Limiting enzyme: Protein X is an enzyme that acts on a substrate to provide bene
ID: 218538 • Letter: L
Question
Limiting enzyme: Protein X is an enzyme that acts on a substrate to provide benefit to the cell. L is the substrate concentration. Calculate the fitness function f(X,L) assuming a cost that is linear in the protein concentration , c~–?X, and benefit that is a Michaelis-Menten function of the protein concentration , b(L,X) = b0 L X/(X+K), appropriate for cases where enzyme X, rather than its substrate, is limiting.
a. Calculate the optimal enzyme level as a function of L and K.?
b. What is the minimal substrate level Lc required for maintenance of the gene for X by the organism? When is the gene that encodes X lost? Explain.
Explanation / Answer
Protein X acts on substrate L.
Assume that the cost function is linear, i.e. cost= aX ( a is an arbitrary constant ).
The benefit is given as benefit=(b0) * (LX) / ( (X+K)
Fitness function f(X, L)= benefit - cost = [(b0 * L * X) / (X + K) ] - [aX]
(a) At optimal enzyme concentration, d(f(X, L))/dX will be zero ( for any system, slope of tangent at maxima / minima is zero ).
d(f(X, L))/dX = [(b0*L) * ( X+K ) - ( b0*L*X )*(1)] / [ (X+K)^2] - a
0 = [(b0*L) ( X + K - X )] / [(X + K)^2] - a
a = bo*L*K / [(X + K)^2]
[(X + K)^2] = b0*L*K / a
X ( optimal enzyme concentration ) = sqrt [ b0*L*K/a] - K
(b) Maintenance of gene will occur only if the benefit to the cell is greater than the cost of producing protein X.
i.e. the fitness function has to be > 0
i.e. (b0*L*X) / ( X+K ) - aX > 0
(b0)*(L) / ( X+K ) > a
L > a * (X + K) / (b0) ( minimal substrate concentration for gene to persist )
Gene will be lost once the benefit to cell is none or the costs exceed the benefits. This will occur when
L <= (a) * (X + K ) / (b0)
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.