# Get PDF Transformations

### Homogenous Coordinates

Their role was established when cobalamine which also uses these channels was found to competitively inhibit DNA uptake. In this poly HB is envisioned to wrap around DNA itself a polyphosphate , and is carried in a shield formed by Ca ions. It is suggested that exposing the cells to divalent cations in cold condition may also change or weaken the cell surface structure, making it more permeable to DNA. The heat-pulse is thought to create a thermal imbalance across the cell membrane, which forces the DNA to enter the cells through either cell pores or the damaged cell wall. Electroporation is another method of promoting competence.

After the electric shock, the holes are rapidly closed by the cell's membrane-repair mechanisms. Most species of yeast , including Saccharomyces cerevisiae , may be transformed by exogenous DNA in the environment. Several methods have been developed to facilitate this transformation at high frequency in the lab. Efficiency — Different yeast genera and species take up foreign DNA with different efficiencies.

Even within one species, different strains have different transformation efficiencies, sometimes different by three orders of magnitude. For instance, when S. A number of methods are available to transfer DNA into plant cells. Some vector -mediated methods are:. Introduction of DNA into animal cells is usually called transfection , and is discussed in the corresponding article. The discovery of artificially induced competence in bacteria allow bacteria such as Escherichia coli to be used as a convenient host for the manipulation of DNA as well as expressing proteins.

Typically plasmids are used for transformation in E. In order to be stably maintained in the cell, a plasmid DNA molecule must contain an origin of replication , which allows it to be replicated in the cell independently of the replication of the cell's own chromosome. The cells are incubated on ice with the DNA, and then briefly heat-shocked e. This method works very well for circular plasmid DNA. In contrast, cells that are naturally competent are usually transformed more efficiently with linear DNA than with plasmid DNA.

The transformation efficiency using the CaCl 2 method decreases with plasmid size, and electroporation therefore may be a more effective method for the uptake of large plasmid DNA. Because transformation usually produces a mixture of relatively few transformed cells and an abundance of non-transformed cells, a method is necessary to select for the cells that have acquired the plasmid.

Antibiotic resistance is the most commonly used marker for prokaryotes. The transforming plasmid contains a gene that confers resistance to an antibiotic that the bacteria are otherwise sensitive to. The mixture of treated cells is cultured on media that contain the antibiotic so that only transformed cells are able to grow. Another method of selection is the use of certain auxotrophic markers that can compensate for an inability to metabolise certain amino acids, nucleotides, or sugars.

This method requires the use of suitably mutated strains that are deficient in the synthesis or utility of a particular biomolecule, and the transformed cells are cultured in a medium that allows only cells containing the plasmid to grow. In a cloning experiment, a gene may be inserted into a plasmid used for transformation. However, in such experiment, not all the plasmids may contain a successfully inserted gene. Additional techniques may therefore be employed further to screen for transformed cells that contain plasmid with the insert.

Cells containing successfully ligated insert can then be easily identified by its white coloration from the unsuccessful blue ones. Other commonly used reporter genes are green fluorescent protein GFP , which produces cells that glow green under blue light, and the enzyme luciferase , which catalyzes a reaction with luciferin to emit light. The recombinant DNA may also be detected using other methods such as nucleic acid hybridization with radioactive RNA probe, while cells that expressed the desired protein from the plasmid may also be detected using immunological methods.

From Wikipedia, the free encyclopedia. Not to be confused with an unrelated process called malignant transformation which occurs in the progression of cancer. Molecular Biology of the Cell. The Journal of Hygiene. Journal of Molecular Biology. The American Phytopathological Society. Retrieved 14 January Retrieved 28 January Journal of Particulate Science and Technology. Infection, Genetics and Evolution. Journal of Bioscience and Bioengineering. Bioscience, Biotechnology, and Biochemistry.

The Journal of General Physiology. Annual Review of Microbiology. Japanese Journal of Cancer Research. DNA repair as the primary adaptive function of sex in bacteria and eukaryotes". In Kimura S, Shimizu S. Genetics of Bacteria PDF. Daniel 1 August Archived from the original on September 3, Archived PDF from the original on 21 December Transfection Chromosomal crossover Gene conversion Fusion gene Horizontal gene transfer Sister chromatid exchange Transposon.

Antigenic shift Reassortment Viral shift. Roundup ready soybean Vistive Gold. Ice-minus bacteria Hepatitis B vaccine Oncolytic virus. Gene therapy Genetic enhancement. Gene knockout Gene knockdown Gene targeting. Below you'll see 3 vectors where each vector is represented with x,y as arrows in a 2D graph. Because it is more intuitive to display vectors in 2D than in 3D you can think of the 2D vectors as 3D vectors with a z coordinate of 0. Since vectors represent directions, the origin of the vector does not change its value. Also, when displaying vectors in formulas they are generally displayed as follows: Because vectors are specified as directions it is sometimes hard to visualize them as positions.

