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general form of complex number is a+ib and we denote it as z. z=a+ib. Examples; Random; Assuming "complex conjugate of" is a math function | Use "complex conjugate" as a function property instead. $2+i\sqrt{5}$ $-\frac{1}{2}i$ Show Solution Analysis of the Solution. Log in. (8) In particular, 1 z = z The nonconjugate transpose operator, A. We find the remaining roots are: Examples of Use. However, you're trying to find the complex conjugate of just 2. (2)\begin{aligned} The complex conjugate is particularly useful for simplifying the division of complex numbers. Complex Conjugates Problem Solving - Intermediate, Complex Conjugates Problem Solving - Advanced, https://brilliant.org/wiki/complex-conjugates-problem-solving-easy/. z^2+\overline{z} &= (a+bi)^2+(a-bi) \\ Consider the complex number z = a~+~ib, z ~+~ \overline {z} = a ~+ ~ib~+ ~ (a~ – ~ib) = 2a which is a complex number having imaginary part as zero. Parameters x array_like. {\displaystyle a-bi.} We find its remaining roots are: Addition of Complex Numbers. $f(x) = \begin{pmatrix}x - 3 \end{pmatrix}.\begin{pmatrix}x + 1 \end{pmatrix}.\begin{pmatrix}x - 3i \end{pmatrix}.\begin{pmatrix}x + 3i \end{pmatrix}$, Given $$2- i$$ is a root of $$f(x) = 2x^4 - 14x^3 + 38x^2 - 46x +20$$, so is $$2 + i$$. The conjugate of the complex number $$a + bi$$ is the complex number $$a - bi$$. \overline { \left( \frac { 2-3i }{ 4+5i } \right) \left( \frac { 4-i }{ 1-3i } \right) } &= \left( \frac { -3x }{ 1+25{ x }^{ 2 } } -\frac { 15{ x }^{ 2 } }{ 1+25{ x }^{ 2 } } i \right) +\left( \frac { 9 }{ 10 } i+\frac { 3 }{ 10 } \right) \\ □\alpha \overline{\alpha}=1. □, Since α2=3−4i,\alpha^2=3-4i,α2=3−4i, we have Observe that these two equations cannot hold simultaneously, then the two complex numbers in the problem cannot be the conjugates of each other for any real value x. We find its remaining roots are: ', performs a transpose without conjugation. For example, if B = A' and A(1,2) is 1+1i, then the element B(2,1) is 1-1i. Example: Conjugate of 7 – 5i = 7 + 5i. \Rightarrow \sin x-i\cos 2x &= \cos x-i\sin 2x, \end{aligned} sinx+icos2x⇒sinx−icos2x=cosx−isin2x=cosx−isin2x, In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. This will allow us to find the zero(s) of a polynomial function in pairs, so long as the zeros are complex numbers. Example 1. \overline{\sin x+i\cos 2x} &= \cos x-i\sin 2x \\ Given a complex number $${\displaystyle z=a+bi}$$ (where a and b are real numbers), the complex conjugate of $${\displaystyle z}$$, often denoted as $${\displaystyle {\overline {z}}}$$, is equal to {\displaystyle a-bi. in root-factored form we therefore have: public: static System::Numerics::Complex Conjugate(System::Numerics::Complex value); The following example displays the conjugate of two complex numbers. Given $$1-i$$ is one of the zeros of $$f(x) = x^3 - 2x+4$$, find its remaining roots and write $$f(x)$$ in root factored form. z2=−1+3i2z3=zz2=1+3i2⋅−1+3i2=−1z4=zz3=1+3i2⋅(−1)=−1−3i2z5=z2z3=−1+3i2⋅(−1)=1−3i2z6=(z3)2=1⋮,\begin{aligned} are examples of complex numbers. To divide complex numbers. Find the cubic polynomial that has roots 555 and 3+i.3+i.3+i. The