Team:BOKU-Vienna/Model

Modeling

Model

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 A_i(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:

This equation is solved for A₂(t):

A pattern was recognized and a general solution for the cell count with i mutations at any specific time t was established.

Solution of the differential equation for i ∈ ℕ:

Depending on the parameters µ and f three different scenarios can be described. The sum of the equations for all numbers of mutation i, however, equals the total cell count:

Scenario I, µ > f:

When the growth rate µ [h^-1] exceeds the mutation rate f [h^-1], more cells are created than mutations are introduced. Thus, the total cell count, as well as the cell count with i mutations, grows exponentially (Fig. 1). As an advantage all cells statistically propagate before they mutate. This means that no clones with a possibly benefiting mutation are lost by further mutation. However, a disadvantage of this scenario is an also exponential growth of unmutated cells, which have to be sorted but do not contain any new information. This scenario is the most probable with D.I.V.E.R.T due to low mutation rates systems like D.I.V.E.R.T. achieve compared to the average growth speed of microorganisms like E. coli and S. cerevisiae.

Figure 1: Growth of the populations with i mutations and the total cell count when µ is bigger than f. Only curves for up to 3 mutations are shown. Total cell count at the beginning, X(0) = 10⁸ cells, µ = 0.2 h^-1, f = 0.1 h^-1.

Scenario II, µ < f:

In this scenario the mutation rate f exceeds the growth rate µ. Therefore, cell counts with i mutations have maxima even though the total cell count is rising exponentially (Fig. 2). This scenario is useful to create libraries with big amounts of clones with a certain number of mutations more quickly. However, cells with mutations can be lost again, because they statistically mutate before they propagate.

Figure 2: Growth of the populations with i mutations and the total cell count when µ is smaller than f. Only curves for up to 3 mutations are shown. Total cell count at the beginning, X(0) = 10⁸ cells, µ = 0.2 h^-1, f = 0.3 h^-1.

Scenario III, µ = f:

When the growth rate µ and the mutation rate f are the same. This means that during every cell propagation one mutation happens. As the total cell count rises exponentially again, the cell counts for i mutations increase to the power of i (Fig. 3). This combines advantages from both previous scenarios. Statistically no cell clones are getting lost, but at the same time the number of cells with no new mutations is not rising. By equalizing µ and f the sorting effort is minimized.

Figure 3: Growth of the populations with i mutations and the total cell count when µ is equal to f. Only curves for up to 3 mutations are shown. Total cell count at the beginning, X(0) = 10⁸ cells, µ = 0.2 h^-1, f = 0.2 h^-1.

Distribution of mutated cells:

Depending on the mutation rate and time the distribution of cell counts with i mutations varies. However, all scenarios are Poisson distributed.

The expected number of occurrences (λ) is calculated as:

Figure 4: Poisson distribution with different numbers of occurrences resulting from different mutation rates f (0.1 h^-1; 0.3 h^-1; 0.5 h^-1) at a time t (10 h). The step function is a result of the discrete property of the Poisson distribution.

Thoughts on the mutation rate f:

The average possibility for a mutation occurring per hour per cell, f, is depending on three different properties. Firstly, the error rate of the reverse transcriptase, which varies widely. For PCRs normally few errors are favored, however, systems like D.I.V.E.R.T. need higher error rates to work feasible. For our proof of concept, we used Moloney murine leukemia virus reverse transcriptase which has an approximate error rate of 2 * 10^-6 mutations per base in vitro¹. However, there are RTs with a lot higher error rates as well, but this would lead to undesired effects, for example introducing 2 mutations in one cycle. This leads to the second property, the target gene length. For longer target genes the possibility of introducing a random mutation through reverse transcription is higher than for shorter ones. The third property that influences the mutation rate f is the number of cycles per hour. It is hard to estimate, because of a lot factors, for example, the speed of transcription, reverse transcription and Flp/FRT mediated recombination (for approach I) all contribute to it. To sum up, the mutation rate f is the product of the error rate of the reverse transcriptase e, the target gene length l and the speed of D.I.V.E.R.T. cycles s.

[1]: Comparison of Moloney murine leukemia virus mutation rate with the fidelity of its reverse transcriptase in vitro. (1992) .