ARTICLE https://doi.org/10.1038/s42003-023-04560-4 OPEN A mathematical model of fibrinogen-mediated erythrocyte–erythrocyte adhesion Catarina S. Lopes 1,8, Juliana Curty2,8, Filomena A. Carvalho1, A. Hernández-Machado3,4,5,6, Koji Kinoshita 7, Nuno C. Santos 1 ✉ & Rui D. M. Travasso 2 ✉ Erythrocytes are deformable cells that undergo progressive biophysical and biochemical changes affecting the normal blood flow. Fibrinogen, one of the most abundant plasma proteins, is a primary determinant for changes in haemorheological properties, and a major independent risk factor for cardiovascular diseases. In this study, the adhesion between 1234567890():,; human erythrocytes is measured by atomic force microscopy (AFM) and its effect observed by micropipette aspiration technique, in the absence and presence of fibrinogen. These experimental data are then used in the development of a mathematical model to examine the biomedical relevant interaction between two erythrocytes. Our designed mathematical model is able to explore the erythrocyte–erythrocyte adhesion forces and changes in erythrocyte morphology. AFM erythrocyte–erythrocyte adhesion data show that the work and detach- ment force necessary to overcome the adhesion between two erythrocytes increase in the presence of fibrinogen. The changes in erythrocyte morphology, the strong cell-cell adhesion and the slow separation of the two cells are successfully followed in the mathematical simulation. Erythrocyte-erythrocyte adhesion forces and energies are quantified and matched with experimental data. The changes observed on erythrocyte–erythrocyte interactions may give important insights about the pathophysiological relevance of fibrinogen and erythrocyte aggregation in hindering microcirculatory blood flow. 1 Instituto de Medicina Molecular, Faculdade de Medicina, Universidade de Lisboa, Lisbon, Portugal. 2 CFisUC, Department of Physics, University of Coimbra, Coimbra, Portugal. 3 Departament de Física de la Matèria Condensada, Facultat de Física, Universitat de Barcelona, Barcelona, Spain. 4 Centre de Recerca Matemàtica, Bellaterra, Spain. 5 Barcelona Graduate School of Mathematics (BGSMath), Barcelona, Spain. 6 Institute of Nanoscience and Nanotechnology (IN2UB), Universitat de Barcelona, Barcelona, Spain. 7 Department of Molecular Medicine, University of Southern Denmark, Odense, Denmark. 8These authors contributed equally: Catarina S. Lopes, Juliana Curty. ✉email: nsantos@fm.ul.pt; ruit@uc.pt COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio 1 ARTICLE COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 E rythrocytes are adaptable cells that undergo progressive between cells and the extracellular matrix46, and cell deformation physical and chemical changes during the inflammatory and motility42–44,46. Interestingly, the model proposed by Non- pathophysiology in the vascular system1. Biochemical omura (2012)42,45 can be applied to investigate cellular mor- changes can be noticed in erythrocytes, such as the decrease on phology alterations due to cell–cell interaction, e.g., in epithelial cell volume with cell aging. Among other factors, this is pre- tissues, in tumour growth, or as a consequence of differential sumably due to the loss of potassium and the loss of membrane adhesion between cell types. patches by microvesiculation, resulting in an increase on cell In this study, we present a theoretical model for the evolution of density1,2. Erythrocyte deformability is an important factor two erythrocytes’ mechanical interaction while they are pushed affecting the high shear blood viscosity. Under normal conditions, against each other, and then pulled apart. We used the experimental erythrocytes deform sufficiently to cross blood vessels of different assays (AFM and micropipette aspiration technique) to measure the calibres3. The erythrocytes’ membrane, forming a complex with erythrocytes typical adhesion force between two cells (Fig. 1). the cytoskeleton, may also exhibit plastic changes under some Afterwards, we used these values to parameterise the model and to pathological circumstances and can be permanently deformed by explore in more detail the cell–cell interactions as a function of time. excessive shear forces4. Mechanical properties of erythrocytes In the Results and Discussion section, we describe the observations strongly affect blood flow. For example, low erythrocyte from the micropipette aspiration assays (Fig. 1a) and the AFM deformability increases blood viscosity and flow resistance, which measurements (Fig. 1b) carried out to explore the adhesion between lead to several cardiovascular and cerebrovascular conditions5. erythrocytes in the presence of fibrinogen. We also show how our Interestingly, erythrocyte adhesion mediated by receptors is mathematical model is able to explore the cell–cell interaction forces crucial in haematological diseases such as sickle cell anaemia6, and cell morphology alterations during the cell–cell interaction. In malaria7 and type 2 diabetes mellitus8. Blood platelets also drive the Conclusions section, we discuss the results of this work and draw erythrocyte aggregation, leading to clot formation. This the final conclusions. We describe the experimental setup and the mechanism is crucial in wound healing or thrombus-induced mathematical model in the “Methods” section. strokes. In healthy donors, erythrocytes have a tendency to form aggregates called rouleaux that look similar to a stack of coins when no flow is applied9. These rouleaux can be separated into Results and discussion smaller units by means of a shear flow, giving rise to the shear Erythrocyte–erythrocyte adhesion evaluation by micropipette thinning behaviour of blood. experiments. Cell–cell adhesion studies were performed with