complex conjugate of a complex number is defined as two complex number having an equal real part and imaginary part equal in magnitude but opposite in sign. \left(\alpha-\overline{\alpha}\right)+\left(\frac{1}{\alpha}-\frac{1}{\overline{\alpha}}\right) &= 0 \\ □f(x)=(x-5+i)(x-5-i)(x+2). The complex conjugate can also be denoted using z. z, z, z, denoted. Complex Conjugate Root Theorem. Given $$2i$$ is one of the roots of $$f(x) = x^3 - 3x^2 + 4x - 12$$, find its remaining roots and write $$f(x)$$ in root factored form. −p=(2+3i)+(2−3i),q=(2+3i)(2−3i).-p=\left(2+\sqrt{3}i\right)+\left(2-\sqrt{3}i\right),\quad q=\left(2+\sqrt{3}i\right)\left(2-\sqrt{3}i\right).−p=(2+3i)+(2−3i),q=(2+3i)(2−3i). The significance of complex conjugate is that it provides us with a complex number of same magnitude‘complex part’ but opposite in direction. The complex conjugate has the same real component aaa, but has opposite sign for the imaginary component bbb. $f(z^*) = 0$. example the choice H = A − 1 and K = I leads to the classical complex conjugate gradient method; with H = A − 1 and K = l H × l (incomplete complex Cholesky factorization), we \ _\squareq=7. z^4 &= zz^3=\frac{1+\sqrt{3}i}{2} \cdot (-1)=\frac{-1-\sqrt{3}i}{2} \\ Use the rationalizing factor 19−7i19-7i19−7i to simplify: 5+14i19+7i⋅19−7i19−7i=193410−231410i. $f(x) = -2.\begin{pmatrix}x - 1 \end{pmatrix}.\begin{pmatrix}x - (2 + 3i) \end{pmatrix}.\begin{pmatrix}x - (2 - 3i) \end{pmatrix}$, Given $$3i$$ is a root of $$f(x) = x^4 - 2x^3 + 6x^2 - 18x - 27$$, so is $$-3i$$. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … (a−bi)2+p(a−bi)+q=(a2−b2+pa+q)−(2ab+pb)i,(a-bi)^2+p(a-bi)+q=\big(a^2-b^2+pa+q\big)-(2ab+pb)i,(a−bi)2+p(a−bi)+q=(a2−b2+pa+q)−(2ab+pb)i, Therefore, it must be true that a−bia-bia−bi is also a root of the quadratic equation. \qquad (1)a2−b2+pa+q=0,2ab+pb=0. When b=0, z is real, when a=0, we say that z is pure imaginary. (2) Tips . Complex Conjugates. &= x^3-11x^2+40x-50. Real parts are added together and imaginary terms are added to imaginary terms. □\alpha \overline{\alpha}=5. They would be: 3-2i,-1+1/2i, and 66+8i. and are told $$2+3i$$ is one of its roots. \qquad (2)\end{aligned}a2−b2+a2ab−b⇒b(2a−1)=0(1)=0=0. In other words if we find, or are given, one complex root, then we can state that its complex conjugate is also a root. Let's look at an example to see what we mean. 104016 Dr. Aviv Censor Technion - International school of engineering The complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude, but the complex value is opposite in sign. □. For example, for ##z= 1 + 2i##, its conjugate is ##z^* = 1-2i##. $f(x) = \begin{pmatrix}x - 2 \end{pmatrix}.\begin{pmatrix}x + 3 \end{pmatrix}.\begin{pmatrix}x + 1 \end{pmatrix}.\begin{pmatrix}x - i \end{pmatrix}.\begin{pmatrix}x + i \end{pmatrix}$, $$z_1 = 3$$ and $$z_2 = 1+2i$$ are roots of the equation: Mathematical articles, tutorial, examples. $-2x^4 + bx^3 + cx^2 + dx + e = -2.\begin{pmatrix}x - (1 - \sqrt{2}i) \end{pmatrix}.\begin{pmatrix}x + (1 + \sqrt{2}i)\end{pmatrix}.\begin{pmatrix}x - (2 - 3i)\end{pmatrix}.