What we basically visualize is we set the origin of the direction to 0,0,0 and then point towards a certain direction that specifies the point, making it a position vector we could also specify a different origin and then say: The position vector 3,5 would then point to 3,5 on the graph with an origin of 0,0. Using vectors we can thus describe directions and positions in 2D and 3D space. Just like with normal numbers we can also define several operations on vectors some of which you've already seen.

A scalar is a single digit or a vector with 1 component if you'd like stay in vector-land. For addition it would look like this: Negating a vector results in a vector in the reversed direction. A vector pointing north-east would point south-west after negation. To negate a vector we add a minus-sign to each component you can also represent it as a scalar-vector multiplication with a scalar value of Addition of two vectors is defined as component-wise addition, that is each component of one vector is added to the same component of the other vector like so: Just like normal addition and subtraction, vector subtraction is the same as addition with a negated second vector: Subtracting two vectors from each other results in a vector that's the difference of the positions both vectors are pointing at.

This proves useful in certain cases where we need to retrieve a vector that's the difference between two points. A vector forms a triangle when you visualize its individual x and y component as two sides of a triangle:. In this case the length of vector 4, 2 equals: There is also a special type of vector that we call a unit vector. A unit vector has one extra property and that is that its length is exactly 1. Unit vectors are displayed with a little roof over their head and are generally easier to work with, especially when we only care about their directions the direction does not change if we change a vector's length.

Multiplying two vectors is a bit of a weird case. Normal multiplication isn't really defined on vectors since it has no visual meaning, but we have two specific cases that we could choose from when multiplying: The dot product of two vectors is equal to the scalar product of their lengths times the cosine of the angle between them.

If this sounds confusing take a look at its formula: Why is this interesting? This would effectively reduce the formula to: You might remember that the cosine or cos function becomes 0 when the angle is 90 degrees or 1 when the angle is 0. This allows us to easily test if the two vectors are orthogonal or parallel to each other using the dot product orthogonal means the vectors are at a right-angle to each other.

In case you want to know more about the sin or the cosine functions I'd suggest the following Khan Academy videos about basic trigonometry. So how do we calculate the dot product? The dot product is a component-wise multiplication where we add the results together. It looks like this with two unit vectors you can verify that both their lengths are exactly 1: We now effectively calculated the angle between these two vectors.

The dot product proves very useful when doing lighting calculations. The cross product is only defined in 3D space and takes two non-parallel vectors as input and produces a third vector that is orthogonal to both the input vectors. If both the input vectors are orthogonal to each other as well, a cross product would result in 3 orthogonal vectors. This will prove useful in the upcoming tutorials. The following image shows what this looks like in 3D space:.

Unlike the other operations, the cross product isn't really intuitive without delving into linear algebra so it's best to just memorize the formula and you'll be fine or don't, you'll probably be fine as well. Below you'll see the cross product between two orthogonal vectors A and B: However, if you just follow these steps you'll get another vector that is orthogonal to your input vectors. Now that we've discussed almost all there is to vectors it is time to enter the matrix!

## Transformation (function)

Each individual item in a matrix is called an element of the matrix. An example of a 2x3 matrix is shown below: This is the opposite of what you're used to when indexing 2D graphs as x,y. To retrieve the value 4 we would index it as 2,1 second row, first column.

Matrices are basically nothing more than that, just rectangular arrays of mathematical expressions. They do have a very nice set of mathematical properties and just like vectors we can define several operations on matrices, namely: Addition and subtraction between a matrix and a scalar is defined as follows: The same applies for matrix-scalar subtraction: Matrix addition and subtraction between two matrices is done on a per-element basis. So the same general rules apply that we're familiar with for normal numbers, but done on the elements of both matrices with the same index.

This does mean that addition and subtraction is only defined for matrices of the same dimensions. A 3x2 matrix and a 2x3 matrix or a 3x3 matrix and a 4x4 matrix cannot be added or subtracted together. Let's see how matrix addition works on two 2x2 matrices: Just like addition and subtraction, a matrix-scalar product multiples each element of the matrix by a scalar. The following example illustrates the multiplication: A scalar basically scales all the elements of the matrix by its value.

In the previous example, all elements were scaled by 2. So far so good, all of our cases weren't really too complicated. That is, until we start on matrix-matrix multiplication. Multiplying matrices is not necessarily complex, but rather difficult to get comfortable with. Matrix multiplication basically means to follow a set of pre-defined rules when multiplying. There are a few restrictions though: You can only multiply two matrices if the number of columns on the left-hand side matrix is equal to the number of rows on the right-hand side matrix.

Let's get started with an example of a matrix multiplication of 2 2x2 matrices: Matrix multiplication is a combination of normal multiplication and addition using the left-matrix's rows with the right-matrix's columns. Let's try discussing this with the following image:. We first take the upper row of the left matrix and then take a column from the right matrix. The row and column that we picked decides which output value of the resulting 2x2 matrix we're going to calculate.