and Despite erythrocytes being a major component of thrombi9, in without the addition of fibrinogen (1.0 mg/mL), evaluating the blood, fibrinogen (one of the most abundant plasma proteins) is a binding between two erythrocytes from healthy donors (Fig. 2). The primary determinant of changes in haemorheological properties, and total amount of erythrocyte pairs analysed were 32 and 34 with and these alterations exacerbate the complications in peripheral blood without fibrinogen addition, respectively. For each erythrocyte pair, circulation during cardiovascular pathologies10. Fibrinogen is we obtained 16 ± 5 adhesion image sequences, with a 15 s cycle each, involved in the coagulation cascade, it has a crucial role in blood and determined if adhesion between the cells was observed. Often it clotting and has been identified as a major independent risk factor for can be visualised that this adhesion leads to a pull of the membrane cardiovascular diseases11–20. An increase in fibrinogen plasma con- of the erythrocyte of the left side (static micropipette) when the centration promotes erythrocyte aggregation21. Using atomic force erythrocyte from right micropipette is being moved away from it. microscopy (AFM)-based force spectroscopy, we previously identi- Moreover, in those situations it can be observed that the adhesion fied the specific receptor for fibrinogen on the erythrocyte between the two erythrocytes forces the erythrocyte on the right to membrane12,17,22. The simultaneous binding of fibrinogen to such bend to the left as it is being pulled away. In Fig. 2 we present seven receptors on two different erythrocytes may transiently bridge these representative images from three cycles to illustrate cell–cell inter- cells, increasing blood viscosity at the microcirculatory level, action in events where no adhesion is observed (with and without impairing a proper blood flow and increasing vascular risk. fibrinogen addition, Fig. 2a, b, respectively), and in an event where Erythrocyte–erythrocyte adhesion has been measured by different adhesion between the cells is present (Fig. 2c). Events with changes in approaches, including AFM12,17,22. Micropipette aspiration, devel- erythrocyte shape were often absent in the sequences without fibri- oped by Evan Evans in the 1980s23–25, may also be used. It has been nogen, while they were frequently observed when fibrinogen was extensively used to measure the mechanical properties of erythrocyte added to the experiment (Fig. 2d). membranes, including membrane elastic modulus and membrane viscosity26,27, to analyse the mechanical properties of bilayers28, the Erythrocyte–erythrocyte adhesion evaluation by AFM experi- mechanical and adhesive properties of single cells on substrates or ments. Cell–cell adhesion was also evaluated by AFM-based force cell–cell adhesion29–31. Cell morphology (cell shape and spreading) spectroscopy experiments (Fig. 3). It is observed that the work can be visualised during experiments and it is also an initial indi- necessary to overcome the adhesion of two erythrocytes is lower in cation of the propensity for adhesion32. the absence of fibrinogen than in the presence of 1.0 mg/mL of Phase-field models have been used by the physical community fibrinogen (0.63 ± 0.12 fJ vs. 1.45 ± 0.34 fJ, respectively; p = 0.030; to study domain formation and growth in phase separating non- Fig. 3a). The erythrocyte–erythrocyte maximum detachment force equilibrium systems33–35. These models are particularly useful in was also quantified in the same conditions (Fig. 3b). Without fibri- describing the dynamical properties of the interface or membrane nogen, the maximum detachment force between erythrocytes was separating different domains. In biology, phase-field models have 168.7 ± 33.9 pN, which was lower than in the presence of fibrinogen been extensively used to describe tumour growth36,37, vessel 1.0 mg/mL (330.5 ± 65.2 pN; p = 0.030; Fig. 3b). Thus, fibrinogen growth38–41, as well as cell deformation and dynamics42–46. promotes a stronger erythrocyte adhesion. Mathematical models based on the minimisation of energy functionals are efficient in determining cell morphology evolu- tion. Phase-field models33–35, in particular, are especially useful in Simulation results. The simulation of the micropipette assay was describing the dynamical properties of the interface or membrane then carried out using the mathematical model described by Eq. separating different domains. In biology, they have been exten- (2) (see Methods section). Initially, the model was used to equi- sively used to describe tumour growth36,37, vessel growth38–41,47, librate two erythrocytes with the target volume and surface area multiscale modelling of thrombus in blood vessels48, interactions from Table 1. 2 COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 ARTICLE Fig. 1 Schematic experimental setup representation of the erythrocyte–erythrocyte adhesion experiments carried out using micropipette aspiration and atomic force microscopy. Using the micropipette aspiration technique (a), a negative pressure (suction) is applied to catch an erythrocyte in each micropipette. One of these micropipettes (depicted as the one on the left side of each panel from I to IV) is kept static. The piezo-driven micropipette, with an attached erythrocyte, approaches the other erythrocyte on the static micropipette, until a certain contact force is reached (I). Then, during the retraction of the right side pipette, cell–cell adhesion may occur, leading to the pulling of the surface of the erythrocyte from the left side when the erythrocyte from right side micropipette is being moved away from it (II and III), until the complete erythrocyte–erythrocyte detachment is reached (IV). Using atomic force microscopy (AFM) (b), the tipless cantilever with an attached erythrocyte is approached to other erythrocyte on the substrate, until a certain contact force is reached (I). Afterwards, during retraction, cell–cell adhesion may occur, causing the cantilever to adhere to the sample up to a distance beyond the initial contact point (II). When the applied force overcomes the cell–cell interaction forces, the cantilever pulls of sharply (III), before totally breaking the cell–cell contact and complete the erythrocyte detachment (IV). According to the minimisation of the Canham-Helfrich energy, of erythrocytes profiles (Fig. 4a), and as a 3D representation S 3=2 (Fig. 4b). The total force exerted on the right erythrocyte along the ratio ν ¼ V target =ð43 πð target 4π Þ Þ between the enclosed volume, the three axes is plotted in Fig. 4c as a function of time. In this Vtarget, and the volume of a sphere with the same surface area, Starget, frontal collision, due to the symmetry of the collision, we observe as the vesicle determines its equilibrium shape49,50. For ν ≳ 0.65, the a zero total force in directions x and y. On the other hand, along vesicle presents a prolate shape, approaching a sphere when ν ~ 1. the collision axis, the erythrocytes initially undergo an attractive For ν ≲ 0.55, the vesicle free-energy is minimised by bending into a force, as they approach each other (first minimum in Fig. 4c). stomatocyte shape. Oblate shapes, on the other hand, are obtained This attractive force leads to an alteration in cell morphology: in at the vicinity of ν ~ 0.60. The values used for the erythrocytes the attraction region the cells flatten to maximise the region of surface area and volume lead to ν = 0.614, and therefore we obtain contact (Fig. 4a, panel 3). As the cells are pushed further against the characteristic erythrocyte oblate morphology. each other, the hard core repulsion term, regulated by γ in Eq. (2), In this simulation, the two erythrocytes were moved towards is responsible for the appearance of a strong repulsion force each other with relative advection velocity 2v = 1.14 μm/s (see between the two erythrocytes (maximum in Fig. 4c), which Table 1) until they collide. Then, the erythrocytes were moved rapidly increases as the cells get closer. At t = 1.18 s in the backwards with the same absolute velocity. During the simula- simulation, the motion of the cell is reversed and the repulsion tion, the forces on the erythrocytes were measured and the force rapidly decreases. As the two erythrocytes break apart, the alteration in erythrocyte morphology was characterised. adhesion force works to pull them together. We observe a rapid We started by simulating an aligned collision between the two increase in adhesion force until the cells are sufficiently apart and erythrocytes. In this collision, the two erythrocytes are parallel to the force between them then decreases to zero. each other and their relative velocity is parallel to the line that The model of Eq. (2) may be used to explore the force profiles connects the erythrocytes’ centres and to the z axis. The value for for different collisions between two erythrocytes. In Fig. 5a, b, we the adhesion coefficient in this simulation was set to plot the timelapse for an off-centred collision. In this case, the η = 8.0 × 10−12 J/m to simulate attractive forces on the range of two erythrocytes are still parallel to each other, but their centres 300 pN, as observed in the AFM assays (Fig. 3b and Fig. 4c). This are not aligned, and they approach with an offset distance d (the adhesion coefficient corresponds pffiffi to an adhesion energy between length of the projection along the y axis of the vector connecting two membranes of 935ϵ2 η ¼ 17:1 aJ/μm2 (this expression is the two cell centres). obtained directly from the integration of the energy adhesion The force applied in the cell along the velocity direction follows term between two interfaces in their equilibrium state). As the the same trend as in the aligned collision (Fig. 4c and Fig. 5c). As two erythrocytes collide, the contact area between them is on the the cells approach, there is an adhesion force that may lead to range of 50 to 100 μm2, and so the simulated energy of adhesion small changes in cell morphology. Then, as the cells are pushed between the two cells will be on the order of 1 to 2 fJ, similarly to against each other, the hard core repulsion force increases rapidly. the values observed experimentally (Fig. 3a). Finally as the cells are pushed away from each other, the adhesion Figure 4 presents the timelapse of the collision between the two force reaches its maximum before decreasing to zero, when the erythrocytes as they approach and then recede, as a 2D mid-slice cells are far from each other. Notably, for higher values of the COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio 3 ARTICLE COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 offset, the overlap between the two cells is smaller and the force magnitude in the z direction decreases. In the direction of the offset (the y direction), there is a non- zero force applied in the cell. As the erythrocyte on the right approaches the other one, it is pushed vertically for the two shapes to better fit each other. In this way, the magnitude of the vertical force depends non-monotonously on the offset. This force is still the sum of the cell–cell attraction force and the cell–cell hard core repulsion. After the initial contact, instead of flattening out, as in the aligned collision, in the collision with an offset the cell membranes adjust to better maximise the contact area. In this way, the thicker part of the erythrocyte attempts to “dock” into the middle dimple of the other erythrocyte. The forces