\begin{pmatrix}x - (2 + 3i)\end{pmatrix}$ Sign up, Existing user? □\ _\square □. The complex conjugate of a + bi is a – bi, and similarly the complex conjugate of a – bi is a + bi.This consists of changing the sign of the imaginary part of a complex number.The real part is left unchanged.. Complex conjugates are indicated using a horizontal line over the number or variable. Up Main page Complex conjugate. When a complex number is multiplied by its complex conjugate, the result is a real number. $x^4 + bx^3 + cx^2 + dx + e = \begin{pmatrix}x - 2i \end{pmatrix}.\begin{pmatrix}x + 2i\end{pmatrix}.\begin{pmatrix}x - (3 + i)\end{pmatrix}.\begin{pmatrix}x - (3 - i)\end{pmatrix}$ Using the fact that $$z_1 = 3$$ and $$z_2 = 1+2i$$ are roots of the equation $$x^3 + bx^2 + cx + d = 0$$, we find the following: Using the fact that: □. then its complex conjugate, $$z^*$$, is also a root: Conjugate of a complex number = is and which is denoted as \overline {z}. Z; Extended Capabilities; See Also Given $$3i$$ is a root of $$f(x) = x^4 - 2x^3 + 6x^2 - 18x - 27$$, find its remaining roots and write $$f(x)$$ in its root-factord form. \Rightarrow b(2a-1) &=0. It can help us move a square root from the bottom of a fraction (the denominator) to the top, or vice versa.Read Rationalizing the Denominator to … Thus the complex conjugate of −4−3i is −4+3i. The real part is left unchanged. Given $$i$$ is a root of $$f(x) = x^5 + 2x^4 - 4x^3 - 4x^2 - 5x - 6$$, find this polynomial function's remaining roots and write $$f(x)$$ in its root-factored form. which implies αα‾=1. Algebra 1M - international Course no. One importance of conjugation comes from the fact the product of a complex number with its conjugate, is a real number!! The operation also negates the imaginary part of any complex numbers. I know how to take a complex conjugate of a complex number ##z##. $b = -12, \ c = 48, \ d = -76, \ e = 78$. $2x^3 + bx^2 + cx + d = 2.\begin{pmatrix}x + 2 \end{pmatrix}.\begin{pmatrix}x - (3 + i)\end{pmatrix}.\begin{pmatrix}x - (3 - i)\end{pmatrix}$ Free math tutorial and lessons. f(x)=(x−5+i)(x−5−i)(x+2). &= (a^2-b^2+a)+(2ab-b)i=0. Examples open all close all. \hspace{1mm} 10. z−z‾=2Im(z)\hspace{1mm} z-\overline { z } =2\text{Im}(z)z−z=2Im(z), twice the imaginary element of z.z.z. Then $b = -5, \ c = 11, \ d = -15$. Complex conjugates are a major part of the conjugate root theorem, so we definitely want to be familiar with them. Hence, POWERED BY THE WOLFRAM LANGUAGE. expanding the right hand side, simplifying as much as possible, and equating the coefficients to those on the left hand side we find: In the following tutorial we further explain the complex conjugate root theorem. If provided, it must have a shape that the inputs broadcast to. Using the fact that $$z_1 = 3$$, $$z_2 = i$$ and $$z_3 = 2-3i$$ are roots of the equation $$x^5 +bx^4 + cx^3 + dx^2 + ex + f = 0$$, we find: Using the fact that: The complex conjugate of a + bi is a – bi , and similarly the complex conjugate of a – bi is a + bi. z^3 &= zz^2=\frac{1+\sqrt{3}i}{2} \cdot \frac{-1+\sqrt{3}i}{2}=-1 \\ If a solution is not possible explain why. &=\frac { 4+3i }{ 5+2i } \cdot \frac { 5-2i }{ 5-2i } \\ Additional overloads are provided for arguments of any fundamental arithmetic type: In this case, the function assumes the value has a zero imaginary component. Thus the complex conjugate of −4−3i is −4+3i. &=\left( \frac { -3x }{ 1+25{ x }^{ 2 } } +\frac { 3 }{ 10 } \right) +\left( \frac { -15{ x }^{ 2 } }{ 1+25{ x }^{ 2 } } i+\frac { 9 }{ 10 } i \right) \\ $f(x) = \begin{pmatrix}x - 3 \end{pmatrix}.