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1.
Blood ; 143(6): 548-560, 2024 Feb 08.
Artículo en Inglés | MEDLINE | ID: mdl-37944157

RESUMEN

ABSTRACT: Nonmuscle cell contractility is an essential feature underlying diverse cellular processes such as motility, morphogenesis, division and genome replication, intracellular transport, and secretion. Blood clot contraction is a well-studied process driven by contracting platelets. Megakaryocytes (MKs), which are the precursors to platelets, can be found in bone marrow and lungs. Although they express many of the same proteins and structures found in platelets, little is known about their ability to engage with extracellular proteins such as fibrin and contract. Here, we have measured the ability of MKs to compress plasma clots. Megakaryocytes derived from human induced pluripotent stem cells (iPSCs) were suspended in human platelet-free blood plasma and stimulated with thrombin. Using real-time macroscale optical tracking, confocal microscopy, and biomechanical measurements, we found that activated iPSC-derived MKs (iMKs) caused macroscopic volumetric clot shrinkage, as well as densification and stiffening of the fibrin network via fibrin-attached plasma membrane protrusions undergoing extension-retraction cycles that cause shortening and bending of fibrin fibers. Contraction induced by iMKs involved 2 kinetic phases with distinct rates and durations. It was suppressed by inhibitors of nonmuscle myosin IIA, actin polymerization, and integrin αIIbß3-fibrin interactions, indicating that the molecular mechanisms of iMK contractility were similar or identical to those in activated platelets. Our findings provide new insights into MK biomechanics and suggest that iMKs can be used as a model system to study platelet contractility. Physiologically, the ability of MKs to contract plasma clots may play a role in the mechanical remodeling of intravascular blood clots and thrombi.


Asunto(s)
Células Madre Pluripotentes Inducidas , Trombosis , Humanos , Megacariocitos/metabolismo , Células Madre Pluripotentes Inducidas/metabolismo , Plaquetas/metabolismo , Trombosis/metabolismo , Fibrina/metabolismo , Plasma
2.
Commun Biol ; 6(1): 869, 2023 08 24.
Artículo en Inglés | MEDLINE | ID: mdl-37620422

RESUMEN

While blood clot formation has been relatively well studied, little is known about the mechanisms underlying the subsequent structural and mechanical clot remodeling called contraction or retraction. Impairment of the clot contraction process is associated with both life-threatening bleeding and thrombotic conditions, such as ischemic stroke, venous thromboembolism, and others. Recently, blood clot contraction was observed to be hindered in patients with COVID-19. A three-dimensional multiscale computational model is developed and used to quantify biomechanical mechanisms of the kinetics of clot contraction driven by platelet-fibrin pulling interactions. These results provide important biological insights into contraction of platelet filopodia, the mechanically active thin protrusions of the plasma membrane, described previously as performing mostly a sensory function. The biomechanical mechanisms and modeling approach described can potentially apply to studying other systems in which cells are embedded in a filamentous network and exert forces on the extracellular matrix modulated by the substrate stiffness.


Asunto(s)
COVID-19 , Trombosis , Humanos , Plaquetas , Simulación por Computador , Fibrina
3.
iScience ; 25(7): 104654, 2022 Jul 15.
Artículo en Inglés | MEDLINE | ID: mdl-35832887

RESUMEN

Although septins have been well-studied in nucleated cells, their role in anucleate blood platelets remains obscure. Here, we elucidate the contribution of septins to human platelet structure and functionality. We show that Septin-2 and Septin-9 are predominantly distributed at the periphery of resting platelets and co-localize strongly with microtubules. Activation of platelets by thrombin causes clustering of septins and impairs their association with microtubules. Inhibition of septin dynamics with forchlorfenuron (FCF) reduces thrombin-induced densification of septins and lessens their colocalization with microtubules in resting and activated platelets. Exposure to FCF alters platelet shape, suggesting that septins stabilize platelet cytoskeleton. FCF suppresses platelet integrin αIIbß3 activation, promotes phosphatidylserine exposure on activated platelets, and induces P-selectin expression on resting platelets, suggesting septin involvement in these processes. Inhibition of septin dynamics substantially reduces platelet contractility and abrogates their spreading on fibrinogen-coated surfaces. Overall, septins strongly contribute to platelet structure, activation and biomechanics.