measured in the y direction are the ones driving this push on the cell on the right to adapt its shape to the cell on the left. This “docking” of the cells implies that they are pushed closer together without overlapping than in the aligned collision described above (Fig. 4), and therefore the peak of hard core repulsion between the cells is smaller than in the aligned scenario (Fig. 5c), but it also has a small component in the negative y direction for the offset values between d = 1.02 μm and d = 4.08 μm, which can be clearly identified in the force profiles (Fig. 5d). More drastic cell deformations can be observed for larger values of adhesion coefficient. For η = 2.7 × 10−11 J/m, for example, we observe a large alteration in cell morphology when the cells are being pulled apart (see Fig. 6 for an offset d = 4.08 μm), akin to the largest changes in erythrocyte morphology observed in our micropipette assay (e.g. in Fig. 2c). In this scenario, the cells adhere strongly to one another. When they are pulled back, the sections of the cells which are not adhered move easily backwards, while the adhered section takes a longer time to separate (Fig. 6a, b). Similarly to the previous cases, as the cells approach, we initially observe the adhesion force between them, and the cells maximise locally the area of contact (Fig. 6c). As the cells are further pushed against each other, the repulsion force prevents the cell–cell overlap. In this simulation the contact region is small (a consequence of the large offset), but a local maximum in the force between the two cells due to the repulsion force at t = 1.18 s can still be observed. As the cells are then pulled apart, we observe a peak in adhesion. As they separate from each other, the force profile indicates a slow decrease in the adhesion force between the cells, consequence of a slow and progressive separation process (Fig. 6c). These strong adhesion and slow separation lead to dramatic changes in erythrocyte morphology (Fig. 6a, b). Fig. 2 Sequence of microscopy images of the micropipette aspiration assay and measured erythrocyte–erythrocyte adhesion frequency. Images at different stages of the erythrocyte–erythrocyte adhesion in the Conclusion absence of fibrinogen (a) and in the presence of 1.0 mg/mL of fibrinogen In this work we presented a coupled experimental and theoretical (b, c). A negative pressure (20 Pa) was applied to catch an erythrocyte in approach to analyse erythrocyte adhesion forces during cell–cell each micropipette (left side pipette is kept static). The arrows indicate the contact. The AFM experimental measurements of the adhesion direction of movement of the erythrocyte on the right. Erythrocyte- between cells enabled the parameterisation of a novel mathema- erythrocyte adhesion was continuously observed on a digital video for tical model, which can be used to explore, in finer detail, the subsequent analysis with LabVIEW. It is possible to observe that in the process of cell–cell adhesion mediated by fibrinogen. presence of fibrinogen there was a higher adhesion between erythrocytes, One of the major achievements of this work was to study leading to a pulling of the membrane of the erythrocyte of the left side (on fibrinogen-mediated erythrocyte aggregation using micropipette the static micropipette) when the erythrocyte from right micropipette is manipulation and atomic force microscopy. Erythrocyte- being moved away from it. d Erythrocyte-erythrocyte adhesion frequency. erythrocyte adhesion was measured by both techniques in the The represented values are the percentage of the adhesion between two absence and presence of fibrinogen. Erythrocytes with higher erythrocytes from healthy donors. The frequency of adhesion between two adhesion (high values of cell–cell detachment work and force) in erythrocytes in the presence of fibrinogen 1.0 mg/mL is higher than on its the presence of fibrinogen could be associated with a higher risk absence (46.7 ± 26.0% vs. 32.0 ± 30.4%, respectively; p = 0.04). The total of thromboembolic events and a worse clinical prognosis. We number of erythrocyte pairs analysed, obtained from five healthy donors, previously reported that the deformability, measured in an were n = 32 and n = 34, with and without fibrinogen addition, respectively. ektacytometer in conditions mimicking low calibre vessels, of the For each erythrocyte pair, we obtained 16 ± 5 adhesion image sequences. erythrocyte of chronic heart failure patients is lower than for Experimental values are presented together with box-and-whiskers plot healthy donors17. Using AFM to assess erythrocytes from these with the median value. patients, we showed that higher forces are needed to break the 4 COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 ARTICLE Fig. 3 AFM data of erythrocyte–erythrocyte detachment work and maximum detachment force. The work necessary to detach two erythrocytes in the presence of fibrinogen 1.0 mg/mL is higher than on its absence (p = 0.030) (a). The maximum detachment force of two erythrocytes in the presence of fibrinogen is also higher (p = 0.030) (b). For comparison, in b we also indicate the values of the erythrocyte–erythrocyte forces of our simulations. The yellow circle corresponds to the adhesion force between the cells in the simulation of Fig. 4 (below). The orange triangles indicate the adhesion forces between the cells in the simulations of Fig. 5 (below), for the different off-centred distances (1.02 μm, 2.04 μm, 3.06 μm and 4.08 μm). The red square depicts the adhesion force between the cells in the simulation of Fig. 6 (below). The experimental data refers to n = 28 biologically independent blood samples. For each blood sample approximately 40 force-distance curves were performed with and without