\begin{pmatrix}x - 2i \end{pmatrix}.\begin{pmatrix}x + 2i \end{pmatrix}$, Given $$1-i$$ is one of the zeros of $$f(x) = x^3 - 2x+4$$, so is $$1+i$$. In the last example (113) the imaginary part is zero and we actually have a real number. (α‾)2=α2‾=3+4i.\left(\overline{\alpha}\right)^2=\overline{\alpha^2}=3+4i.(α)2=α2=3+4i. \end{aligned}z2+z=(a+bi)2+(a−bi)=(a2−b2+a)+(2ab−b)i=0. Z; Extended Capabilities; See Also Sum of two complex numbers a + bi and c + di is given as: (a + bi) + (c + di) = (a + c) + (b + d)i. &=\frac { 26 }{ 29 } +\frac { 7 }{ 29 } i\\\\ Input value. For calculating conjugate of the complex number following z=3+i, enter complex_conjugate (3 + i) or directly 3+i, if the complex_conjugate button already appears, the result 3-i is returned. For example, for a polynomial f(x)f(x)f(x) with real coefficient, f(z=a+bi)=0f(z=a+bi)=0f(z=a+bi)=0 could be a solution if and only if its conjugate is also a solution f(z‾=a−bi)=0f(\overline z=a-bi)=0f(z=a−bi)=0. If we represent a complex number z as (real, img), then its conjugate is (real, -img). We then need to find all of its remaining roots and write this polynomial in its root-factored form. If not provided or None, a freshly-allocated array is returned. (1)a^2-b^2+pa+q=0, \quad 2ab+pb=0. Performing the necessary operations, and using the properties of complex numbers and their conjugates, we have, (2−3i4+5i)(4−i1−3i)‾=(2−3i4+5i)‾⋅(4−i1−3i)‾=2−3i‾4+5i‾⋅4−i‾1−3i‾=2+3i4−5i.4+i1+3i=5+14i19+7i.\begin{aligned} 1.1, in the process of rationalizing the denominator for the division algorithm. in root-factored form we therefore have: We find the remaining roots are: □. If the coefficients of a polynomial are all real, for example, any non-real root will have a conjugate pair. It is like rationalizing a rational expression. The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. The complex conjugate z* has the same magnitude but opposite phase When you add z to z*, the imaginary parts cancel and you get a real number: (a + bi) + (a -bi) = 2a When you multiply z to z*, you get the real number equal to |z|2: (a + bi)(a -bi) = a2 –(bi)2 = a2 + b2. Already have an account? The conjugate … Written, Taught and Coded by: For example, if we have ‘a + ib’ as a complex number, then the conjugate of this will be ‘a – ib’. This means that the equation has two roots, namely iii and −i-i−i. \end{aligned}(αα)2⇒αα=α2(α)2=(3−4i)(3+4i)=25=±5. Comment on sreeteja641's post “general form of complex … Y = pagectranspose(X) applies the complex conjugate transpose to each page of N-D array X.Each page of the output Y(:,:,i) is the conjugate transpose of the corresponding page in X, as in X(:,:,i)'. since the values of sine or cosine functions are real numbers. Using the fact that: How do you take the complex conjugate of a function? z^2 &= \frac{-1+\sqrt{3}i}{2} \\ 4 years ago. $f(x) = 2.\begin{pmatrix}x - 2 \end{pmatrix}.\begin{pmatrix}x - 1 \end{pmatrix}.