Dependence of dielectrophoretic force on the size of linear erythrocyte aggregates in suspension

For modeling of erythrocyte rouleaux (linear cell aggregates) we develop an approximation procedure for the dipole moment in short cylinders, which contains the case of ellipsoidal bodies as a first approximation, but allows corrections for short cylinders, more representative for such particles. In dependence on the number of erythrocytes in an aggregation, i.e., on different but discrete rouleaux lengths, the dielectrophoretic force is calculated and represented against the frequency of the applied AC field. Predictions are made for frequency regions in the 107–108 Hz range where the magnitude and the direction of dielectrophoretic forces is different for different rouleaux sizes. This property can be used for the detection and spatial separation of rouleaux populations of different length in a microelectronic array.     

Dielectric approach to investigation of erythrocyte aggregation. II. Kinetics of erythrocyte aggregation-disaggregation in quiescent and flowing blood

A method based on dielectric properties of dispersed systems was applied to investigate the kinetics of RBC aggregation and the break-up of the aggregates. Experimentally, this method consists of measuring the capacitance at a frequency in the beginning of the beta-dispersion. Two experimental protocols were used to investigate the aggregation process. In the first case, blood samples were fully dispersed and then the flow was decreased or stopped to promote RBC aggregation. It was found that the initial phases of RBC aggregation are not affected by the shear rate. This finding indicates that RBC aggregation is a slow coagulation process. In the second case, RBCs aggregated under flow conditions at different shear rates and after the capacitance reached plateau levels, the flow was ceased. The steady-state capacitance of the quiescent blood and the kinetics of RBC aggregation after stoppage of shearing depend on the prior shear rate. To clarify the reasons for this effect, the kinetics of the disaggregation process was studied. In these experiments, time courses of the capacitance were recorded under different flow conditions and then a higher shear stress was applied to break up RBC aggregates. It was found that the kinetics of the disaggregation process depend on both the prior and current shear stresses. Results obtained in this study and their analysis show that the kinetics of RBC aggregation in stasis consists of two consecutive phases: At the onset, red blood cells interact face-to-face to form linear aggregates and then, after an accumulation of an appropriate concentration of these aggregates, branched rouleaux are formed via reactions of ends of the linear rouleaux with sides of other rouleaux (face-to-side interactions). Branching points are broken by low shear stresses whereas dispersion of the linear rouleaux requires significantly higher energy.     

Segregation into separate rouleaux of erythrocytes from different species. Evidence against the agglomerin hypothesis of rouleaux formation.

Erythrocytes from one species were labelled with fluorescein isothiocyanate and mixed with unlabelled erythrocytes from another species. Albumin polymers were added to generate rouleaux. The species of origin of erythrocytes in rouleaux was determined by fluorescence microscopy. Erythrocytes from different species segregated into independent rouleaux. However, fluorescent and non-fluorescent erythrocytes from one individual were mixed randomly in rouleaux. These results confirm, using a novel experimental approach, previous observations of Sewchand & Canham [(1976) Can. J. Physiol. Pharmacol. 54, 437-442]. Since rouleaugenic agents are not species-specific, under the 'agglomerin' hypothesis of rouleau formation they would be expected to form bridges between cells from different species. It follows that either the agglomerin hypothesis is incorrect, or additional species-specific surface components are involved in the aggregation of agglomerin-cross-bridged cells.     

Effects of shear rate on rouleau formation in simple shear flow

A kinetic equation for rouleau formation in a simple shear flow is derived, based on several assumptions. These are (a) colliding rouleaux stick to one another with a certain probability to form a single rouleau; (b) simultaneous collisions between more than two rouleaux are negligible; (c) rouleaux are broken by a viscous force exerted by the suspending fluid on the surfaces of rouleaux; (d) when a rouleau is broken by viscous forces, only two fragments are formed. Based on a simple mathematical model, collision rate, sticking probability and degradation rate are obtained as functions of applied shear rate. From the solution of the kinetic equation, the average size of rouleaux is obtained as a function of time with shear rate as a parameter. It is shown that the average size of rouleaux increases monotonically with increasing time and tends to an equilibrium size. The average size of rouleaux in a dynamical equilibrium decreases monotonically with increasing shear rate and tends to one cell as shear rate approaches infinity. It is also found that the initial rate of rouleau formation increases with increasing shear rate at very low shear rate, but this trend is reversed at higher shear rates. The theoretical results are compared quantitatively with experimental data.     