4.
Artículo en Inglés | MEDLINE | ID: mdl-35386550

RESUMEN

Fibrin deformation and interaction of fibrin with other blood components play critical roles in hemostasis and thrombosis. In this review, computational and mathematical biomechanical models of fibrin network deformation and contraction at different spatio-temporal scales as well as challenges in developing and calibrating multiscale models are discussed. There are long standing challenges. For instance, applicability of models to identify and test potential mechanisms of the biomechanical processes mediating interactions between platelets and fiber networks in blood clot stretching and contraction needs to be examined carefully. How the structural and mechanical properties of major blood clot components influences biomechanical responses of the entire clot subjected to external forces, such as blood flow or vessel wall deformations needs to be investigated thoroughly.

5.
Blood Adv ; 5(23): 4901-4909, 2021 12 14.
Artículo en Inglés | MEDLINE | ID: mdl-34570183

RESUMEN

Blood clot contraction is driven by traction forces generated by the platelet cytoskeleton that are transmitted to fibrin fibers via the integrin αIIbß3. Here we show that clot contraction is impaired by inhibitors of the platelet cytosolic protease calpain. We used subtiligase-mediated labeling of amino termini and mass spectrometry to identify proteolytically cleaved platelet proteins involved in clot contraction. Of 32 calpain-cleaved proteins after TRAP stimulation, 14 were cytoskeletal, most prominently talin and vinculin. A complex of talin and vinculin constitutes a mechanosensitive clutch connecting integrins bound to the extracellular matrix with the actin cytoskeleton. Accordingly, we focused on talin and vinculin. Talin is composed of an N-terminal head domain and a C-terminal rod domain organized into a series of 4- and 5-helix bundles. The bundles contain 11 vinculin binding sites (VBSs), each of which is an α-helix packed into a bundle interior and requiring structural rearrangement to initiate vinculin binding. We detected 8 calpain-mediated cleavages in talin, 2 previously identified in unstructured regions and 6 in α-helical regions in proximity to a VBS. There is evidence in vitro that applying mechanical force across talin enables vinculin binding to the talin rod. However, we found that inhibiting platelet cytoskeletal contraction had no effect on talin cleavage, indicating that talin cleavage by calpain in platelets does not require cytoskeleton-generated tensile force. Therefore, it is likely that calpain acts in the later stages of clot retraction through focal adhesion disassembly.


Asunto(s)
Talina , Trombosis , Sitios de Unión , Calpaína , Fibrina , Humanos , Talina/metabolismo
6.
Biochemistry ; 60(21): 1722-1730, 2021 06 01.
Artículo en Inglés | MEDLINE | ID: mdl-34010565

RESUMEN

The fluorescent reporters commonly used to visualize proteins can perturb both protein structure and function. Recently, we found that 4-cyanotryptophan (4CN-Trp), a blue fluorescent amino acid, is suitable for one-photon imaging applications. Here, we demonstrate its utility in two-photon fluorescence microscopy by using it to image integrins on cell surfaces. Specifically, we used solid-phase peptide synthesis to generate CHAMP peptides labeled with 4-cyanoindole (4CNI) at their N-termini to image integrins on cell surfaces. CHAMP (computed helical anti-membrane protein) peptides spontaneously insert into membrane bilayers to target integrin transmembrane domains and cause integrin activation. We found that 4CNI labeling did not perturb the ability of CHAMP peptides to insert into membranes, bind to integrins, or cause integrin activation. We then used two-photon fluorescence microscopy to image 4CNI-containing integrins on the surface of platelets. Compared to a 4CNI-labeled scrambled peptide that uniformly decorated cell surfaces, 4CNI-labeled CHAMP peptides were present in discrete blue foci. To confirm that these foci represented CN peptide-containing integrins, we co-stained platelets with integrin-specific fluorescent monoclonal antibodies and found that CN peptide and antibody fluorescence coincided. Because 4CNI can readily be biosynthetically incorporated into proteins with little if any effect on protein structure and function, it provides a facile way to directly monitor protein behavior and protein-protein interactions in cellular environments. In addition, these results clearly demonstrate that the two-photon excitation cross section of 4CN-Trp is sufficiently large to make it a useful two-photon fluorescence reporter for biological applications.


