What my research is about

[luzula]

sionnain asked what my research is about. My first answer was that I study partial differential equations, and the kind of things I ask are: If an operator has a particular regularity, what regularity will the solution have? How does the solution behave at the boundary?

At which point sionnain asked what those terms mean. *g*

First, you need to remember what an equation is. This is an example:

x + 1 = 2

The important thing with an equation is that there's something that's unknown, and your goal is to find out what the unknown thing is equal to. In this case the unknown thing is the "x", and it's easy to see that it must be equal to 1. So x=1 is the solution of the equation.

In differential equations, which is what I do, the unknown thing isn't a number any more, but a function. A function is something where you enter one thing (often a number, which is called the variable), and the function does something with the variable according to a rule, and gives you a result. For example, the function

f(x) = 2x

takes the number x and doubles it. More generally, you can think of a function kind of as a recipe--you enter in all the ingredients, the recipe tells you what to do with them, and then you get a cake as the output.

Often a function isn't defined everywhere, that is, there's a rule that says we can only enter certain numbers. For example, we could decide that we only allow numbers between 0 and 1. In this example, the boundary of the area where the function is defined is the 0 and the 1, because they are at the "edges" of the area.

The regularity of a function has to do with how much the output changes if you only change the input a little bit. If the output jumps around a lot when you change the input just a little bit, it has low regularity. This is generally undesirable in applications.

An operator is sort of like a meta-function. Ordinary functions take a number as input and give you another number as output. Instead, an operator takes a function as input and gives you another function as output. Operators are often used to construct differential equations (also, they often involve taking the derivative of the function, if you know what that is). You get an equation like this: if A is an operator, f(x) is an unknown function and g(x) is a known function, then

A(f(x)) = g(x)

is an equation. Often, one also specifies the values of f(x) at the boundary. We want to find the solution, i e figure out what kind of function f(x) is.

Hopefully these sentences will make more sense now: If an operator has a particular regularity, what regularity will the solution have? How does the solution behave at the boundary?

So what do you use these things for? Well, partial differential equations have lots of applications. For example, you can think of heat as a function of three space variables and one time variable. We use a particular point in space and time as the input, and the heat function tells us the temperature at that point. We can specify the temperature in a room at a specific time, and also specify what the walls do (maybe they are insulating?). This is equivalent to specifying the values of the heat function at the boundary. Then, solving the heat equation is the same as figuring out how the temperature inside the room changes as time passes.

Comments

[meresy.livejournal.com]

Basics of basics, since I have not studied this very closely myself. My thoughts on RNAi are somewhat holey.

So, there is something called the central dogma of biology, that is genes (DNA) are transcribed into copies or RNA messages (mRNA) which are then translated into proteins (the translation is done by a big RNA enzyme or ribozyme, which is important later). Proteins are basically the tools the cell uses to do almost everything -- from making more DNA, to producing chemicals and digesting others, infecting hosts. Now, depending on the type of cell, its life cycle, etc. there will be a different repertoire of proteins produced at any given time. This is controlled at all three levels -- by blocking or inducing transcription, blocking or inducing translation, or by modifying the produced protein.

And, proteins being the go-to for most everything, the things that modulate gene expression tend to be effector proteins -- inducers, repressors, enzymes that digest RNA or proteins, etc. Insulin is a protein. Hemoglobin is a protein. Their production is controlled by other proteins. You get the idea. For a very long time, one assumed that if a modification or modulation was happening, it was about how proteins were affecting the gene or the message or the product.

But! In genomes there are many little bits of DNA that are transcribed into short RNAs that do not go on to be translated into proteins, or that are trimmed off of larger pieces. So if it's not a message, why is the cell wasting time and energy producing these little bits? The way you test the utility of anything in biology is by knock-out -- removing it and seeing what happens. Someone deleted these "extras"... and the result was that in some cases crazy things happened to the progression of the cell's life cycle and processes, even though no protein-encoding gene had been affected.

It turned out that the cell can control the RNA messages with other RNAs -- complementary bits and pieces that can block translation of proteins because the stick to essential parts of the messenger RNA. Since then they have discovered whole systems of small interfering RNAs, microRNA, satelliteRNA and other things that are manipulating processes in ways we didn't even know about. It's turned out to be very, very important for understanding developmental biology, and is even useful clinically.

Also it helps explain and confirm some of the ideas about the recursive issues in cell biology (which came first, DNA or protein?) Turns out RNA, which was always sort of thought as an adaptor between DNA and proteins, may be the precusor molecule. (Look up "RNA World Hypothesis" for good time).