Kinetics of linear rouleaux formation studied by visual monitoring of red cell dynamic organization

Red blood cells (RBCs) in the presence of plasma proteins or other macromolecules may form aggregates, normally in rouleaux formations, which are dispersed with increasing blood flow. Experimental observations have suggested that the spontaneous aggregation process involves the formation of linear rouleaux (FLR) followed by formation of branched rouleaux networks. Theoretical models for the spontaneous rouleaux formation were formulated, taking into consideration that FLR may involve both "polymerization," i.e., interaction between two single RBCs (e + e) and the addition of a single RBC to the end of an existing rouleau (e + r), as well as "condensation" between two rouleaux by end-to-end addition (r + r). The present study was undertaken to experimentally examine the theoretical models and their assumptions, by visual monitoring of the spontaneous FLR (from singly dispersed RBC) in plasma, in a narrow gap flow chamber. The results validate the theoretical model, showing that FLR involves both polymerization and condensation, and that the kinetic constants for the above three types of intercellular interactions are the same, i.e., k(ee) = k(er) = k(rr) = k, and for all tested hematocrits (0.625-6%) k < 0.13 +/- 0.03 s(-1).     

Theory of long-range coherence in biological systems. I. the anomalous behaviour of human erythrocytes

In this paper, an explicit expression of the interaction potential has been obtained, based on the Fröhlich model of long-range coherence in biological cells. These theoretical expressions are used to correlate with the experimental data obtained from the light scattering and the Brownian motion for the red blood cells (human erythrocytes). The necessary conditions are derived for the formation of rouleaux in human erythrocytes in the presence of long range interactions. The kinetics of rouleaux formation in terms of the properties of the diffusion coefficient has been investigated. It is concluded that both types of experimental data can be interpreted satisfactorily by this model.     

Rouleaux formation as a measure of the phase separating ability of plasma

The increased erythrocyte sedimentation rate (ESR) occurring in various diseases reflects a change in certain plasma proteins which causes erythrocytes to aggregate into rapidly sedimenting “piles of coins”, or rouleaux. In spite of some 300 years of study, the mechanism of the phenomenon and the adaptive significance, if any, of the change in properties of plasma, remain elusive. One hypothesis on the mechanism is that rouleaugenic agents in plasma act as multivalent “agglomerins”, which react with the erythrocyte surface and cross-link cells. Alternatively, a “phase separation” hypothesis can be derived by analogy with the phase separation of high molecular weight polymers from polymer mixtures at high polymer concentrations. Considering the cell surface to resemble that of a large polymer, it is postulated that there should be a separation of a cell phase from a solution phase in the presence of high concentrations of a high molecular weight polymer. In keeping with the available data, both hypotheses predict that rouleaux formation should increase with increasing polymer size. In addition, the phase separation hypothesis predicts (i) no need for reaction of a rouleaugenic agent with the erythrocyte surface, and (ii) that cells from different species, when mixed in the presence of a rouleaugenic agent, will form independent rouleaux. Current evidence is consistent with both predictions. We suggest that an increased ESR reflects a change in the physical properties of plasma such that particulate material (e.g. viruses, tumour cells), will tend to spontaneously aggregate in regions of stagnant blood flow (such as the sinusoids of the spleen). This change in the physical properties of plasma, tending to favour phase separations, will also favour antigen-antibody and other intermolecular reactions.     

Kinetics of rouleau formation. II. Reversible reactions.