Asunto(s)
Integrinas/metabolismo , Microscopía de Fluorescencia por Excitación Multifotónica/métodos , Triptófano/análogos & derivados , Aminoácidos/metabolismo , Plaquetas/metabolismo , Membrana Celular/metabolismo , Integrinas/fisiología , Péptidos/síntesis química , Péptidos/química , Complejo GPIIb-IIIa de Glicoproteína Plaquetaria/metabolismo , Unión Proteica/fisiología , Dominios Proteicos/fisiología , Triptófano/síntesis química , Triptófano/química
8.
Haematologica ; 104(9): 1866-1878, 2019 09.
Artículo en Inglés | MEDLINE | ID: mdl-30792211

RESUMEN

Platelets play a key role in the formation of hemostatic clots and obstructive thrombi as well as in other biological processes. In response to physiological stimulants, including thrombin, platelets change shape, express adhesive molecules, aggregate, and secrete bioactive substances, but their subsequent fate is largely unknown. Here we examined late-stage structural, metabolic, and functional consequences of thrombin-induced platelet activation. Using a combination of confocal microscopy, scanning and transmission electron microscopy, flow cytometry, biochemical and biomechanical measurements, we showed that thrombin-induced activation is followed by time-dependent platelet dysfunction and disintegration. After ~30 minutes of incubation with thrombin, unlike with collagen or ADP, human platelets disintegrated into cellular fragments containing organelles, such as mitochondria, glycogen granules, and vacuoles. This platelet fragmentation was preceded by Ca2+ influx, integrin αIIbß3 activation and phosphatidylserine exposure (activation phase), followed by mitochondrial depolarization, generation of reactive oxygen species, metabolic ATP depletion and impairment of platelet contractility along with dramatic cytoskeletal rearrangements, concomitant with platelet disintegration (death phase). Coincidentally with the platelet fragmentation, thrombin caused calpain activation but not activation of caspases 3 and 7. Our findings indicate that the late functional and structural damage of thrombin-activated platelets comprise a calpain-dependent platelet death pathway that shares some similarities with the programmed death of nucleated cells, but is unique to platelets, therefore representing a special form of cellular destruction. Fragmentation of activated platelets suggests that there is an underappreciated pathway of enhanced elimination of platelets from the circulation in (pro)thrombotic conditions once these cells have performed their functions.


Asunto(s)
Plaquetas/inmunología , Muerte Celular , Activación Plaquetaria/efectos de los fármacos , Trombina/farmacología , Adenosina Difosfato/metabolismo , Adenosina Trifosfato/metabolismo , Coagulación Sanguínea/efectos de los fármacos , Plaquetas/metabolismo , Calcio/metabolismo , Colágeno/metabolismo , Citoesqueleto/metabolismo , Citometría de Flujo , Humanos , Microscopía Confocal , Microscopía Electrónica de Rastreo , Microscopía Electrónica de Transmisión , Agregación Plaquetaria/efectos de los fármacos , Plasma Rico en Plaquetas/metabolismo , Especies Reactivas de Oxígeno/metabolismo
9.
Nat Commun ; 8(1): 1274, 2017 11 02.
Artículo en Inglés | MEDLINE | ID: mdl-29097692

RESUMEN

Blood clot contraction plays an important role in prevention of bleeding and in thrombotic disorders. Here, we unveil and quantify the structural mechanisms of clot contraction at the level of single platelets. A key elementary step of contraction is sequential extension-retraction of platelet filopodia attached to fibrin fibers. In contrast to other cell-matrix systems in which cells migrate along fibers, the "hand-over-hand" longitudinal pulling causes shortening and bending of platelet-attached fibers, resulting in formation of fiber kinks. When attached to multiple fibers, platelets densify the fibrin network by pulling on fibers transversely to their longitudinal axes. Single platelets and aggregates use actomyosin contractile machinery and integrin-mediated adhesion to remodel the extracellular matrix, inducing compaction of fibrin into bundled agglomerates tightly associated with activated platelets. The revealed platelet-driven mechanisms of blood clot contraction demonstrate an important new biological application of cell motility principles.


