components, and blood flow in the development of arterial and venous thrombosis. Given the tissue-specific bleeding observed in hemophilia patients, it may be instructive to consider the principles of Virchow's triad when investigating mechanisms operant in hemostatic disorders as well. Blood composition (the function of circulating blood cells and plasma proteins) is the most well studied component of the triad. For example, increased levels of plasma procoagulant proteins such as prothrombin and fibrinogen are established risk factors for thrombosis, whereas deficiencies in plasma factors VIII and IX result in bleeding (hemophilia A and B, respectively). Vessel wall (cellular) components contribute adhesion molecules that recruit circulating leukocytes and platelets to sites of vascular damage, tissue factor, which provides a procoagulant signal of vascular breach, and a surface upon which coagulation complexes are assembled. Blood flow is often characterized by 2 key variables: shear rate and shear stress. Shear rate affects several aspects of coagulation, including transport rates of platelets and plasma proteins to and from the injury site, platelet activation, and the kinetics of fibrin monomer formation and polymerization. Shear stress modulates adhesion rates of platelets and expression of adhesion molecules and procoagulant activity on endothelial cells lining the blood vessels. That no one abnormality in any component of Virchow's triad fully predicts coagulopathy a priori suggests coagulopathies are complex, multifactorial, and interactive. In this review, we focus on contributions of blood composition, vascular cells, and blood flow to hemostasis and thrombosis, and suggest that cross-talk among the 3 components of Virchow's triad is necessary for hemostasis and determines propensity for thrombosis or bleeding. Investigative models that permit interplay among these components are necessary to understand the operant pathophysiology, and effectively treat and prevent thrombotic and bleeding disorders.

A previously validated mathematical model of intravascular platelet deposition and tissue factor (TF)-initiated coagulation under flow is extended and used to assess the influence on thrombin production of the activation of factor XI (fXI) by thrombin and of the activation of factor IX (fIX) by fXIa. It is found that the importance of the thrombin-fXIa-fIXa feedback loop to robust thrombin production depends on the concentration of platelets in the blood near the injury. At a near-wall platelet concentration of ~250,000/μL, typical in vessels in which the shear rate is <200 s(-1), thrombin activation of fXI makes a significant difference only at low densities of exposed TF. If the near-wall platelet concentration is significantly higher than this, either because of a higher systemic platelet count or because of the redistribution of platelets toward the vessel walls at high shear rates, then thrombin activation of fXI makes a major difference even for relatively high densities of exposed TF. The model predicts that the effect of a severe fXI deficiency depends on the platelet count, and that fXI becomes more important at high platelet counts., (Copyright © 2012 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

The body's response to vascular injury involves two intertwined processes: platelet aggregation and coagulation. Platelet aggregation is a predominantly physical process, whereby platelets clump together, and coagulation is a cascade of biochemical enzyme reactions. Thrombin, the major product of coagulation, directly couples the biochemical system to platelet aggregation by activating platelets and by cleaving fibrinogen into fibrin monomers that polymerize to form a mesh that stabilizes platelet aggregates. Together, the fibrin mesh and the platelet aggregates comprise a thrombus that can grow to occlusive diameters. Transport of coagulation proteins and platelets to and from an injury is controlled largely by the dynamics of the blood flow. To explore how blood flow affects the growth of thrombi and how the growing masses, in turn, feed back and affect the flow, we have developed the first spatial-temporal mathematical model of platelet aggregation and blood coagulation under flow that includes detailed descriptions of coagulation biochemistry, chemical activation and deposition of blood platelets, as well as the two-way interaction between the fluid dynamics and the growing platelet mass. We present this model and use it to explain what underlies the threshold behaviour of the coagulation system's production of thrombin and to show how wall shear rate and near-wall enhanced platelet concentrations affect the development of growing thrombi. By accounting for the porous nature of the thrombus, we also demonstrate how advective and diffusive transport to and within the thrombus affects its growth at different stages and spatial locations.

