Dynamics of a microsphere in an anisotropic gel ... - UCSB Engineering
Dynamics of a microsphere in an anisotropic gel: a frontier in intracellular microrheology Manuel Gómez-González, Kathryn Osterday and Juan C. del Álamo Mechanical and Aerospace Engineering Department, University of California at San Diego, San Diego, CA 92093-0411 Tel: 858-822-1367 E-mail: [email protected]
Summary: Directionality in the mechanical properties of the cell is key to cell function and relation with its environment. We have developed a new technique to calculate the Directional Shear Moduli of the cell cytoplasm from the measured displacements of embedded particles. The mechanical properties of the cell cytoplasm play a determinant role in many cell functions. Not only the magnitude but also the directionality is key to understand the cell behavior. For example, dysfunctional cells present different mechanical properties than healthy cells, stem cells are known to differently specialize according to the environment stiffness, etc. Microrheological techniques have the potential to be used, among others, to assess the ability of individual or groups of cells to perform certain functions and interact with their environment, as a screening technique for cell diseases such as cancer, or as a control test for tissue engineering. Standard Particle Tracking Microrheology [1] measures the motion of microparticles embedded in the cell cytoplasm and estimates its mechanical properties, but it relies on the isotropy of the medium. However, the motion of particles in the cell cytoplasm (Fig. 1a) is not isotropic but it follows the cytoplasm alignment. Current lack of understanding of the dynamics of particles in this complex environment challenges the application of microrheology to live cells. To overcome this difficulty, we studied the drag force experienced by a microsphere in an anisotropic viscoelastic network (the cytoskeleton), permeated by a background liquid (the cytosol). In the limit of strong frictional coupling between the network and the liquid, the flow around the sphere is modeled with a generalized Stokes equation using several viscosity parameters. We solve this equation analytically to provide new closed-form microrheology formulae that relate the resistance measured experimentally to the anisotropic properties of the network. a)
b)
Fig. 1. a) Particles inside the cell don’t move as in an isotropic liquid [2]. b) Ratio of shear modulus calculated with our new technique (red) and comparison with the isotropic (blue) and effective viscosity approach (green): they underestimate the directionality of the medium.
In Fig. 1b we represent the results of our new technique and compare them to previous methods. The x-axis represents the ratio of Mean Squared Displacements of a probe along principal directions, and the y-axis the anisotropy of the medium defined as the ratio of principal shear moduli calculated. Due to the incompressibility of the background liquid, the motion of a particle is coupled with the shear moduli along every direction, and the measurements of previous techniques ([1] and [2]) render an average of the actual shear moduli. As a result, previous techniques work well for quasi-isotropic media, but highly underestimate the directionality of the mechanical properties for moderately anisotropic media. In the range of values of interest in cell mechanics, previous techniques render errors in the order of 200% for the Directional Shear Moduli. [1] Mason T. G., Weitz D. A., “Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids”, Physical Review Letters, vol: 74, 1995. [2] del Alamo J. C., et al, “Anisotropic rheology and directional mechanotransduction in vascular endothelial cells”, PNAS, vol: 105, 2008.
Summary: Directionality in the mechanical properties of the cell is key to cell function and relation with its environment. We have developed a new technique to calculate the Directional Shear Moduli of the cell cytoplasm from the measured displacements of embedded particles. The mechanical properties of the cell cytoplasm play a determinant role in many cell functions. Not only the magnitude but also the directionality is key to understand the cell behavior. For example, dysfunctional cells present different mechanical properties than healthy cells, stem cells are known to differently specialize according to the environment stiffness, etc. Microrheological techniques have the potential to be used, among others, to assess the ability of individual or groups of cells to perform certain functions and interact with their environment, as a screening technique for cell diseases such as cancer, or as a control test for tissue engineering. Standard Particle Tracking Microrheology [1] measures the motion of microparticles embedded in the cell cytoplasm and estimates its mechanical properties, but it relies on the isotropy of the medium. However, the motion of particles in the cell cytoplasm (Fig. 1a) is not isotropic but it follows the cytoplasm alignment. Current lack of understanding of the dynamics of particles in this complex environment challenges the application of microrheology to live cells. To overcome this difficulty, we studied the drag force experienced by a microsphere in an anisotropic viscoelastic network (the cytoskeleton), permeated by a background liquid (the cytosol). In the limit of strong frictional coupling between the network and the liquid, the flow around the sphere is modeled with a generalized Stokes equation using several viscosity parameters. We solve this equation analytically to provide new closed-form microrheology formulae that relate the resistance measured experimentally to the anisotropic properties of the network. a)
b)
Fig. 1. a) Particles inside the cell don’t move as in an isotropic liquid [2]. b) Ratio of shear modulus calculated with our new technique (red) and comparison with the isotropic (blue) and effective viscosity approach (green): they underestimate the directionality of the medium.
In Fig. 1b we represent the results of our new technique and compare them to previous methods. The x-axis represents the ratio of Mean Squared Displacements of a probe along principal directions, and the y-axis the anisotropy of the medium defined as the ratio of principal shear moduli calculated. Due to the incompressibility of the background liquid, the motion of a particle is coupled with the shear moduli along every direction, and the measurements of previous techniques ([1] and [2]) render an average of the actual shear moduli. As a result, previous techniques work well for quasi-isotropic media, but highly underestimate the directionality of the mechanical properties for moderately anisotropic media. In the range of values of interest in cell mechanics, previous techniques render errors in the order of 200% for the Directional Shear Moduli. [1] Mason T. G., Weitz D. A., “Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids”, Physical Review Letters, vol: 74, 1995. [2] del Alamo J. C., et al, “Anisotropic rheology and directional mechanotransduction in vascular endothelial cells”, PNAS, vol: 105, 2008.