Asunto(s)
Plaquetas/metabolismo , Fibrina/metabolismo , Seudópodos/metabolismo , Trombosis/metabolismo , Abciximab , Actomiosina/metabolismo , Anticuerpos Monoclonales/farmacología , Fenómenos Biomecánicos , Plaquetas/efectos de los fármacos , Plaquetas/patología , Plaquetas/fisiología , Adhesión Celular/fisiología , Humanos , Procesamiento de Imagen Asistido por Computador , Fragmentos Fab de Inmunoglobulinas/farmacología , Microscopía Confocal , Microscopía Fluorescente , Inhibidores de Agregación Plaquetaria/farmacología , Complejo GPIIb-IIIa de Glicoproteína Plaquetaria/antagonistas & inhibidores , Complejo GPIIb-IIIa de Glicoproteína Plaquetaria/metabolismo , Seudópodos/efectos de los fármacos , Seudópodos/patología , Seudópodos/fisiología , Trombosis/patología
10.
J R Soc Interface ; 14(136)2017 11.
Artículo en Inglés | MEDLINE | ID: mdl-29142014

RESUMEN

Thromboembolism, one of the leading causes of morbidity and mortality worldwide, is characterized by formation of obstructive intravascular clots (thrombi) and their mechanical breakage (embolization). A novel two-dimensional multi-phase computational model is introduced that describes active interactions between the main components of the clot, including platelets and fibrin, to study the impact of various physiologically relevant blood shear flow conditions on deformation and embolization of a partially obstructive clot with variable permeability. Simulations provide new insights into mechanisms underlying clot stability and embolization that cannot be studied experimentally at this time. In particular, model simulations, calibrated using experimental intravital imaging of an established arteriolar clot, show that flow-induced changes in size, shape and internal structure of the clot are largely determined by two shear-dependent mechanisms: reversible attachment of platelets to the exterior of the clot and removal of large clot pieces. Model simulations predict that blood clots with higher permeability are more prone to embolization with enhanced disintegration under increasing shear rate. In contrast, less permeable clots are more resistant to rupture due to shear rate-dependent clot stiffening originating from enhanced platelet adhesion and aggregation. These results can be used in future to predict risk of thromboembolism based on the data about composition, permeability and deformability of a clot under specific local haemodynamic conditions.


Asunto(s)
Modelos Cardiovasculares , Resistencia al Corte , Tromboembolia/metabolismo , Trombosis/metabolismo , Humanos , Permeabilidad , Tromboembolia/patología , Tromboembolia/fisiopatología , Trombosis/patología , Trombosis/fisiopatología
11.
Sci Rep ; 7(1): 13001, 2017 10 11.
Artículo en Inglés | MEDLINE | ID: mdl-29021578

RESUMEN

The formation of platelet thrombi is determined by the integrin αIIbß3-mediated interactions of platelets with fibrinogen and fibrin. Blood clotting in vivo is catalyzed by thrombin, which simultaneously induces fibrinogen binding to αIIbß3 and converts fibrinogen to fibrin. Thus, after a short time, thrombus formation is governed by αIIbß3 binding to fibrin fibers. Surprisingly, there is little understanding of αIIbß3 interaction with fibrin polymers. Here we used an optical trap-based system to measure the binding of single αIIbß3 molecules to polymeric fibrin and compare it to αIIbß3 binding to monomeric fibrin and fibrinogen. Like αIIbß3 binding to fibrinogen and monomeric fibrin, we found that αIIbß3 binding to polymeric fibrin can be segregated into two binding regimes, one with weaker rupture forces of 30-60 pN and a second with stronger rupture forces >60 pN that peaked at 70-80 pN. However, we found that the mechanical stability of the bimolecular αIIbß3-ligand complexes had the following order: fibrin polymer > fibrin monomer > fibrinogen. These quantitative differences reflect the distinct specificity and underlying molecular mechanisms of αIIbß3-mediated reactions, implying that targeting platelet interactions with fibrin could increase the therapeutic indices of antithrombotic agents by focusing on the destabilization of thrombi rather than the prevention of platelet aggregation.


