Transcription Network Basics: Part Three

Logic approximations, handling multiple inputs, and the dynamics of transcription networks

In Part Two, we looked at the two basic types of transcription factors — activators and repressors — and introduced an input function that mathematically models their behavior: the Hill function.

In this article, we'll introduce approximating this input function — capturing its essence rather than its details — so that we can easily analyze more complex input functions, like what happens when multiple transcription factors regulate a gene. Finally, we'll introduce the topic of transcription network dynamics, or how a cell's response (to produce or not produce a protein) looks like over time.

Logic approximations

The essence of an input function is a transition between two values — from high to low for an activator, and from low to high for a repressor — and a threshold that defines when the transition occurs.

The Hill functions we introduced in Part Two achieve this transition with a smooth, sigmoidal shape. In its simplest form, however, this function can be like a switch: either on or off, with a threshold for when turning on or off occurs. This kind of approximation is known as a logic approximation, and it greatly simplifies the behavior of the Hill function, or any other input function.

Mathematically, off can be represented as no proteins being produced, or , on as proteins being produced at the maximum level, or , and the threshold as . Then, we can write this logic approximation using a step function. For an activator, this looks like:

where , the step function, is either 0 or 1 depending on whether the active concentration is greater than the threshold concentration or not.

For a repressor, the logic is reversed: being on until the threshold is reached, after which it is off. This can be written as:

Let's take a look at this graphically:

Multiple Inputs

The primary benefit of logical approximations is that we can evaluate more complex input functions more easily. We can begin to look at genes that are regulated by multiple transcription factors and gauge their behavior without doing unnecessarily long calculations.

Let's look at the example of genes that are regulated by two activators. One gene might require that both activators are bound to the promoter in order to enable high levels of expression; just one is not enough. This represents an AND gate:

Here, both and need to be 1 for the gene to express at the level . If any one is 0, the gene is not expressed.

For another gene, either one binding will suffice — like an OR gate:

The combinations are not limited to Boolean operations. Transcription factors can also combine in an additive way:

Genes in multi-cellular organisms can have many inputs, occasionally more than a dozen, so you might see how quickly this can get complicated.

Even with just two transcription factors, the combinations above look pretty abstract. This section just introduces the concept; in future articles, we'll look at how exactly these logical operations — the AND and OR gates — are performed in organisms, and the benefits they provide.i

Dynamics and Response Time

So far, we've looked at how protein production is affected by transcription factors, but we haven't yet considered how it does so over time. When a cell receives membrane damage signals, how quickly does it make the repair proteins? The speed and way in which a cell responds to signals is extremely vital, which makes analyzing the dynamics and response time of transcription networks an important task in systems biology.

Let's start with a basic transcription interaction in a network, → . Recollect that → describes gene regulation; a short way of saying that when signal appears, it transforms into its active state and binds to the promoter of gene , which enables transcription of , and then translation, resulting finally in protein accumulating in the cell.

Simple diagram of gene expression, described by X → Y.

But how quickly does protein accumulate? We know, from the step function, that after a certain concentration threshold is crossed, the cell produces the maximal amount of protein. If it is produced over a certain time period, we can say that it is produced at a rate .i

This only covers protein production, but cells lose proteins as well. There are two processes that balance production: protein degradation (intentional destruction by other specialized proteins) and dilution (reduction in concentration due to an increase in cell volume).

Both processes reduce the concentration of in a cell. The degradation rate is , and the dilution rate is , both with units of . The total removal rate is the just the sum of the two:

Now let's look at how concentration changes. A change in the concentration of in a cell over time, , is given by the difference between its production and its removal:i

When the production rate matches the removal rate, a steady state is reached. The cell maintains a consistent concentration of , denoted by . Consistent concentration means of a change of , so this can be found by solving for :

The equation shows that the steady state is a ratio of the production rate and the removal rate. This makes sense: if the production rate is high, a high removal rate is required to keep the same steady state concentration.

Decay and accumulation

We now have the basic equations that describe the dynamics of simple regulation, or how the concentration of a protein behaves over time.

Let's use these to analyze some cellular scenarios. Scenario one: what happens when the input signal is gone, and so the production of stops ()? How does concentration deplete over time?

So, the loss of a signal leads to an exponential decay in protein concentration. Now let's consider the reverse scenario: what happens when a signal is introduced to a cell where protein production hasn't yet started ()? How does protein increase over time?

These equations set the baseline for how quickly a cell can respond to signals. In future articles, we'll look at how cells can speed up or slow down this response with different network structures, and why that might be important, if not crucial, in an organism's survival.

Summary

In these three introductory articles, we've explored several important concepts in transcription networks:

1. Transcription factors and gene expression: We introduced the concept of transcription factors as proteins that regulate gene expression in response to signals.

2. Transcription networks: We showed how transcription factors and genes form complex networks of interactions within cells and introduced simple notation and diagrams.

3. Activators and repressors: We explored the two main types of transcription factors and how they increase or decrease gene expression.

4. Mathematical models: We examined the Hill function as a way to mathematically model the relationship between transcription factor concentration and protein production rate, as well as the step function as a simplified representation of it.

5. Dynamics and response time: We explored how protein concentrations change over time with different production and removal rates, and how quickly a cell can respond to signals.

These concepts form the foundation for understanding more complex transcription networks and their behavior in living cells. They help us analyze how quickly cells can respond to environmental changes and how they maintain appropriate protein levels.

In future articles, we'll build upon these basics to explore more intricate network motifs like auto-regulation and feedback loops, and how cells use these principles to create sophisticated regulatory systems. We'll also look at real-world examples of how these networks function in various biological processes and how understanding them can lead to advances in fields like medicine and biotechnology.

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