Prions: From good to bad with a copper switch

At the molecular level, the difference between Doctor Jekyll and Mr Hyde lies in a metal, copper. In its physiological form, the prion protein (PrPC ) is ‘good’ and is involved in normal body processes. It can happen, however, that because of some as yet unknown mechanism, it changes form and turns into a threat for the health of humans and animals (it is responsible for neurodegenerative diseases such as spongiform encephalopathies). According to a new SISSA study, the mechanism underlying this change is a metal, copper, or rather a particular region of the protein to which the metal binds, which acts as a sort of 'switch’ that turns PrPC into its terrible alter ego.

“We still don’t know what complex molecular mechanisms cause the prion protein to become bad,” explains Giuseppe Legname, professor at the International School for Advanced Studies (SISSA) in Trieste who coordinated the new study, “nor do we know any treatments to cure prion diseases. Our research has finally uncovered a critical cofactor, which is capable of triggering the transformation of prions proteins from good to bad. And this cofactor is copper which binds to an amino acid sequence of the prion protein, known as 'fifth copper binding site’, which has so far been poorly studied”

Caption:This is a photograph showing how PrPC turns into a Prion. Credit: SISSA

A Formula For the Inverse

The determinant of a matrix can also provide a way to find the inverse of a matrix.

Definition of the Cofactor Matrix
Let A = [aᵢⱼ] be an n x n matrix. Then the cofactor matrix of A, denoted as cof(A), is defined as cof(A) = [cof(A)ᵢⱼ], where cof(A)ᵢⱼ is the ijth cofactor of A.

Note that cof(A)ᵢⱼ denotes the ijth entry of the cofactor matrix.

This cofactor matrix is used in creating the formula for the inverse of A.

Definition of the Adjoint of Matrix A
The adjugate of A is the transpose of its cofactor matrix, denoted as adj(A). It is also called the classical adjoint of A.

If matrix A was a 2 x 2 matrix, its adjoint, adj(A), is the following:

In general, adj(A) can always be found by taking the transpose of the cofactor matrix of A.

Theorem of the Inverse and the Determinant
Let A be an n x n matrix. Then A is invertible if and only if det(A) ≠ 0. If det(A) ≠ 0, then the following is true:

Given the following matrix A, find the inverse using the formula for the inverse:

To begin, find the determinant of A using row operations to obtain the upper triangular matrix B.

Since only multiplies of rows were added to other rows, det(B) = det(A).

By the theorem of determinants and triangular matrices:
det(B) = det(A) = (1)(-6)(-2) = 12

Since det(A) ≠ 0, it is invertible.

The following is the cofactor matrix of A:

The following is using the formula of the inverse to find the inverse of A:

Checking if the answer is correct:

This method for finding the inverse of A is useful, because it can be used with complicated matrices that contain entries of functions rather than numbers.

Given the following matrix, find the inverse using the formula of the inverse:

The following is calculating the determinant of A(t) by expanding along the first column or row:

Since det(A(t)) ≠ 0, A(t) is invertible.

The following is the cofactor matrix of A(t):

Calculating the inverse of A(t) using the formula of the inverse:

thecynicaltank replied to your post “Alright topic for discussion, let’s hope this can stay pretty civil…”

You need to remember that black people also commit more crime. They make up about 13% of the U.S. population, but commit about 52% of the murders. However, it’s really irritating that black people have a better chance of going to jail.

You can’t necessarily use that as proof though. It’s a self fulfilling prophecy: people expect black people to commit more crimes, so black people are arrested/accused more easily leading to people to continue to assume black people commit more crimes, etc.

Though honestly, the bigger cofactor of crime is poverty. So the better question is “why are black communities poorer than white communities?”