Emmanuil E. Shnol

Finite Platelet Size Could Be Responsible for Platelet Margination Effect

Blood flows through vessels as a segregated suspension. Erythrocytes distribute closer to the vessel axis, whereas platelets accumulate near vessel walls. Directed platelet migration to the vessel walls promotes their hemostatic function. The mechanisms underlying this migration remain poorly understood, although various hypotheses have been proposed to explain this phenomenon (e.g., the available volume model and the drift-flux model). To study this issue, we constructed a mathematical model that predicts the platelet distribution profile across the flow in the presence of erythrocytes. This model considers platelet and erythrocyte dimensions and assumes an even platelet distribution between erythrocytes. The model predictions agree with available experimental data for near-wall layer margination using platelets and platelet-modeling particles and the lateral migration rate for these particles. Our analysis shows that the strong expulsion of the platelets from the core to the periphery of the blood vessel may mainly arise from the finite size of the platelets, which impedes their positioning in between the densely packed erythrocytes in the core. This result provides what we believe is a new insight into the rheological control of platelet hemostasis by erythrocytes.

Blood Coagulation and Propagation of Autowaves in Flow

This study analyses the effect of flow and boundary reactions on spatial propagation of waves of blood coagulation. A simple model of coagulation in plasma consisting of three differential reaction-diffusion equations was used for numerical simulations. The vessel was simulated as a two-dimensional channel of constant width, and the anticoagulant influence of thrombomodulin present on the undamaged vessel wall was taken into account. The results of the simulations showed that this inhibition could stop the coagulation process in the absence of flow in narrow channels. For the used mathematical model of coagulation this was the case if the width was below 0.2 mm. In wider vessels, the process could be stopped by the rapid blood flow. The required flow rate increased with the increase of the damage region size. For example, in a 0.5-mm wide channel with 1-mm long damage region, the propagation of coagulation may be terminated at the flow rate of more than 20 mm/min.

Continuous mathematical model of platelet thrombus formation in blood flow

An injury of a blood vessel requires quick repairing of the wound in order to prevent a loss of blood. This is done by the hemostatic system. The key point of its work is the formation of an aggregate from special blood elements, namely, platelets. The construction of a mathematical model of the formation of a thrombocyte aggregate with an adequate representation of its physical, chemical, and biological processes is an extremely complicated problem. A large size of platelets compared to that of molecules, strong inhomogeneity of their distribution across the blood flow, high shear velocities, the moving boundary of the aggregate, the interdependence of its growth and the blood flux hamper the construction of closed mathematical models convenient for biologists. We propose a new PDE-based model of a thrombocyte aggregate formation. In this model, the movement of its boundary due to the adhesion and detachment of platelets is determined by the level set method. The model takes into account the distribution inhomogeneity of erythrocytes and platelets across the blood flow, the invertible adhesion of platelets, their activation, secretion, and aggregation. The calculation results are in accordance with the experimental data concerning the kinetics of the ADP-evoked growth of a thrombus in vivo for different flow velocities.

Some relations between different characteristics of selection

Consider the action of selection with fitness w(x) on a quantitative trait x. What selection, among those that produce the same value of selection differential, leads to minimal values of (a) genetic load, (b) variance of the relative fitness, and (c) variance of the trait after selection? We have shown that for (a) and (c) the answer is strict truncation, whereas for (b) the answer is linear selection. The results for (a) and (b) are true for any selection, while the result for (c) is true only for directional selection. Implications of these findings are discussed.

On Propagation of Excitation Waves in Moving Media: The FitzHugh-Nagumo Model

Background Existence of flows and convection is an essential and integral feature of many excitable media with wave propagation modes, such as blood coagulation or bioreactors. Methods/Results Here, propagation of two-dimensional waves is studied in parabolic channel flow of excitable medium of the FitzHugh-Nagumo type. Even if the stream velocity is hundreds of times higher that the wave velocity in motionless medium (), steady propagation of an excitation wave is eventually established. At high stream velocities, the wave does not span the channel from wall to wall, forming isolated excited regions, which we called “restrictons”. They are especially easy to observe when the model parameters are close to critical ones, at which waves disappear in still medium. In the subcritical region of parameters, a sufficiently fast stream can result in the survival of excitation moving, as a rule, in the form of “restrictons”. For downstream excitation waves, the axial portion of the channel is the most important one in determining their behavior. For upstream waves, the most important region of the channel is the near-wall boundary layers. The roles of transversal diffusion, and of approximate similarity with respect to stream velocity are discussed. Conclusions These findings clarify mechanisms of wave propagation and survival in flow.

On the relationship between the load and the variance of relative fitness

BackgroundOperation of natural selection can be characterized by a variety of quantities. Among them, variance of relative fitness V and load L are the most fundamental.ResultsAmong all modes of selection that produce a particular value V of the variance of relative fitness, the minimal value Lminof load L is produced by a mode under which fitness takes only two values, 0 and some positive value, and is equal to V/(1+V).ConclusionsAlthough it is impossible to deduce the load from knowledge of the variance of relative fitness alone, it is possible to determine the minimal load consistent with a particular variance of relative fitness. The concept of minimal load consistent with a particular biological phenomenon may be applicable to studying several aspects of natural selection.ReviewersThe manuscript was reviewed by Sergei Maslov, Alexander Gordon, and Eugene Koonin.