If we take the first row of the left matrix the resulting value will end up in the first row of the result matrix, then we pick a column and if it's the first column the result value will end up in the first column of the result matrix. This is exactly the case of the red pathway. To calculate the bottom-right result we take the bottom row of the first matrix and the rightmost column of the second matrix.

To calculate the resulting value we multiply the first element of the row and column together using normal multiplication, we do the same for the second elements, third, fourth etc. The results of the individual multiplications are then summed up and we have our result. Now it also makes sense that one of the requirements is that the size of the left-matrix's columns and the right-matrix's rows are equal, otherwise we can't finish the operations!

### More Guides

The result is then a matrix that has dimensions of n,m where n is equal to the number of rows of the left-hand side matrix and m is equal to the columns of the right-hand side matrix. Don't worry if you have difficulties imagining the multiplications inside your head. Just keep trying to do the calculations by hand and return to this page whenever you have difficulties.

Over time, matrix multiplication becomes second nature to you. Let's finish the discussion of matrix-matrix multiplication with a larger example. Try to visualize the pattern using the colors. As a useful exercise, see if you can come up with your own answer of the multiplication and then compare them with the resulting matrix once you try to do a matrix multiplication by hand you'll quickly get the grasp of them.

As you can see, matrix-matrix multiplication is quite a cumbersome process and very prone to errors which is why we usually let computers do this and this gets problematic real quick when the matrices become larger. If you're still thirsty for more and you're curious about some more of the mathematical properties of matrices I strongly suggest you take a look at these Khan Academy videos about matrices.

Anyways, now that we know how to multiply matrices together, we can start getting to the good stuff. Up until now we've had our fair share of vectors these tutorials. We used vectors to represent positions, colors and even texture coordinates. Let's move a bit further down the rabbit hole and tell you that a vector is basically a Nx1 matrix where N is the vector's number of components also known as an N-dimensional vector. If you think about it, it makes a lot of sense. Vectors are just like matrices an array of numbers, but with only 1 column.

So, how does this new piece of information help us? Well, if we have a MxN matrix we can multiply this matrix by our Nx1 vector, since the columns of our matrix are equal to the number of rows of our vector, thus matrix multiplication is defined. But why do we care if we can multiply matrices with a vector? In case you're still a bit confused, let's start with some examples and you'll soon see what we mean. In OpenGL we usually work with 4x4 transformation matrices for several reasons and one of them is that most of the vectors are of size 4.

The most simple transformation matrix that we can think of is the identity matrix. The identity matrix is an NxN matrix with only 0s except on its diagonal. As you'll see, this transformation matrix leaves a vector completely unharmed: This becomes obvious from the rules of multiplication: Since each of the row's elements are 0 except the first one, we get: When we're scaling a vector we are increasing the length of the arrow by amount we'd like to scale, keeping its direction the same.

Since we're working in either 2 or 3 dimensions we can define scaling by a vector of 2 or 3 scaling variables, each scaling one axis x , y or z. We will scale the vector along the x-axis by 0. Let's see what it looks like if we scale the vector by 0. Keep in mind that OpenGL usually operates in 3D space so for this 2D case we could set the z-axis scale to 1 thus leaving it unharmed.

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The scaling operation we just performed is a non-uniform scale, because the scaling factor is not the same for each axis. If the scalar would be equal on all axes it would be called a uniform scale. Let's start building a transformation matrix that does the scaling for us. We saw from the identity matrix that each of the diagonal elements were multiplied with its corresponding vector element. What if we were to change the 1 s in the identity matrix to 3 s?

In that case, we would be multiplying each of the vector elements by a value of 3 and thus effectively scale the vector by 3.

## Transformation (genetics) - Wikipedia

The w component is used for other purposes as we'll see later on. Translation is the process of adding another vector on top of the original vector to return a new vector with a different position, thus moving the vector based on a translation vector. We've already discussed vector addition so this shouldn't be too new. Just like the scaling matrix there are several locations on a 4-by-4 matrix that we can use to perform certain operations and for translation those are the top-3 values of the 4th column.

This wouldn't have been possible with a 3-by-3 matrix. With a translation matrix we could move objects in any of the 3 directions x , y , z we'd like, making it a very useful transformation matrix for our transformation toolkit. The last few transformations were relatively easy to understand and visualize in 2D or 3D space, but rotations are a bit trickier. If you want to know exactly how these matrices are constructed I'd recommend that you watch the rotation items of Khan Academy's linear algebra videos.

First let's define what a rotation of a vector actually is. A rotation in 2D or 3D is represented with an angle. An angle could be in degrees or radians where a whole circle has degrees or 2 PI radians. I personally prefer to work in degrees, since they seem to make more sense to me. Most rotation functions require an angle in radians, but luckily degrees are easily converted to radians: Rotations in 3D are specified with an angle and a rotation axis. The angle specified will rotate the object along the rotation axis given.

Try to visualize this by spinning your head a certain degree while continually looking down a single rotation axis.