Problem: end-to-end analysis of biological interactions at all timescales is hard; womp womp. No relationship explicitly between sequence, crystallography, md, etc. Also, some of them have time, some of them are frozen, etc.

Solution: use ML to glue multiple scales’ analysis together, using ML to

proteins can be encoded as hierarchies

1. protein functional behavior
2. secondary structure/primary structure
3. amino acids
4. sequences!

Slicing through the embedding space of GenSLMs can be used to identify these larger scale things from just the sequence by looking at the “general area” it exists in the latest space.

Problem: end-to-end analysis of biological interactions at all timescales is hard; womp womp. No relationship explicitly between sequence, crystallography, md, etc. Also, some of them have time, some of them are frozen, etc.

Solution: use ML to glue multiple scales’ analysis together, using ML to

story 1: proteins can be encoded as hierarchies

1. protein functional behavior
2. secondary structure/primary structure
3. amino acids
4. sequences!

Slicing through the embedding space of GenSLMs can be used to identify these larger scale things from just the sequence by looking at the “general area” it exists in the latest space.

For vector \(v\) in the span of orthogonal basis \(v_1, ..v_{n}\):

\begin{equation} v = c_1 v_1 + \dots + c_{n} v_{n} \end{equation}

we can write:

\begin{equation} c_{j} = \frac{v \cdot v_{j}}{ v_{j} \cdot v_{j}} \end{equation}

Proof:

\begin{equation} \langle v, v_{j} \rangle = c_{n} \langle v_{1}, v_{j} \rangle \dots \end{equation}

which is \(0\) for all cases that’s not \(\langle v_{j}, v_{j} \rangle\) as the \(v\) are orthogonal, and \(\mid v_{j} \mid^{2}\) for the case where it is.

Fourier Series and how to find them.

For a function given at some interval of length \(l\), then the function can be written at:

\begin{equation} f(x) = \sum_{k=1}^{\infty} a_{k} \sin \qty( \frac{k\pi x}{l}) \end{equation}

or

\begin{equation} f(x) = \sum_{k=1}^{\infty} b_{k} \cos \qty( \frac{k\pi x}{l}) \end{equation}

Recall that because sin and cos are even and odd parts, the functions above force an even and oddness to your expansions. They will be particularly helpful for Dirichlet Conditions and Neumann Conditions.

requirements

Consider a function that has no periodicity, but that:

\begin{equation} f(x), -\infty < x < \infty \end{equation}

And assume that:

\begin{equation} \int_{\infty}^{\infty} |f(x)| \dd{x}, < \infty \end{equation}

important: look up! the integral of \(f(x)\) has to converge AND this means that the \(f(x)\) goes to \(0\) actually at boundaries.

(meaning the function decays as you go towards the end)

definition

a Fourier transform is an invertible transformation:

\begin{equation} f(x) \to \hat{f}(\lambda) \end{equation}