Math 19. Lecture 20Separation of Variables (II)T. JudsonFall 20051 Modeling the Density of ProteinIt is known that the concentration of certain proteins at any cell in an embryodetermines whether or not a particular gene is expressed in that cell. We willconsider a cell model of an embryo whereu(t, x, y)is the density of protein at time t and position (x, y). We will consider ourembryo to be square, [0, L] × [0, L], where Protein is produced along theleft-hand edge according tou(t, 0, y) = sinπyL.Observe that this function is zero at (0, 0) and (0, L). Assume also thatu(t, x, 0) = 0u(t, x, L) = 0u(t, L, y) = 0.1The protein will diffuse according to the equation∂u∂t= µ∂2u∂x2+∂2u∂y2− ru .Eventually, we will reach a steady-stateµ∂2u∂x2+∂2u∂y2− ru = 0. (1)2 Separation of VariablesIfu(x, y) = A(x)B(y),then equation (1) becomesµ(A′′(x)B(y) + A(x)B′′(y)) − A(x)B(y) = 0orµA′′(x)A(x)+B′′(y)B(y)= r.The first term of the expression inside the parentheses of the last equationis a function of x and t he second term is a function of y. Since x and yare independent variables and the equation is equal to a constant r, both ofthese terms must be constant. Therefore, we can assume that1AA′′=rµ− λ (2)1BB′′= λ. (3)2The boundary conditions now becomeA(0)B(y) = sinπyL,A(L)B(y) = 0,A(x)B(0) = 0,A(x)B(L) = 0.We first solve B′′= λB. There are three cases.• If λ > 0, thenB = αe√λ y+ βe−√λ y.• If λ = 0, thenB = α + βy.• If λ < 0, thenB = α cosp|λ| y + β cosp|λ| y.Applying the boundary condition A(0)B(y) = sin π/L, the only consistentcase occurs when λ < 0. If we let α = 0 and β = 1, thenA(0)B(y) = sinπyL,and λ = −π2/L2.Equation (2) now becomesd2Adx2=rµ+π2L2A.To simplify matters, we will letc =rµ+π2L2.Thus, we need to solve the equationd2Adx2= cA.In this case, c > 0. so the solutions must be of the formA(x) = αe√c x+ βe−√c x.3SinceA(0)B(y) = sinπyL,α + β = 1. Since A(L)B(y) = 0,αe√c L+ βe−√c L.Thus,α = −1e2√c L− 1β =e2√c Le2√c L− 1Thus,u(x, y) =−e√c x+ e2√c Le−√c xe2√c L− 1sinπyLwherec =rµ+π2L2.00.20.40.60.8100.20.40.60.8100.250.50.75100.20.40.60.8Readings and References• C. Taubes. Modeling Differential Equations in Biology. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter
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