During cell migration, cells become polarized, change their shape, and move in response to various cues, both internal and external . Many existing mathematical models of cell polarization are formulated in  one or two spatial dimensions and hence cannot accurately capture the effect of cell shape, as well as the response of the cell to signals from different directions in a three-dimensional environment. To study those effects , we introduce a three-dimensional reaction-diffusion model of a cell. As some key molecules in cell polarization, such as the small GTPases, can exist both membrane bound and soluble in the cytosol, we first look at the role of cell geometry on the membrane binding/unbinding dynamics of such molecules. We derive quite general conditions under which effective existing one or two-dimensional computational models are valid, and find novel renormalizations of parameters in the effective model. We then extend an established one-dimensional cell polarization pathway in our three-dimensional framework. Our simulations indicate that even in some quasi-one-dimensional scenarios, such as polarization of a cell along a linear growth factor gradient, the cell shape can influence the polarization behavior of the cell, with cells of some shape polarizing more efficiently than those of other shapes. We also investigate the role of the previously ignored membrane unbinding rate on polarization. Furthermore, we simulate the response of the cell when the external signal is changing directions, and we find that more symmetric cells can change their polarized state more effectively towards the new stimulus than cells which are elongated along the direction of the original stimulus  .
During cell migration, cells become polarized, change their shape, and move in response to various internal and external cues. Cell polarization is defined through the URLanization of molecules such as PI3K or small GTPases, and is determined by intracellular signaling networks. It results in directional forces through actin polymerization and myosin contractions. Many existing mathematical models of cell polarization are formulated in terms of reaction-diffusion systems of interacting molecules, and are often defined in one or two spatial dimensions . In this paper , we introduce a 3D reaction-diffusion model of interacting molecules in a single cell, and find that cell geometry has an important role affecting the capability of a cell to polarize, or change polarization when an external signal changes direction. Our results suggest a geometrical argument why more roundish cells can repolarize more effectively than cells which are elongated along the direction of the original stimulus , and thus enable roundish cells to turn faster, as has been observed in experiments. On the other hand, elongated cells preferentially polarize along their main axis even when a gradient stimulus appears from another direction. Furthermore, our 3D model can accurately capture the effect of binding and unbinding of important regulators of cell polarization to and from the cell membrane. This spatial separation of membrane and cytosol, not possible to capture in 1D or 2D models, leads to marked differences of our model from comparable lower-dimensional models .
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cues. Cell polarization is defined through the URLanization of molecules such as PI3K or small GTPases, and is determined by intracellular signaling networks. It results in directional forces through actin polymerization and myosin contractions.
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terms of reaction-diffusion systems of interacting molecules, and are often defined in
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and hence cannot accurately capture the effect of cell shape, as well as the response of the cell to signals from different directions in a three-dimensional environment. To study those effects
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three-dimensional
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a cell. As some key molecules in
interacting molecules in a single cell, and find that cell geometry has an important role affecting the capability of a
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polarization, such as the small GTPases, can exist both membrane bound and soluble in the cytosol, we first look at the role of cell geometry on the membrane binding/unbinding dynamics of such molecules. We derive quite general conditions under which effective existing one or two-dimensional computational models are valid, and find novel renormalizations of parameters in the effective model. We then extend an established one-dimensional cell polarization pathway in our three-dimensional framework. Our simulations indicate that even in some quasi-one-dimensional scenarios, such as polarization of a cell along a linear growth factor gradient, the cell shape can influence the polarization behavior of the cell, with cells of some shape polarizing more efficiently than those of other shapes. We also investigate the role of the previously ignored membrane unbinding rate on polarization. Furthermore, we simulate the response of the cell when the external signal is changing directions, and we find that more symmetric cells can change their polarized state more effectively towards the new stimulus
to polarize, or change polarization when an external signal changes direction. Our results suggest a geometrical argument why more roundish cells can repolarize more effectively
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, and thus enable roundish cells to turn faster, as has been observed in experiments. On the other hand, elongated cells preferentially polarize along their main axis even when a gradient stimulus appears from another direction. Furthermore, our 3D model can accurately capture the effect of binding and unbinding of important regulators of cell polarization to and from the cell membrane. This spatial separation of membrane and cytosol, not possible to capture in 1D or 2D models, leads to marked differences of our model from comparable lower-dimensional models