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Classical Mechanics

# Kinematics in gene expression

Cellular identity is a function of the sets of proteins (or enzymes) produced by the cell. In all cells, protein production proceeds according to the central dogma. Information permanently stored in the genome (DNA) is "transcribed" into messenger RNA (mRNA) by polymerases. This mRNA is then "translated" into the corresponding protein by the ribosome.

There is plenty of interesting chemistry associated with this process, but here we'll focus on its kinematics.

The first step of gene expression is making an RNA copy out of the information in the genome, called a messenger RNA. Polymerases are the molecular machines that carry out this process.

For simplicity, let's assume that our genome has just one copy of the gene that we're interested in, so that $$g=1$$. The gene has a sequence (start site) that attracts polymerase molecules to bind and start making the messenger RNA. The overall rate of mRNA production is a function of the number of polymerase enzymes in the cell as well as the speed at which any given polymerase works, called the transcription rate.

We can wrap all this information into a single rate $$\alpha_m$$, and we can say that $\frac{dm}{dt}= \alpha_m.$

Suppose $$\alpha_m = \frac13$$ molecules per second. What is the number of mRNA molecules produced in $$\SI{1}{\minute}$$?

The second step of gene expression is the translation of each mRNA molecule into protein by the ribosome. As the ribosome proceeds along the mRNA, the resultant protein molecule starts to fold onto itself, adopting the three dimensional structure that gives it its specific chemical properties.

Similar to the case of transcription, the rate of translation $$\alpha_p$$ is a function of the number of ribosomes and the concentration of the protein building blocks. The overall rate of production is given by the translation rate per mRNA, multiplied by the total number of mRNAs $$m$$. We can write $\frac{dp}{dt} = \alpha_p m,$ or equivalently $p = p_0 + \int\alpha_p m\ dt.$

You may notice that the system of equations we've written down closely resembles the case of 1D kinematics. In fact, we can make a one-to-one mapping from velocity and acceleration to the variables in our system.

Which of the following is the correct relationship between the two systems?

\begin{align} \textrm{A.} \quad &\textrm{Transcription} \longleftrightarrow\textrm{Acceleration} \\ &\textrm{Translation} \longleftrightarrow\textrm{Velocity} \\ &\\ \textrm{B.} \quad &\textrm{Translation} \longleftrightarrow\textrm{Acceleration} \\ &\textrm{Transcription} \longleftrightarrow\textrm{Velocity} \end{align}

Up to now, we've considered the case where RNA polymerase has a built-in attraction for each gene in the genome. If this were the whole story, cells would produce every possible messenger RNA all of the time, which we know is not the case!

To prevent this wasteful production, cells have a mechanism that can guard access to the RNA polymerase called induction. The basic logic of induction is that RNA polymerase will not be able to produce messenger RNA unless a specific molecule, the inducer, is present inside the cell.

For example, bacteria will not produce the enzyme that help them eat the sugar lactose unless lactose is present inside the cell (specifically its breakdown product, allolactose).

Suppose that the transcription rate is given by $$\alpha_m = \frac13$$ messenger RNA per second and the translation rate by $$\alpha_p = \frac{1}{10}$$ molecules per messenger RNA per second.

How many molecules of the protein are produced by $$\SI{50}{\second}$$ post-induction, assuming that the inducer molecule is not present in the culture until $$t=0$$?

Details and Assumptions:

• Assume that protein production starts as soon as the inducer is introduced to the culture.

What can we change to accelerate the rate of production of a given enzyme?

In general, there will be a delay, $$\Delta T$$, between the time that we introduce the inducer molecule to the medium and the time that we can first observe enzyme molecules. This lag reflects the time required to synthesize the first messenger RNA molecule and translate it into protein. This time is of great interest as it captures the quantitative behavior of the molecular machinery.

We can find this lag time by fitting our data for $$p$$ vs. $$T$$ to find its intercept.

However, because we're biologists, we only know how to fit lines!

Which of the following transformations can we apply to our data in order to find the point of intercept $$\Delta T$$?

Details and Assumptions:

• Suppose that we have a way to easily measure the number of completed molecules of the enzyme over time.

In this quiz, we analyzed the central dogma of gene expression and proposed a set of governing equations based on some elementary facts from biology. We realized that these equations are equivalent to the equations of 1D kinematics, and that we can therefore apply them directly to our system.

Finally, we showed that we can use this model to measure the molecular timing of gene expression.

In general, the fundamental problems of nature are limited, and keep reappearing in various contexts. It is to our advantage to see through these disguises so as to map our knowledge from one domain directly to another, when possible.

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