# 25 this element is under pressure p we wish to obtain

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25 This element is under pressure p; we wish to obtain a relationship between gravity and pressure. In x-direction: p dy dz-(p +𝜕𝜕𝑝𝑝𝜕𝜕𝑥𝑥) dy dz = 0𝜕𝜕𝑝𝑝𝜕𝜕𝑥𝑥= 0In z-direction: p dx dy-(p +𝜕𝜕𝑝𝑝𝜕𝜕𝑧𝑧dz) dx dy - 𝜌𝜌g dx dy dz = 0𝜕𝜕𝑝𝑝𝜕𝜕𝑧𝑧= - 𝜌𝜌𝜌𝜌For constant fluid density ρat some elevation h the pressure is 𝛥𝛥𝑝𝑝= - 𝜌𝜌g Going back to the micropipette… We will use this relationship between gravity and pressure in the micropipette aspiration problem. The smallest suction pressures produced by the micropipette aspiration experiment are on the order of a few µm of water. Example: a pressure head of 1µm of H2O creates a pressure p = - (1 g/ml) (9.81 m/ s2) 1 x 10-6-0.01 Pa (or 0.01 pN/µm2) Law of Laplace Let us now calculate the cell membrane's (cortical) tension using the Law of Laplace. As I have indicated previously, I would like you to think of this cortical tension as either the tension that develops in the cortical actin or the equivalent surface tension that develops if we treat the cell as a liquid drop. So, if we consider the cell to be modeled with a liquid drop, the law of Laplace is a relationship between the surface tension and pressure within a fluid drop that has a membrane with surface tension in it. Here we will derive the Law of Laplace for the simple case of a spherical drop of fluid with an internal pressure (pi, with units of force per area) and a uniform surface tension (Tc, with units of force per length). A free body diagram for half of the drop, including both the surface and the internal fluid, is shown in the figure below. Let us consider a summation of forces. The cell has an internal pressure, pi, and an external pressure, po (pressure has units of force per area). Its radius is Rc. Its membrane develops (cortical) tension Tc(units of force per length). Balance forces due to pi, po, TFι= pι(πRc2) Fo= po (πRc2) [note: the y-components cancel out and x-components are integrated over surface] (pι- po) πRc2= 2πRcTc[note: the unit for Tcis pN/µm] 𝑝𝑝𝑖𝑖− 𝑝𝑝0=2𝑇𝑇𝑐𝑐𝑅𝑅𝑐𝑐(1)
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c
26 Force due to 𝑝𝑝𝑝𝑝Force due to 𝑝𝑝This formula says that the cortical tension causes the inside of the cell to be pressurized with respect to the pressure outside of the cell. This derivation assumes the existence of a membrane surrounding the interior of the cell and a uniform state of stress, pι, inside of the cell. One can use a thermodynamic analysis to obtain an identical result. Now we will proceed with an analysis of micropipette aspiration. We assume that the cell has been aspirated such that the aspiration length (say Lp) is equal to the pipette radius (Rp). First, we examine a free body diagram, noting that the aspirated region is a hemisphere. The free body diagram for the cell portion inside the micropipette is shown below: Let us now create another free-body diagram for the cell portion inside the micropipette, assuming that the aspirated region is a hemisphere. From 𝛴𝛴𝐹𝐹= 0𝑝𝑝𝑖𝑖(𝜋𝜋𝑅𝑅𝑝𝑝2) +𝑝𝑝𝑝𝑝(𝜋𝜋𝑅𝑅𝑝𝑝2)− 𝑇𝑇𝑇𝑇(2𝜋𝜋𝑅𝑅𝑝𝑝) = 0𝑝𝑝𝑖𝑖+𝑝𝑝𝑝𝑝=2𝑇𝑇𝑐𝑐𝑅𝑅𝑝𝑝(2) 𝑖𝑖
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