fibrinogen addition. Experimental values are presented together with box-and-whiskers plots with the median value. Methods Table 1 Model parameters. Donors. Five healthy blood volunteers’ donors were included in this study to perform the micropipette aspiration technique, following a protocol with Odense University Hospital (Denmark). This study was approved by the ethical committee Parameter Symbol Value of Odense University Hospital and by the University of Southern Denmark (SDU). Bending rigidity KB 2.0 aJ Blood samples were obtained from the donors through the cooperation agreement Bending rigidity coefficient κB 432 J/m3 on blood for research or quality assurance (Project number DP059, Application for Interface width (lattice size) ϵ 0.17 μm blood for research or quality assurance, Document-620.8), at the South Danish Transfusion Service and Tissue Centre, Department of Clinical Immunology, Hard-core repulsion coefficient γ 138 J/m3 Odense University Hospital. For the AFM studies, we used a control group of 28 Area conservation coefficient αS 1.8 × 10−2 J/mm4 healthy blood donors, following a protocol with Instituto Português do Sangue e da Volume conservation coefficient αV 5.9 × 104 J/mm6 Transplantação (IPST, Lisbon, Portugal). The study was approved by the joint Target area Starget 197 μm2 Ethics Committee of Faculdade de Medicina da Universidade de Lisboa and Centro Target volume Vtarget 160 μm3 Hospitalar Universitário Lisboa Norte. All the participants gave and signed their Mobility M 7.4 × 10−2 m3J−1s−1 informed consent. Velocity k!vk 0.57 μm/s Erythrocyte isolation and sample preparation. Erythrocytes were isolated from the blood of healthy donors, as previously described12,17,19. The blood, collected interaction between fibrinogen and its erythrocyte membrane into K3EDTA anticoagulant tubes, was centrifuged at 1040 × g for 10 min, at 22 °C. receptor than on erythrocytes from healthy donors17. We also Plasma and buffy-coat were removed and the erythrocytes washed three times with used AFM to show that higher work and force values are buffered saline glucose citrate (BSGC) pH 7.3, supplemented with CaCl2 1 mM. Erythrocyte final suspension was prepared at 0.1% (v/v) haematocrit, in BSGC. For necessary to completely detach two erythrocytes from essential AFM experiments, the erythrocyte suspension was placed on a poly-L-lysine coated arterial hypertension patients, when compared with cells from glass slide, and erythrocytes were allowed to deposit and adhere for 30 min, at healthy donors, both in the absence and presence of fibrinogen room temperature. Non-adherent cells were removed by sequential washes with molecules in solution19. The erythrocytes from these hyperten- buffer. Cells were allowed to equilibrate for 15 min before the AFM measurements. sion patients were also stiffer and had a significant increase in aggregation, when compared with the control group19. Micropipette aspiration assay. The micropipette aspiration system consisted of The mathematical model permitted to visualise in detail the time micron-sized glass pipettes operated by micromanipulators, a manometer system controlling the aspiration pressure, and a chamber on a microscope stage from dependence of the interaction forces and cell morphology alterations which erythrocytes are aspirated into the micropipette26,30,31. The setup was built between two colliding erythrocytes. The adhesion forces and adhe- on an inverted optical microscope (Axiovert 100, Carl Zeiss, Jena, Germany) with sion energies between the two cells have been quantitatively matched 3D micrometres (Newport, Irvine, CA, USA) mounted on the stage that holds the to the experimental assays. As the two erythrocytes approach each micropipettes and the experimental chamber. Micropipettes were produced with single-barrel borosilicate capillary glass (0.75 mm × 0.4 mm 6″ glass capillary, A-M other, the adhesion works to maximise the contact area between the Systems, Sequim, WA, USA), heated and pulled using a pipette puller (P-47, Sutter cells. The simulation enables then to follow the decrease in adhesion Instruments, Novato, CA, USA). Tips were cut down to the desired inner size force as the cells are pulled away from each other. In situations of (1.5–2 μm) by using a micro forge (MF-900, Narishige, Setagaya, Japan)51. A large adhesion, the alterations in morphology can be very significant. chamber was placed on the slide-holder of the microscope for observation with an With the simulation, we can better understand, and explore in detail, eyepiece between the camera and the microscope, using a 40 × magnification. The micropipettes were filled with BSGC supplemented with 1 mM CaCl2 and con- the process of erythrocyte collision observed in the micropipette nected to a water reservoir. Pressure control was monitored using a pressure assay. Starting from the experimental mechanical characterisation of transducer (Valydine Engineering, Northridge, CA, USA). After placing the erythrocytes from a patient, we will be able to use this model to chamber with buffer on the microscope and adjusting the micropipette, the size of quantitatively predict the aggregation and flow dynamics of a group its tip was recorded and the pressure adjusted according to the micrometre. The condition of no flow was considered as zero pressure. After introducing the ery- of those cells in a blood vessel. throcyte suspension into the glass chamber, a small negative pressure (15 Pa) was The results obtained, in the previous works and here, high- applied, leading to the slow flow of an erythrocyte towards the pipette, enabling the lighting the changes on erythrocyte cell–cell interactions, may cell to be caught at the pipette tip. The video was digitally recorded and operated give