Data File 1: Mature red blood cells: from optical model to inverse light-scattering problem

Coefficients of polynomial approximation Originally published in Biomedical Optics Express on 01 April 2016 (boe-7-4-1305

Method for the simulation of blood platelet shape and its evolution during activation

<div><p>We present a simple physically based quantitative model of blood platelet shape and its evolution during agonist-induced activation. The model is based on the consideration of two major cytoskeletal elements: the marginal band of microtubules and the submembrane cortex. Mathematically, we consider the problem of minimization of surface area constrained to confine the marginal band and a certain cellular volume. For resting platelets, the marginal band appears as a peripheral ring, allowing for the analytical solution of the minimization problem. Upon activation, the marginal band coils out of plane and forms 3D convoluted structure. We show that its shape is well approximated by an overcurved circle, a mathematical concept of closed curve with constant excessive curvature. Possible mechanisms leading to such marginal band coiling are discussed, resulting in simple parametric expression for the marginal band shape during platelet activation. The excessive curvature of marginal band is a convenient state variable which tracks the progress of activation. The cell surface is determined using numerical optimization. The shapes are strictly mathematically defined by only three parameters and show good agreement with literature data. They can be utilized in simulation of platelets interaction with different physical fields, e.g. for the description of hydrodynamic and mechanical properties of platelets, leading to better understanding of platelets margination and adhesion and thrombus formation in blood flow. It would also facilitate precise characterization of platelets in clinical diagnosis, where a novel optical model is needed for the correct solution of inverse light-scattering problem.</p></div

Method for the simulation of blood platelet shape and its evolution during activation - Fig 1

<p>(A) One-to-one comparison of experimental images of coiled platelets marginal band (<i>reused with permissions from: Diagouraga et al. Journal of Cell Biology</i>. <i>204:177–185</i>. <i>DOI</i>: <a href="https://doi.org/10.1083/jcb.201306085" target="_blank">10.1083/jcb.201306085</a>) and manually-fitted overcurved circles. Overcurvature is listed near each plotted curve. (B) Examples of the overcurved circles. (C) Illustration of circular tube modeling the initial flat peripheral ring of microtubules. (D,E) Models of coiled marginal band in platelets with different overcurvature and non-zero thickness.</p

All possible axisymmetric profiles of resting platelets as obtained by the solution of variational problem (4).

<p>Black circles correspond to the marginal band cross-sections. Volume increases from top to bottom, curvature increases when going from the first to the fifth row, but decreases for the last row.</p

Possible mechanism of the overcurvature formation.

<p>Molecular motors induce relative sliding of microtubules, which are additionally cross-linked by bridge proteins. This leads to the formation of excessive curvature in the microtubules bundle, which manifests itself by the out-of-plane coiling of marginal band.</p

Overcurvature (<i>O</i>_p)–Dimensionless volume (<i>v</i>) phase diagram of platelets model morphologies.

<p>Area I corresponds to resting platelets with overcurvature = 1; Area II stands for activated platelets (hypothetically reversible activation); Area III is where platelets become sphered (irreversible activation).</p

Example of platelet model with 20 pseudopodia.

<p>Their total relative volume is 0.01.</p

Illustration of platelet surface optimization problem.

<p>The surface is made by the revolution of profile {<i>x</i>(<i>t</i>), <i>y</i>(<i>t</i>)} (thick line) around the ordinate axis. It consists of two parts: free (light red) and attached to the marginal band (green). Point <i>t</i> = <i>u</i> corresponds to the contact between one of the mobile subdomain and the fixed part, and points <i>t</i> = 0 and <i>t</i> = <i>T</i> are where the the surface intersects the axis <i>x</i> = 0.</p

Method for the simulation of blood platelet shape and its evolution during activation - Fig 8

<p>A. Light scattering model obtained by filling of platelet model with dipoles; incident wave propagates from below. B. Result of light scattering simulation—controur plot of <i>S</i>₁₁ element of the Mueller matrix in logarithmic scale versus polar and azimuthal scattering angles.</p

Scheme of the construction of blood platelet shape model (A–E) and the soap bubble supported by the overcurved wire ring (F), which presents the real-world approximation of the optimization problem solution for the mobile surface parts.

<p>Scheme of the construction of blood platelet shape model (A–E) and the soap bubble supported by the overcurved wire ring (F), which presents the real-world approximation of the optimization problem solution for the mobile surface parts.</p