Red blood cells aggregate face-to-face to form long, cylindrical, straight chains and sometimes branched structures called rouleaux. Here we extend a kinetic model developed by R. W. Samsel and A. S. Perelson (1982, Biophys. J. 37:493-514) to include both the formation and dissociation of rouleaux. We examine thermodynamic constraints on the rate constants of the model imposed by the principle of detailed balance. Incorporation of reverse reactions allows us to compute mean sizes of rouleaux and straight chain segments within rouleaux, as functions of time and at equilibrium. Using the Flory - Stockmayer method from polymer chemistry, we obtain a closed-form solution for the size distribution of straight chain segments within rouleaux at any point in the evolution of the reaction. The predictions of our theory compare favorably with data collected by D. Kernick , A.W.L. Jay , S. Rowlands , and L. Skibo (1973, Can. J. Physiol. Pharmacol. 51:690-699) on the kinetics of rouleau formation. When rouleaux grow large, they may contain rings or loops and take on the appearance of a network. We demonstrate the importance of including the kinetics of ring closure in the development of realistic models of rouleaux formation.     

Kinetics of rouleau formation. I. A mass action approach with geometric features.

In the presence of certain macromolecules, such as fibrinogen, immunoglobulin, dextran, and polylysine, erythrocytes tend to aggregate and form cylindrical clusters called "rouleaux" in which cells resemble coins in a stack. The aggregates may remain cylindrical or they may branch, forming tree, and networklike structures. Using the law of mass action and notions from polymer chemistry, we derive expressions describing the kinetics of the early phase of aggregation. Our models generalize work initiated by Ponder in 1927 who used the Smoluchowski equation to predict the concentration of rouleaux of different sizes. There are two novel features to our generalization. First, we allow erythrocytes that collide near the end of a stack of cells to move to the end of the cylinder and elongate it. Second, we incorporate geometric information into our models and describe the kinetics of branched rouleau formation. From our models we can predict the concentration of rouleaux with n cells and b branches, the mean number of cells per rouleau, the mean number of branches per rouleau, and the average length of a branch. Comparisons are made with the available experimental data.     

Effects of hematocrit on thixotropic properties of human blood

The rheological properties of whole human blood exhibit thixotropic behavior at low shear rates up to about ten reciprocal seconds (1). The accepted cause of this shear rate-dependent and time-dependent behavior is the progressive breakdown of rouleaux into individual red cells. Huang developed a rheological equation which incorporates the kinetics of rouleau breakdown in his models (2). This five-parameter equation was used successfully to represent the hysteresis loop and the torque-decay curve of whole human blood. Numerical values of these five thixotropic parameters, which characterize the rheological behavior of the blood from apparently healthy human subjects, were established (3). In this communication, we examined the effect of hematocrit on each of the above mentioned parameters. The results show that the following parameters will increase their values with an increase in hematocrit: the yield stress, Newtonian contribution of viscosity, non-Newtonian contribution of viscosity, apparent viscosity and the equilibrium value of the structural parameter which indicates the relative amount of rouleaux in blood. Mathematical equations were developed to give the relationship between parameters and hematocrit. Two other thixotropic parameters, viz. the kinetic rate constant of rouleaux breakdown into individual red cells and the order of the breakdown reaction, were found to be independent of the hematocrit. It is consistent with reaction kinetic theory that the rate constant and the order of reaction are independent of the concentration of reactants.     

Rheology of Human Blood, near and at Zero Flow: Effects of Temperature and Hematocrit Level

Static normal human blood possesses a distinctive yield stress. When the yield stress is exceeded, the same blood has a stress-shear rate function under creeping flow conditions closely following Casson's model, which implies reversible aggregation of red cells in rouleaux and flow dominated by movement of rouleaux. The yield stress is essentially independent of temperature and its cube root varies linearly with hematocrit value. The dynamic rheological properties in the creeping flow range are such that the relative viscosity of blood to water is almost independent of temperature. Questions raised by these data are discussed, including red cell aggregation promoted by elements in the plasma.     

Comparison and content of the Wright-Fisher model of random genetic drift, the diffusion approximation, and an intermediate model

We investigate the detailed connection between the Wright-Fisher model of random genetic drift and the diffusion approximation, under the assumption that selection and drift are weak and so cause small changes over a single generation. A representation of the mathematics underlying the Wright-Fisher model is introduced which allows the connection to be made with the corresponding mathematics underlying the diffusion approximation. Two ‘hybrid’ models are also introduced which lie ‘between’ the Wright-Fisher model and the diffusion approximation. In model 1 the relative allele frequency takes discrete values while time is continuous; in model 2 time is discrete and relative allele frequency is continuous. While both hybrid models appear to have a similar status and the same level of plausibility, the different nature of time and frequency in the two models leads to significant mathematical differences. Model 2 is mathematically inconsistent and has to be ruled out as being meaningful. Model 1 is used to clarify the content of Kimura's solution of the diffusion equation, which is shown to have the natural interpretation as describing only those populations where alleles are segregating. By contrast the Wright-Fisher model and the solution of the diffusion equation of McKane and Waxman cover populations of all categories, namely populations where alleles segregate, are lost, or fix.     

