Modeling.

With D.I.V.E.R.T. it is possible to create a large library of a mutant gene in vivo. An important part of creating a library is the amount of mutations introduced in this gene. Few mutations may not be enough to change a gene’s property, a huge amount of mutations, however, can lead to unfavorable effects regardless of small positive influences. Thus, the exact knowledge of the distribution of mutations is vital for creating functional libraries with D.I.V.E.R.T. Underneath, we are modeling how the distribution of clones containing a different amount of mutations changes over time and how this distribution is affected by a different growth rate µ and mutation rate f.

Introduction – total cell count:

The growth of microorganisms, or in other words the total cell count, in liquid culture can be described as:

The alteration of the total cell count X(t) over a specific time t depends on the specific growth rate µ and the total cell count at this specific time X(t). By solving this differential equation, the total cell count at any time can be calculated by:

This formula shows the well-known exponential growth of microorganisms in liquid culture in excess of substrate. Whereas the total cell count at a specific time t is annotated as X(t), the cell count of clones with i mutations is written as Ai(t).

Solution of the differential equation for i = 0:

The alteration of the cell count with 0 mutations, A0(t), over the time t depends on two terms. Firstly, A0(t) declines due to the constant mutation of cells with a rate of f [h-1]. Secondly, A0(t) rises because some of the cells propagate with a rate of µ [h-1]. By solving this differential equation, the count of cells with 0 mutations at any time t [h] can be described as:

A0(0) is the total cell count in the moment when D.I.V.E.R.T is induced.

Solution of the differential equation for i = 1:

Contrary to the equation (3), the alteration of the cell count with 1 mutation, A1(t), over the time t depends now on three terms. The first two terms, the decline of A1(t) due to constant mutations of cells with a rate of f and a rise of A1(t) due to cell propagation with a rate of µ are selfsame. However, an additional term describes the amount of cells with previously 0 mutations that mutate in the time t with the exact same rate f. Hence equation (5) contains A0(t) which was already solved in (4).

The equation (6) is solved:

Solution of the differential equation for i = 2:

The equation for the cell count with 2 mutations, (8), works completely similar to the equation for the cell count with 1 mutation (5). The alteration depends on the three same terms: decline due to mutation of cells with previously 2 mutations, rise due to cell propagation and rise due to mutation of cells with previously 1 mutation. By applying the solution of A1(t) from equation (7) into equation (8), it can be written as: