Question

The model of the Poisson promoter considered in the chapter assumed that the number of copies of the gene of interest was fixed at one. However, as a result of the replication of the chromosomal DNA, during some part of the cell cycle there will be two (or even more for rapidly dividing cells) copies of the gene of interest. In this problem, we imagine that during a fraction \(f\) of the cell cycle, there is one copy of our gene of interest and during the rest of the cell cycle there are two such copies. (a) Write down the appropriate distribution \(p(m)\) for \(m\) mRNA molecules as a function of the parameter \(f\) (b) Find \(\langle m\rangle\) (c) Find \(\left\langle m^{2}\right\rangle\) and use it to find the Fano factor. (d) Plot the Fano factor as a function of \(f\) for different choices of the mean mRNA copy number for a single promoter. How "Poissonian" do you expect an unregulated promoter to be? (Problem courtesy of Rob Brewster and Daniel Jones.)

Short answer

Depending on the parameters and their frequency distribution within a cell cycle, the mRNA production can vary in the 'Poissonness' with the Fano Factor indicating the variability. Large deviations from 1 (Fano factor) indicate non-Poissonian behavior and thus possibly a regulated process.

Step-by-step solution

Modelling mRNA Distribution with regard to gene copies

First, the appropriate distribution for \(m\) mRNA molecules as a function of \(f\) needs to be written down. Since the gene can either be present in single copy or double copy, two Poisson distributions should be used as follows: \(p(m) = (1-f) \cdot p_{1}(m) + f \cdot p_{2}(m)\), where \(p_{1}(m)\) and \(p_{2}(m)\) are the Poisson distributions for one and two gene copies, respectively.

Finding the mean mRNA number

The mean \(\langle m \rangle\) is found by multiplying each possible outcome \(m\) by its likelihood and summing: \(\langle m \rangle = (1-f) \cdot \langle m_{1} \rangle + f \cdot \langle m_{2} \rangle \). Since the Poisson parameter equals the mean for each part of the cycle, this expression simplifies to \(\langle m \rangle = (1-f) \cdot \mu_{1} + f \cdot \mu_{2}\), where \(\mu_{1}\) and \(\mu_{2}\) are the mean values for one and two gene copies, respectively.

Finding the second moment and Fano factor

The second moment \(\langle m^{2} \rangle\) is found similarly to the mean. The Fano factor is the variance-to-mean ratio, which for a Poisson distribution equals 1. Given that the variance of a Poisson distribution equals the mean, the Fano factor can thus be written as: Fano factor = \( \frac{(1-f) \cdot \mu_{1} + f \cdot 4 \cdot \mu_{2}}{(1-f) \cdot \mu_{1} + f \cdot 2 \cdot \mu_{2}} \) Note the factors 4 and 2 in the numerator and denominator that reflect the variance for one and two gene copies.

Plotting the Fano factor

To draw the Fano factor as a function of \(f\) for different choices of the mean mRNA copy number for a single promoter, one needs to compute the Fano factor for a range of \(f\) values and different \(\mu_{1}\) and \(\mu_{2}\) values, and then plot the results. If the mean copy number equals the noise (that is, the Fano factor equals 1), the promoter is 'Poissonian'. Large deviations indicate non-Poissonian behavior, usually due to regulation or other complex effects on gene expression.

Key concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time, space or other continua, when these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution is a very useful model for many different types of discrete data, especially in the field of biology, such as modeling the number of mutations in a strand of DNA, or, as in our exercise, the number of mRNA molecules produced by a cell.To model mRNA distribution based on gene replication, we must take into account that a cell can have different numbers of gene copies during its cycle. For instance, during the replication phase of the cell cycle (the ‘S’ phase), there are temporarily double the number of gene copies. This fluctuation is where our f parameter comes into play, representing the fraction of the cell cycle with double gene copies. Thus, we adjust our Poisson distribution to accommodate the varying gene copy numbers, which leads to a weighted sum of two distributions for the different copy states.

mRNA Distribution Modeling

In the context of cell cycle and gene replication, mRNA distribution modeling is a statistical tool that helps us predict the number of messenger RNA (mRNA) molecules transcribed from a gene at any point in time. This type of modeling is essential in understanding gene expression because mRNA levels provide a snapshot of the dynamic processes within a cell. The example exercise provided is one application of how mRNA distribution can be modeled, taking into account the varying number of gene copies due to the cell cycle.By integrating the idea that a gene can exist in one or two copies within a cycle, the modeling becomes more sophisticated than the standard single-copy assumption. We utilize the variable f to represent the fraction of the cell cycle where two copies exist, and consequently, the mRNA distribution is determined by combining two distinct Poisson distributions, each representing the gene copy number state. In real-world scenarios, gene expression is not static, and modeling the distribution this way allows us to capture the inherent variability in mRNA populations in cells.

Fano Factor

The Fano factor is a measure of dispersion or variability of a probability distribution and is especially useful in quantifying the noise in biological systems. In the simplest terms, if the Fano factor is equal to 1, the distribution is Poissonian which suggests that the observed variability (or noise) is solely due to random processes. If the Fano factor is greater or less than 1, it indicates additional sources of variability beyond what is expected from a Poisson process.In the step-by-step solution, the Fano factor calculation provides insight into how 'Poissonian' or stochastic the gene expression is under different cell cycle conditions. This is significant because biological processes are influenced by both randomness and regulation. If we observe a Fano factor that deviates from 1 when the gene copies vary during the cell cycle, it suggests that there are other factors influencing mRNA production than just random chance, potentially pointing to deeper regulatory mechanisms. Thus, the Fano factor extends beyond mere calculation to become a window into the complexity of gene expression regulation.

Cell Cycle Gene Replication

Gene replication during the cell cycle is a fundamental process that ensures that each daughter cell receives a complete set of genetic instructions. However, this replication also affects the regulation of gene expression. In the context of the given exercise, understanding how gene replication influences mRNA production during the cell cycle is necessary to predict mRNA levels accurately. The fraction of the cell cycle with two gene copies, denoted as f in our model, is a key parameter because it reflects the changes in the gene copy number that occur during cell division.The fact that there are two gene copies at some stage in the cell cycle significantly affects the statistical nature of mRNA production. As the exercise suggests, the simple one-copy Poisson model is adapted to a more complex situation where the temporal variation in gene copy number during the cell cycle is accounted for. This level of detail is needed if we are to understand and predict gene expression patterns that result from the natural process of cell division and replication, which in turn provides a deeper understanding of both normal cell function and the abnormalities that can lead to diseases like cancer.