Asunto(s)
Coagulación Sanguínea , Fibrina/metabolismo , Complejo GPIIb-IIIa de Glicoproteína Plaquetaria/metabolismo , Trombosis/patología , Coagulación Sanguínea/efectos de los fármacos , Humanos , Cinética , Manganeso/farmacología , Modelos Biológicos , Plasma Rico en Plaquetas/metabolismo , Polimerizacion , Probabilidad , Unión Proteica/efectos de los fármacos
12.
Biomech Model Mechanobiol ; 15(1): 213-228, 2016 Feb.
Artículo en Inglés | MEDLINE | ID: mdl-25982442

RESUMEN

The rheological properties of fibrin networks have been of long-standing interest. As such there is a wealth of studies of their shear and tensile responses, but their compressive behavior remains unexplored. Here, by characterization of the network structure with synchronous measurement of the fibrin storage and loss moduli at increasing degrees of compression, we show that the compressive behavior of fibrin networks is similar to that of cellular solids. A nonlinear stress-strain response of fibrin consists of three regimes: (1) an initial linear regime, in which most fibers are straight, (2) a plateau regime, in which more and more fibers buckle and collapse, and (3) a markedly nonlinear regime, in which network densification occurs by bending of buckled fibers and inter-fiber contacts. Importantly, the spatially non-uniform network deformation included formation of a moving "compression front" along the axis of strain, which segregated the fibrin network into compartments with different fiber densities and structure. The Young's modulus of the linear phase depends quadratically on the fibrin volume fraction while that in the densified phase depends cubically on it. The viscoelastic plateau regime corresponds to a mixture of these two phases in which the fractions of the two phases change during compression. We model this regime using a continuum theory of phase transitions and analytically predict the storage and loss moduli which are in good agreement with the experimental data. Our work shows that fibrin networks are a member of a broad class of natural cellular materials which includes cancellous bone, wood and cork.


Asunto(s)
Fuerza Compresiva , Fibrina/metabolismo , Fenómenos Biomecánicos , Módulo de Elasticidad , Humanos , Dinámicas no Lineales
13.
Adv Exp Med Biol ; 844: 85-98, 2014.
Artículo en Inglés | MEDLINE | ID: mdl-25480638

RESUMEN

Platelets are small, anucleated cells that participate in primary hemostasis by forming a hemostatic plug at the site of a blood vessel's breach, preventing blood loss. However, hemostatic events can lead to excessive thrombosis, resulting in life-threatening strokes, emboli, or infarction. Development of multi-scale models coupling processes at several scales and running predictive model simulations on powerful computer clusters can help interdisciplinary groups of researchers to suggest and test new patient-specific treatment strategies.


Asunto(s)
Plaquetas/fisiología , Vasos Sanguíneos/fisiología , Comunicación Celular , Biología de Sistemas , Animales , Coagulación Sanguínea/fisiología , Plaquetas/citología , Vasos Sanguíneos/citología , Hemostasis/fisiología , Humanos , Activación Plaquetaria , Adhesividad Plaquetaria
14.
Biomaterials ; 35(25): 6739-49, 2014 Aug.
Artículo en Inglés | MEDLINE | ID: mdl-24840618

RESUMEN

Fibrin is a protein polymer that forms a 3D filamentous network, a major structural component of protective physiological blood clots as well as life threatening pathological thrombi. It plays an important role in wound healing, tissue regeneration and is widely employed in surgery as a sealant and in tissue engineering as a scaffold. The goal of this study was to establish correlations between structural changes and mechanical responses of fibrin networks exposed to compressive loads. Rheological measurements revealed nonlinear changes of fibrin network viscoelastic properties under dynamic compression, resulting in network softening followed by its dramatic hardening. Repeated compression/decompression enhanced fibrin clot stiffening. Combining fibrin network rheology with simultaneous confocal microscopy provided direct evidence of structural modulations underlying nonlinear viscoelasticity of compressed fibrin networks. Fibrin clot softening in response to compression strongly correlated with fiber buckling and bending, while hardening was associated with fibrin network densification. Our results suggest a complex interplay of entropic and enthalpic mechanisms accompanying structural changes and accounting for the nonlinear mechanical response in fibrin networks undergoing compressive deformations. These findings provide new insight into the fibrin clot structural mechanics and can be useful for designing fibrin-based biomaterials with modulated viscoelastic properties.


