che359_sp10_hw8_solution

# che359_sp10_hw8_solution - Page 1 of 7 ChE 359 Homework #8...

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Unformatted text preview: Page 1 of 7 ChE 359 Homework #8 due Thursday, April 8, 2010 at 2:30 pm 1. (45 points) Purified embryonic cells in culture aggregate via cell-cell adhesion into spheres, or “spheroids”; Malcolm Steinberg pioneered the view that cells could be viewed thermodynamically as analogs of liquids, such that their assembly is driven by interfacial tension. The formation of an interface results in a surface free energy of: F interface = cells,medium a , where cells,medium is the interfacial tension between the cell aggregate and the surrounding culture medium, and a is the surface area of the aggregate. What happens when two different cell types, having different interfacial tensions, are mixed? The cells will “sort” to minimize the surface contributions to the total free energy of the system. Let’s call the first cell type “A”, the second cell type “B”, and the surrounding culture medium “C”. Thus, we have three possible outcomes to consider, as shown to the right. a. (5 points) We will derive the surface free energies of all three systems. Let’s start with the m iddle system, where the interfacial area of the small sphere (both are small spheres) is a’ . Derive an expression for the surface free energy. The middle system has two cells, A and B, both of which have an interfacial area a’. Thus, if we say that ?? is the interfacial tension between cell A and culture medium C, and ?? is the interfacial tension between cell B and culture medium, then F= ?? ¡ ′ + ?? ¡ ′ b. (10 points) Given that the large spherical shell has the same volume as the small sphere, derive an expression for the (outer) surface area of the spherical shell as a function of a’ in the other two cases. Then, derive expressions for the surface free energies. We need to calculate the interfacial area of the larger circle including the shell and the inside cell. Since the shell has the same volume as the cell, so let’s give the small cell a radius of r s and the larger circle including the shell and the inside cell a radius of r L . Then 4 ¢ £ 3 3 − 4 ¢ ¤ 3 3 = 4 ¢ ¤ 3 3 , ?¢ ¢ £ 3 = 2 ¢ ¤ 3 and ¢ ¤ ¢ £ = 0.5 1 3 ChE 359 SP2010 - Homework #8 Page 2 of 7 Since a’ = 4 ? ? 2 , the interfacial area we want to calculate is 4 ? ? 2 = 4 0.5 1 3 ? ? ¡ 2 = 2 2/3 4 ¢? ? £ 2 = 2 2/3 a’. Use ¤¥ to represent the interfacial tension between cell B and A, then For the case on the left, F= ¤¥ ¦ ′ + ¤§ 2 2/3 ¦ ′ For the case on the right, F= ¤¥ ¦ ′ + ¥§ 2 2/3 ¦ ′ c. (10 points) Foty and Steinberg (R. A. Foty et al. Surface tensions of embryonic tissues predict their mutual envelopment behavior....
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## This note was uploaded on 02/09/2011 for the course CHE 359 taught by Professor Cynthialo during the Spring '10 term at Washington University in St. Louis.

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che359_sp10_hw8_solution - Page 1 of 7 ChE 359 Homework #8...

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