Linear single-cell kinetics. Consider an organism for which division occurs at cell mass M, birth at

Linear single-cell kinetics. Consider an organism for
which division occurs at cell mass M, birth at cell mass M/2, and the
single-cell mass growth rate follow = k1mþk2.

(a) Find the normalized, steady-state cell mass
distribution  for this organism in a chemo stat with dilution rate D.

(b) Find the relation amongD,k1, and k2.

(c) Because there is more than one parameter in the rate
expression for the single-cell kinetics, it is not possible to use the last
result to simplify the expression for the distribution of states and write it
solely as a function of the dilution rate in the

Linear single-cell kinetics. Consider an organism for
which division occurs at cell mass M, birth at cell mass M/2, and the
single-cell mass growth rate follow = k1mþk2.

(a) Find the normalized, steady-state cell mass
distribution  for this organism in a chemo stat with dilution rate D.

(b) Find the relation amongD,k1, and k2.

(c) Because there is more than one parameter in the rate
expression for the single-cell kinetics, it is not possible to use the last
result to simplify the expression for the distribution of states and write it
solely as a function of the dilution rate in the chemo stat. There are simply
not enough equations to solve fork1andk2in terms of D. This problem can be
resolved if one postulates kinetic expressions for both parameters in terms of
the concentration of the limiting substrate. From these equations one can
eliminate the sub strate concentration and obtain one of the parameters in
terms of the other. This result can then be used with the expression obtained
in

(a) to find all the parameter sin terms of the dilution
rate. Demonstrate this procedure for the model, above assuming that

S is the concentration of the growth-limiting substrate
and the parameters ki; max and K must be determined from experiments. Express k2
as a function of k1, and substitute the result into the expression
obtained in

(b), solve for k1 as a function of D, and
eliminate k2 and k1 from the expression.