important insights regarding health status, as well as the with home-built LabVIEW programme. Videos were analysed with ImageJ soft- ware v. 1.53e (NIH, USA)52. pathophysiological relevance of fibrinogen concentration and The two-pipettes technique was applied to visualise the adhesion between two erythrocyte aggregation, since an increment on either of them erythrocytes. We used this technique to observe the effects in erythrocyte shape as a may induce a reduction of the microcirculatory blood flow. consequence of erythrocyte–erythrocyte adhesion, both in the absence and COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio 5 ARTICLE COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 Fig. 4 Simulation time-lapse of two aligned erythrocytes being pushed against each other and then pulled apart. a Mid-slice profile, and b the corresponding 3D representation. The images correspond to the times t = 0, 0.62, 1.25, 1.88 s. c Total force applied on the right erythrocyte by the left erythrocyte as a function of time. presence of fibrinogen (1 mg/mL). On all cell–cell adhesion experiments, a pressure cycles of the erythrocyte. Using a “ramp mode” with the time period of 15 s, of 20 Pa was maintained. A piezoelectrical driven linear translator (LISA, velocity 1.14 μm/s, amplitude 3 V and symmetric regarding forward and backward P-753.12C, Physik Instrumente, Germany, with modular piezo controller E500.00, motion, the observed deformation of the erythrocytes permitted to analyse the Physik Instrumente, Germany) coupled to one of the micropipettes was used to effects of cell–cell adhesion in the assay. These measurements were done in the control the movement of the erythrocytes (approach, adhesion and retraction). absence and presence of 1.0 mg/ml fibrinogen, in BSGC with 1 mM of CaCl2. Real- Using an arbitrary waveform generator (33220A, Agilent, Santa Clara, CA, USA), it time video images were acquired and further analysed to determine the erythrocyte was possible to programme the time course for piezo approach and retraction adhesion frequency. 6 COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 ARTICLE Fig. 5 Simulation time-lapse of two erythrocytes being pushed against each other off centred and then pulled apart. a Mid-slice profile and b corresponding 3D representation with offset distance d = 3.06 μm. The images correspond to the times t = 0, 0.62, 1.25, 1.88 s. Total force applied on the right erythrocyte by the left erythrocyte as a function of time (c) in the z direction and d in the y direction. COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio 7 ARTICLE COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 Fig. 6 Simulation time-lapse of two offset erythrocytes being pushed against each other off centred and then pulled apart for large adhesion. a Mid- slice profile and b corresponding 3D representation with offset distance d = 4.08 μm (at each time instance, we present the erythrocytes observed from two different perspectives: observed along the y direction, in the larger images, and when observed along the x direction, in the respective smaller images). The images correspond to the times t = 0, 1.25, 3.13, and 4.69 s. c Total force applied on the right erythrocyte by the left erythrocyte as a function of the time. Atomic force microscopy. AFM studies were performed on a JPK NanoWizard IV 15 min. Then, the cantilevers were incubated overnight in a 0.5 mg/mL biotinylated atomic force microscope with a CellHesion 200 module (JPK Instruments, Berlin, albumin solution, at 37 °C, in a humidified incubator. The cantilever was washed Germany), mounted on an Axiovert 200 inverted optical microscope (Zeiss, Jena, three times in phosphate buffered saline (PBS) pH 7.4 to remove unbound protein. Germany). To conduct the AFM cell adhesion assays, tipless cantilevers Arrow TL2 After that, the cantilever was incubated in 0.5 mg/mL streptavidin (30 min, at room (NanoSensors, Neuchatel, Switzerland), with 1 μm thickness, 0.03 N/m spring temperature) and, finally, in 0.4 mg/mL biotinylated-concanavalin A solution constant and 7 kHz resonance frequency were chemically functionalised, as pre- (30 min, at room temperature). Functionalised cantilevers were mounted on the viously described18,22,53. AFM cantilevers were cleaned with intense UV light for microscope, the erythrocytes in suspension were injected, and one erythrocyte was 8 COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 ARTICLE captured by positioning the cantilever above the cell centre and pressing onto the The first term of the free energy describes the bending energy of the erythrocyte cell for ~30 s. The erythrocyte attached to the cantilever was then placed on top of and, in the sharp interface limit, becomes identical to the mean curvature term of other erythrocytes previously adhered to a poly-L-lysine glass slide, and the cell–cell the Cahnam-Helfrish energy, i.e., to ∫dsKBC2, where the integral is done over the adhesion experiments were performed. In this way, one erythrocyte was securely membrane and C is the membrane local mean curvature. The Gaussian curvature adhered to the cantilever, while the other was adhered to the surface, and in all term of the Cahnam-Helfrish energy is not included in our phase-field description force-distance curves of measurements on the erythrocyte–erythrocyte system of erythrocyte dynamics, since these cells conserve their integrity in our (with and without the presence of fibrinogen) we never observed cell detachment experiments without changing their topology, and since, due to the Gauss-Bonnet from the tipless cantilever. In fact, according to53, the strength of the concanavalin theorem, the integral of the Gaussian curvature in a closed vesicle is a topological A attachment