Asunto(s)
Materiales Biocompatibles/química , Coagulación Sanguínea/fisiología , Fibrina/química , Estrés Mecánico , Fenómenos Biomecánicos , Humanos , Procesamiento de Imagen Asistido por Computador , Microscopía Confocal , Polímeros/química , Reología , Sustancias Viscoelásticas/química
15.
Proc Natl Acad Sci U S A ; 110(51): 20449-54, 2013 Dec 17.
Artículo en Inglés | MEDLINE | ID: mdl-24284166

RESUMEN

Regulation of microtubule dynamic instability is crucial for cellular processes, ranging from mitosis to membrane transport. Stathmin (also known as oncoprotein 18/Op18) is a prominent microtubule destabilizer that acts preferentially on microtubule minus ends. Stathmin has been studied intensively because of its association with multiple types of cancer, but its mechanism of action remains controversial. Two models have been proposed. One model is that stathmin promotes microtubule catastrophe indirectly, and does so by sequestering tubulin; the other holds that stathmin alters microtubule dynamics by directly destabilizing growing microtubules. Stathmin's sequestration activity is well established, but the mechanism of any direct action is mysterious because stathmin binds to microtubules very weakly. To address these issues, we have studied interactions between stathmin and varied tubulin polymers. We show that stathmin binds tightly to Dolastatin-10 tubulin rings, which mimic curved tubulin protofilaments, and that stathmin depolymerizes stabilized protofilament-rich polymers. These observations lead us to propose that stathmin promotes catastrophe by binding to and acting upon protofilaments exposed at the tips of growing microtubules. Moreover, we suggest that stathmin's minus-end preference results from interactions between stathmin's N terminus and the surface of α-tubulin that is exposed only at the minus end. Using computational modeling of microtubule dynamics, we show that these mechanisms could account for stathmin's observed activities in vitro, but that both the direct and sequestering activities are likely to be relevant in a cellular context. Taken together, our results suggest that stathmin can promote catastrophe by direct action on protofilament structure and interactions.


Asunto(s)
Microtúbulos/química , Simulación de Dinámica Molecular , Estatmina/química , Tubulina (Proteína)/química , Animales , Depsipéptidos/química , Humanos , Microtúbulos/metabolismo , Unión Proteica , Estatmina/metabolismo , Porcinos , Tubulina (Proteína)/metabolismo
16.
PLoS Comput Biol ; 9(6): e1003095, 2013.
Artículo en Inglés | MEDLINE | ID: mdl-23785270

RESUMEN

Thromboembolic disease is a leading cause of morbidity and mortality worldwide. In the last several years there have been a number of studies attempting to identify mechanisms that stop thrombus growth. This paper identifies a novel mechanism related to formation of a fibrin cap. In particular, protein transport through a fibrin network, an important component of a thrombus, was studied by integrating experiments with model simulations. The network permeability and the protein diffusivity were shown to be important factors determining the transport of proteins through the fibrin network. Our previous in vivo studies in mice have shown that stabilized non-occluding thrombi are covered by a fibrin network ('fibrin cap'). Model simulations, calibrated using experiments in microfluidic devices and accounting for the permeable structure of the fibrin cap, demonstrated that thrombin generated inside the thrombus was washed downstream through the fibrin network, thus limiting exposure of platelets on the thrombus surface to thrombin. Moreover, by restricting the approach of resting platelets in the flowing blood to the thrombus core, the fibrin cap impaired platelets from reaching regions of high thrombin concentration necessary for platelet activation and limited thrombus growth. The formation of a fibrin cap prevents small thrombi that frequently develop in the absence of major injury in the 60000 km of vessels in the body from developing into life threatening events.


Asunto(s)
Fibrina/metabolismo , Proteínas/metabolismo , Trombosis/patología , Animales , Hemodinámica , Ratones , Microfluídica/instrumentación , Transporte de Proteínas
17.
Langmuir ; 26(20): 15807-13, 2010 Oct 19.
Artículo en Inglés | MEDLINE | ID: mdl-20839855

RESUMEN

Diameter, velocity, and charge measurements of progeny droplets produced in-flight by a millimeter-size parent drop subjected to electric and ionic fields are reported. Different drop breakup modes were studied using phase doppler anemometry and high-speed digital photography. Drop breakup occurred in applied electric (∼1 kV/cm to ∼10 kV/cm) and ionic (∼10(13)/m(3) to ∼10(15)/m(3)) fields that were generated using a DC-corona discharge in a needle-plate configuration. Effects of the external electric field and the diameter of the parent drop are considered. Several models are summarized, including simulations of the electrohydrodynamics of the corona discharge, electrocapillary stability analysis of the jet, and progeny droplets mobility analysis. Using experimental and model results, the charge of progeny drops is shown to vary as the three-halves power of their diameter.

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