The equilibrium size distribution of rouleaux.

Rouleaux are formed by the aggregation of red blood cells in the presence of macromolecules that bridge the membranes of adherent erythrocytes. We compute the size and degree of branching of rouleaux for macroscopic systems in thermal equilibrium in the absence of fluid flow. Using techniques from statistical mechanics, analytical expressions are derived for (a) the average number of rouleaux consisting of n cells and having m branch points; (b) the average number of cells per rouleau; (c) the average number of branch points per rouleau; and (d) the number of rouleaux with n cells, n = 1, 2, ..., in a system containing a total of N cells. We also present the results of numerical evaluations to establish the validity of asymptotic expressions that simplify our formal analytic results.     

Aggregation and disaggregation of red blood cells

The aggregation of red blood cells may be analyzed as an interaction of an adhesive surface energy and the elastic stored energy which results from deformation of the cell. The adhesive surface energy is the work required to separate a unit adhered area and is the resultant of adhesive forces due to the bridging molecules that bind the cells together and the electrostatic repulsion due to surface charge. The elastic strain energy in the case of the red blood is associated with the membrane elasticity only since the interior of the cell is liquid. The membrane elasticity is due both to bending stiffness and shear. The area expansion is small and may be neglected. These assumptions allow realistic computation of red cell shapes in rouleaux. The disaggregation of rouleaux requires an external force which must overcome the adhesive energy and also supply additional elastic energy of deformation. Depending on the geometry, the initial effect of elastic energy may tend to aid disaggregation. In a shear flow, the stresses on a suspended rouleau alternately tend to compress and to disaggregate the cells if they are free to rotate. This introduces a time dependence so that viscous effects due to the viscosity of the cell membrane, the cell cytoplasm and the external fluid may play a role in determining whether disaggregation proceeds to completion or not.     

A model for rouleaux pattern formation of red blood cells

Human red blood cells (RBCs) in a solution form rouleaux patterns under various conditions. The degree of rouleaux formation depends on, for example, the concentration and molecular weight of added large molecules. We present a two-dimensional discrete cellular space model in which an RBC is represented by a rectangle and differential adhesion is assumed among the longer (a-site), the shorter (b-site) sides of the rectangle and the solvent. The total sum of the adhesion energy is assumed to guide the step-by-step change of the model cell configuration and also define absolutely stable patterns. We compare the set of absolutely stable patterns and cell aggregate patterns for both actual and computer-simulated cases to obtain the basic validity of our framework. Then we proceed to assess the effects of added high polymers to the adhesion parameters. We first note that under suitable conditions, decrease in a-site-solvent affinity is necessary to have complex patterns rather than increase of a-a affinity. The hypothesis that addition of high polymers reduce the a-site-solvent affinity is concomitant with a newly proposed osmotic stress theory. The parameter fitting results for the experimental phase change curves can also be interpreted as supporting more the new theory than existing traditional explanations.     

Changes of RBC aggregation in oxygenation-deoxygenation: pH dependency and cell morphology

The effects of the oxygenation-deoxygenation process on red blood cell (RBC) aggregation were examined in relation to morphological changes in RBCs and the contribution of CO(2). A low-shear rheoscope was used to measure the rate of rouleaux (one-dimensional aggregate) formation in diluted autologous plasma exposed to gas mixtures with different Po(2) and Pco(2). RBC indexes and RBC suspension pH were measured for the oxygenated or the deoxygenated condition, and the cell shape was observed with a scanning electron microscope. In the oxygenation-deoxygenation process, the rate of rouleaux formation increased with rising pH of the RBC suspension, which was lowered in the presence of CO(2). The rate increased with increasing mean corpuscular hemoglobin concentration (thus the cells shrank), which increased with rising pH and decreased in the presence of CO(2). With rising pH, cell diameter increased and cell thickness decreased (thus the cell flattened). In addition, slight echinocytosis was induced in the presence of CO(2), and the aggregation was reduced by the morphological change. In conclusion, RBC aggregation in the oxygenation-deoxygenation process is mainly influenced by the pH-dependent change in the surface area-to-volume ratio of the cells, and the aggregation is modified by CO(2)-induced acidification and the accompanying changes in mean corpuscular hemoglobin concentration and cell shape.     