force is close to 2 nN, thus much stronger than the interaction force invariant67. Therefore, this term would only affect the cell dynamics if the Gaussian between two erythrocytes (below 1 nN, see Fig. 3b above). Moreover, poly-L-lysine curvature rigidity coefficient was not constant throughout the cell membrane, or if is a positively charged amino acid residue polymer commonly adsorbed to surfaces topological transformations were present during the dynamics. To obtain the for strong attachment to the negatively charged glycocalyx of cells, as erythrocytes Fbending term, the local curvature at a specific point of the phase field interface is and, it can provide a firm adherence of erythrocytes to the surface without written as a function of the order parameter and its derivatives, and then integrated changing their characteristic morphology 54. over the system volume (see ref. 50 for a detailed derivation). The applied force was adjusted to 0.3 nN before retraction. Data collection for The hard-core repulsion term between the two erythrocytes leads to an increase in each force-distance cycle was performed at 2 μm/s, with a z-length of 30 μm. A the system’s energy when the erythrocytes overlap in space. In this term, the hard-core retract and an extend delay of 5 s was applied. Measurements were performed in repulsion coefficient γ will be set to a value large enough to prevent the overlap of the the presence or the absence of human fibrinogen (Sigma Aldrich, St. Louis, MO, cells. The function hðϕi Þ ¼ 14 ð1 þ ϕi Þ2 ð2  ϕi Þ is approximately equal to 1/2 at the USA). For each blood sample, approximately 40 force-distance curves were vicinity of ϕi = 0, and has extrema at h( − 1) = 0 and h(1) = 1. This function moderates performed for each concentration (5 force-distance curves × 8 cells attached to a the hard-core repulsion term, which would otherwise be responsible for pushing the poly-L-lysine glass slides). Force-distance curves were analysed using JPK data order parameter values away from the interval [−1, 1]42. processing software v. 6.055 (JPK Instruments, Berlin, Germany). Values of the The second term of the interaction part of the system’s free energy (1) describes maximum detachment force (by the minimum value of adhesion force) and of the the surface adhesion between the erythrocytes. This term indicates that the energy work (energy) necessary to detach two erythrocytes were obtained after analysing decreases when the two order parameter gradients are anti-parallel, i.e., when the each cell–cell adhesion retraction curves. cells are in contact. Finally, the surface and volume terms penalise the system energetically when Statistics and reproducibility. Descriptive statistics are given as mean ± standard the erythrocyte surface and volume is different from their target values Starget and error of the mean (SEM). Results were analysed with Mann–Whitney U test. Vtarget. The surface and volume of each erythrocyte can be directly obtained from R R the following functionals, S½ϕi  ¼ 2p3 ffiffi2 ϵ ð∇ϕð! r ÞÞ d! r and V½ϕi  ¼ hðϕÞd! 2 p < 0.05 was considered as significant. Statistical analyses were performed using r, GraphPad Prism v. 8.02. The simulation results from Figs. 4–6 are the dirct result respectively. The parameters αS and αV in Eq. (1) are penalty parameters, which from the integration of Eq. (2). will be set large enough that both the volume and surface of the erythrocytes are kept constant during the system’s evolution (see Eq. (2) below). Mathematical model. In the present mathematical model, we assign a scalar order From this free energy, we can obtain the following equation for the time parameter to each erythrocyte ϕi ð! r Þ, where i identifies the cell, such that ϕi ≈ + 1 evolution of the order parameter ϕ1 (the equation for the evolution of the other inside the respective erythrocyte and ϕi ≈ − 1 outside. Near the cell membrane, erythrocyte is obtained from (2) by exchanging the indices 1 and 2): ϕ i ð! r Þ varies continuously, and ϕi = 0 at the centre of the interface33. While this ∂ϕ1 ! δF      choice for equilibrium values of the scalar order parameter is standard in phase- þ v  ∇ϕ1 ¼  M ¼ M 2 3ϕ21  1 κB ϕ1 þ ϕ31  ϵ2 ∇2 ϕ1 field models that describe the bending rigidity of biological membranes and of ∂t δϕ1   other systems50,55–57, the choice of ϕi ≈ + 1 inside the cell and ϕi ≈ 0 outside may  2κB ϵ2 ∇2 ϕ1 þ ϕ31  ϵ2 ∇2 ϕ1 be found in other works in the field42,44,45. 3   3   The most relevant energies involved in determining erythrocytes’ shapes and þ γ 1  ϕ21 hðϕ2 Þ  η 1  ϕ21 ∇2 hðϕ2 Þ ð2Þ 4  4  dynamics are the bending energy due to curvature, the adhesion energy when the pffiffiffi erythrocytes approach, and the hard core repulsion to prevent the inter-penetration  3 2MϵαS Starget  S½ϕ1  ∇2 ϕ1 between two erythrocytes. The cell rigidity provided by the erythrocyte 3    cytoskeleton provides further stability to the erythrocyte shape, reducing þ MαV V target  V½ϕ1  1  ϕ21 ; 2 membrane fluctuations and influencing the measured values of the bending rigidity and membrane tension58. As in this work the focus is in exploring the time where ! v is the advection velocity of the cell and M is the system’s mobility. A dependence of the cell–cell interaction forces at the surface of the erythrocytes, we will focus the model on the description of cell adhesion and hard core repulsion. dimensionless version of Eq. (2) can be written by noting that τ ¼ ðMκB Þ1 defines We observe that the bending rigidity energy term is sufficient to stabilise the oblate the simulation timescale, and the interface width, ϵ, gives the simulation length erythrocyte characteristic shape50,59,60 without needing the cytoskeleton scale. The time and space variables in Eq. (2) can be then expressed