Flow through the endothelial surface layer (the glycocalyx and adsorbed plasma proteins) plays an important but poorly understood role in cell signaling through a process known as mechanotransduction. Characterizing the flow rates and shear stresses throughout this layer is critical for understanding how flow-induced ionic currents, deformations of transmembrane proteins, and the convection of extracellular molecules signal biochemical events within the cell, including cytoskeletal rearrangements, gene activation, and the release of vasodilators. Previous mathematical models of flow through the endothelial surface layer are based upon the assumptions that the layer is of constant hydraulic permeability and constant height. These models also assume that the layer is continuous across the endothelium and that the layer extends into only a small portion of the vessel lumen. Results of these models predict that fluid shear stress is dissipated through the surface layer and is thus negligible near endothelial cell membranes. In this paper, such assumptions are removed, and the resultant flow rates and shear stresses through the layer are described. The endothelial surface layer is modeled as clumps of a Brinkman medium immersed in a Newtonian fluid. The width and spacing of each clump, hydraulic permeability, and fraction of the vessel lumen occupied by the layer are varied. The two-dimensional Navier-Stokes equations with an additional Brinkman resistance term are solved using a projection method. Several fluid shear stress transitions in which the stress at the membrane shifts from low to high values are described. These transitions could be significant to cell signaling since the endothelial surface layer is likely dynamic in its composition, density, and height.

The path of a macromolecule on a cell membrane is modeled by a sum of independent identically distributed random variables. Random variables with simple discrete distribution functions capture some important aspects of the jump or hop diffusion reported from single particle tracking experiments that measure the motion of single molecules on a cell membrane. The detail provided by the distribution function for the random variables is critical for accurate simulations of the motion and interactions of macromolecules on the cell membrane. Additionally, the probability distribution for the random variables is easily estimated from single-particle tracking data. The diffusion constant is given by the second moment of the probability distribution, which agrees with the diffusion constant estimated from the mean-square displacement, and thus represents far less information than the distribution function.

The flow of information through a cell requires the constant remodeling of cell signaling networks. Thus, spatially and temporally resolved microscopy of signaling components is needed to understand the behavior of normal cells as well as to uncover abnormal behavior leading to human disease. Nanoprobe labeling and transmission electron microscopy of cytoplasmic face-up sheets of cell membrane have been developed as a high-resolution approach to map the interactions of proteins and lipid during cell signaling. Membrane sheets are labeled with 3-15 nm electron-dense probes for receptors, signaling proteins and lipids and micrographs record the distributions of the probes relative to each other and to surface features. Here, we establish computational methods to extract spatial coordinates of probes from micrographs, to analyze and statistically validate the clustering and co-clustering of these probes and to integrate results between experiments in order to establish the relative spatial distributions of single and multiple probes. Our analyses, and the resulting programs for automating data collection and for carrying out statistical and clustering analyses provide toolboxes specialized for the spatiotemporal analysis and modeling of signal transduction pathways.

Ligand binding to membrane receptors initiates cascades of biochemical events leading to physiological responses. Hundreds of proteins and lipids are implicated in signaling networks and programs in genomics and proteomics are continuously adding new components to the signaling "parts lists". Here, we generate high resolution maps of signaling networks using cytoplasmic face-up membrane sheets that can be labeled with immunogold probes (3-10 nm) and imaged in the transmission electron microscope. Our model system is the mast cell and we focus on mapping the topography of the high affinity IgE receptor, Fc(epsilon)RI, its associated tyrosine kinases, Lyn and Syk, and the signaling proteins that propagate signals from these kinases. Crosslinked receptors and their signaling partners segregate during signaling to multiple, dynamic membrane domains, including a transient Fc(epsilon)RI-Lyn domain and at least two other distinct domains, one characterized by the presence of receptor, Syk and multiple signaling proteins, but not Lyn (primary signaling domains), and one characterized by the presence of LAT and PLCgamma1 but not receptor (secondary signaling domains). PI 3-kinase associates with both primary and secondary signaling domains and may help to recruit specific signaling proteins through the local remodeling of inositol phospholipids. The lipid raft markers, GM1 and Thy-1, fail to localize in native membrane sheets either with each other or with signaling domains. We introduce new probes to localize multiple signaling molecules on the same membrane sheet and new computational tools to capture and analyze their topographical relationships. In the future, we expect that high resolution maps of signaling networks will be integrated with chemical kinetic analyses, with cell fractionation data and with a range of real-time fluorescence measurements, into mathematical models with power to predict mechanisms that regulate the efficiency, specificity, amplitude and duration of signaling pathways.