Math 19. Lecture 20Separation of Variables (II)T. JudsonFall 20041 Modeling the Density of ProteinIt is known t hat 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(A00(x)B(y) + A(x)B00(y)) − A(x)B(y) = 0orµA00(x)A(x)+B00(y)B(y)= r.The first term of the expression inside the parentheses of the last equationis a function of x and the 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 that1AA00=rµ− λ (2)1BB00= 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 B00= λB. There a r e t hree cases.• If λ > 0, thenB = α e√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 Referen ces• C. Taubes. Modeling Differential Eq uations in Biology. Prentice Hall,Upper Saddle River, NJ, 200 1. Chapter
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