as t ¼ t 0 τ, mechanics. Hence, we will consider in the model the bending rigidity KB = 2.0 aJ, x ¼ x0 ϵ, y ¼ y0 ϵ, and z ¼ z 0 ϵ, where t 0 , x0 , y0 , and z 0 are dimensionless variables. which is inside the typical range of erythrocytes bending rigidities, from 0.2 to We finally observe that the relevant parameters that describe the system dynamics 3 aJ61–65, but for the sake of simplicity, we will not add an explicit description of are the dimensionless advection velocity, v0 ¼ Mκv ϵ, the dimensionless repulsion B the cell’s rigidity provided by the cytoskeleton. coefficient, γ0 ¼ κγ , and the dimensionless adhesion coefficient, η0 ¼ κ ηϵ2 . We initially implement a free energy functional F½ϕ1 ð!r Þ; ϕ2 ð! B B r Þ that takes into Nevertheless, in this work we will be using Eq. (2) in its dimensional form for a account the considered energetic contributions 42,44,46,50,59,59 (derived from the more direct substitution of the experimental parameters. Cahnam-Helfrish bending energy66): The parameters for the model are indicated in Table 1. Z Finally, the interaction force in one of the cells due to the other erythrocyte, 2  2 given by ref. 33: F½ϕ1 ð~ rÞ; ϕ2 ð~ rÞ ¼ κB ∑ ϕi þ ϕ3i þ ϵ2 ∇2 ϕi d~ r i¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Z ! δF interaction ! Fbending F1 ¼  ϕ1 ∇ dr ¼ Z δϕ1   Z ð3Þ þ γhðϕ1 Þhðϕ2 Þ þ η∇hðϕ1 Þ  ∇hðϕ2 Þ d~ r       ϕ ∇ γ 1  ϕ21 hðϕ2 Þ  η 1  ϕ21 ∇2 hðϕ2 Þ d! 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ r: 4 1 Finteraction  2 ð1Þ 2 þ αS ∑ Starget  S½ϕi  i¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Numerical methods. Initially, the shape of each isolated cell was obtained from the FSurface integration of Eq. (2) by setting the velocity and cell–cell interaction terms to zero. 2  2 Then, with the full equation, the collision between the two cells was simulated in a þ αV ∑ Vtarget  V½ϕi  ; 3D box of size 15.3 × 15.3 × 15.3 μm3. We have used a fixed grid spectral semi- i¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} implicit method to integrate Eq. (2). The non-linear terms were initially calculated Fvolume in real space, and the fast-Fourier transform algorithm was used to obtain the 3D Fourier transform of these terms. The next iteration values for the order parameters where κB, γ and η are coefficients proportional to the bending rigidity of the cell ϕ1 and ϕ2 were calculated directly in Fourier space. This was done semi-implicitly membrane50, to the hard-core repulsion between cells and to the cell–cell surface with the linear time derivative and fourth spatial derivative (squared Laplacian) adhesion42,46,50, respectively. The width of the interface in the model is given by terms being written as a function of the order parameters in the next time step, and the coefficient ϵ33, which also relates κB to the membrane bending rigidity in the the other terms being written as a function of the order parameters in the present sharp interface limit, K B ¼ 34ϵpffiffi2 κB 50. 3 time step. COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio 9 ARTICLE COMMUNICATIONS BIOLOGY | https://doi.org/10.1038/s42003-023-04560-4 Reporting summary. Further information on research design is available in the Nature 21. Carvalho, F. A. et al. 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Molecularly based analysis of deformation of Peer review information Communications Biology thanks the anonymous reviewers for spectrin network and human erythrocyte. Materi. Sci. Eng.: C 26, 1232–1244 their contribution to the peer review of this work. Primary Handling Editor: Joao (2006). Valente. Peer reviewer reports are available. 63. Fedosov, D. A., Caswell, B. & Karniadakis, G. E. A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys. J. 98, Reprints and permission information is available at http://www.nature.com/reprints 2215–2225 (2010). Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in 64. Tan, J., Keller, W., Sohrabi, S., Yang, J. & Liu, Y. Characterization of published maps and institutional affiliations. nanoparticle dispersion in red blood cell suspension by the lattice-Boltzmann immersed boundary method. Nanomaterials 6, 30 (2016). 65. Fedosov, D. A. Multiscale Modeling of Blood Flow and Soft Matter. (Brown University, 2010). Open Access This article is licensed under a Creative Commons 66. Safran, S. Curvature elasticity of thin films. Adv. Phys. 48, 395–448 (1999). Attribution 4.0 International License, which permits use, sharing, 67. Rueda-Contreras, M. D., Gallen, A. F., Romero-Arias, J. R., Hernandez- adaptation, distribution and reproduction in any medium or format, as long as you give Machado, A. & Barrio, R. A. On gaussian curvature and membrane fission. appropriate credit to the original author(s) and the source, provide a link to the Creative Sci. Rep. 11, 1–10 (2021). Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the Acknowledgements article’s Creative Commons license and your intended use is not permitted by statutory This work was supported by Fundação para a Ciência e a Tecnologia – Ministério da regulation or exceeds the permitted use, you will need to obtain permission directly from Ciência, Tecnologia e Ensino Superior (FCT-MCTES, Portugal) projects PTDC/EMD- the copyright holder. To view a copy of this license, visit http://creativecommons.org/ TLM/7289/2020 (all authors), PTDC/BBB-BMD/6307/2014 and UID/BIM/50005/2019 licenses/by/4.0/. (CSL, FAC, NCS), through Fundos do Orçamento de Estado. JC and RT also thank the support of FEDER funds through the Operational Programme Competitiveness Factors – © The Author(s) 2023 COMPETE and Fundação para a Ciência e a Tecnologia through the strategic projects COMMUNICATIONS BIOLOGY | (2023)6:192 | https://doi.org/10.1038/s42003-023-04560-4 | www.nature.com/commsbio 11