Thixotropic properties of whole blood from healthy human subjects

The steady state non-Newtonian viscosity of whole human blood has been widely studied as a function of the shear rate; and used to characterize the blood in various pathological disorders. In our previous studies, we demonstrated that blood is a thixotropic fluid. Its time-dependency and shear rate dependency of rheological behavior can be represented by an equation developed by Huang. Parameters of the equation can be used for the characterization of an individual's blood. They provide information, such as the kinetic rate constant of breakdown of RBC rouleaux to individual erythrocytes and the relative amount of rouleau formation in the dynamic equilibrium between rouleaux and individual erythrocytes. In this communication, the thixotropic parameters from blood samples of fifteen apparently healthy human subjects were investigated. When compared to the use of apparent viscosity values for the correlation with a pathological disorder, thixotropic parameters are preferable. The mean values of thixotropic parameters obtained from apparently healthy human subjects provide a base for comparison with the same parameters as obtained from blood samples of patients with certain pathological disorders involving the circulation.     

Lysophosphatidic acid and human erythrocyte aggregation

The effect of lysophosphatidic acid (LPA) on the shape and aggregation of human erythrocytes in autologous plasma was studied. The morphology of erythrocytes and their aggregates were studied by light microscopy. It is shown that the addition of plasma with a high LPA content to erythrocytes leads to a change of their shape: discocytes are transformed into echinocytes. There is practically no aggregation of erythrocytes in the form of rouleaux. At the same time, there is observed a strong aggregation of echinocytes. This is accompanied by the formation of microvesicles. The addition of normal blood plasma to echinocytes restores their shape and aggregation of red blood cells in the form of rouleaux. A possible mechanism of action of lysophosphatidic acid on erythrocytes is discussed.     

On the genealogy of a sample of neutral rare alleles

This paper concerns the genealogical structure of a sample of chromosomes sharing a neutral rare allele. We suppose that the mutation giving rise to the allele has only happened once in the history of the entire population, and that the allele is of known frequency q in the population. Within a coalescent framework C. Wiuf and P. Donnelly (1999, Theor. Popul. Biol. 56, 183-201) derived an exact analysis of the conditional genealogy but it is inconvenient for applications. Here, we develop an approximation to the exact distribution of the conditional genealogy, including an approximation to the distribution of the time at which the mutation arose. The approximations are accurate for frequencies q<5-10%. In addition, a simple and fast simulation scheme is constructed. We consider a demography parameterized by a d-dimensional vector alpha=(alpha(1), em leader, alpha(d)). It is shown that the conditional genealogy and the age of the mutation have distributions that depend on a=qalpha and q only, and that the effect of q is a linear scaling of times in the genealogy; if q is doubled, the lengths of all branches in the genealogy are doubled. The theory is exemplified in two different demographies of some interest in the study of human evolution: (1) a population of constant size and (2) a population of exponentially decreasing size (going backward in time).     

The RBC morphological dependence of the RBC disaggregability

The aim of this study was to determine the red blood cell (RBC) disaggregability dependence upon the RBC shape. The study concentrated on stored blood during bank storage and on suspensions of artificially induced echinocytes. Measurements was performed in autologous plasma of hematocrit 0.45 and at constant plasmatic content. Rheological studies using stationary viscometry, nonstationary viscometry and rheoscopy were made in order to assess different stages of the disaggregability process. Whatever the method of measurement used, the morphological interpretation of the results reveal that beyond 75% of echinocytes within the sample, the disaggregation process is altered. The shear stresses required to dissociate the echinocyte aggregates are significantly higher than those required to disaggregate normal RBC rouleaux.