\begin{pmatrix}x - (2 - i) \end{pmatrix}.\begin{pmatrix}x - (2 + i) \end{pmatrix}$, Given $$i$$ is a root of $$f(x) = x^5 + 2x^4 - 4x^3 - 4x^2 - 5x - 6$$, so is $$-i$$. The complex conjugate of $$z$$, denoted by $$\overline{z}$$, is given by $$a - bi$$. □\begin{aligned} We also work through some typical exam style questions. Thus, the conjugate of the complex number. Complex Conjugate. Let's look at an example: 4 - 7 i and 4 + 7 i. For a non-real complex number α,\alpha,α, if α+1α \alpha+\frac{1}{\alpha}α+α1 is a real number, what is the value of αα‾?\alpha \overline{\alpha}?αα? It is found by changing the sign of the imaginary part of the complex number. For example, the complex conjugate of z =3 z = 3 is ¯z = 3 z ¯ = 3. □. We will also derive from the complex roots the standard solution that is typically used in this case that will not involve complex numbers. These are the top rated real world C++ (Cpp) examples of Complex::conjugate from package articles extracted from open source projects. Find Complex Conjugate of Complex Number; Find Complex Conjugate of Complex Values in Matrix; Input Arguments. $\left \{ -2i,\ 2i, \ 3 \right \}$ □\ _\square □, Let cosx−isin2x\cos x-i\sin 2xcosx−isin2x be the conjugate of sinx+icos2x,\sin x+i\cos 2x,sinx+icos2x, then we have α+α1=(α+α1)=α+α1. z = a + b i ( a, b ∈ R) z = a + bi \, (a, b \in \mathbb {R}) z = a+bi(a,b∈ R), the complex conjugate of. #include #include int main () { std::complex mycomplex (50.0,2.0); std::cout << "The conjugate of " << mycomplex << " is " << std::conj(mycomplex) << '\n'; return 0; } The sample output should be like this −. &=x^2-10x+26\end{aligned}(x−(5−i))(x−(5+i))=((x−5)+i)((x−5)−i)=x2−10x+26, is a real factor of f(x).f(x).f(x). Thus, by Vieta's formular. For two complex numbers zand w, the following properties exist: © Copyright 2007 Math.Info - All rights reserved. Subscribe Now and view all of our playlists & tutorials. in root-factored form we therefore have: 1−1αα‾=0,1-\frac{1}{\alpha \overline{\alpha}}=0,1−αα1=0, Example. \Rightarrow \alpha \overline{\alpha} &= \pm 5. A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i 2 = −1.For example, 2 + 3i is a complex number. then nnn must be a multiple of 3 to make znz^nzn an integer. sinx=cosx and cos2x=sin2x\sin x=\cos x \text{ and } \cos 2x=\sin 2xsinx=cosx and cos2x=sin2x For example, setting c = d = 0 produces a diagonal complex matrix representation of complex numbers, and setting b = d = 0 produces a real matrix representation. Using the fact that $$z_1 = 2i$$ and $$z_2 = 3+i$$ are both roots of the equation $$x^4 + bx^3 + cx^2 + dx + e = 0$$, we find: The other roots are: $$z_3 = -2i$$ and $$z_4 = 3 - i$$. The conjugate of a complex number z = a + bi is: a – bi. Multiply both the numerator and denominator with the conjugate of the denominator, in a way similar to when rationalizing an expression: 4+3i5+2i=4+3i5+2i⋅5−2i5−2i=(4+3i)(5−2i)52+22=20−8i+15i−6i229=2629+729i⇒a=2629,b=729. Syntax: template complex conj (const complex& Z); Parameter: z: This method takes a mandaory parameter z which represents the complex number. \ _\squareαα=5. □q=7. \big(x-(5-i)\big)\big(x-(5+i)\big) &= \big((x-5)+i\big)\big((x-5)-i\big) \\ &= (x-5)\big(x^2-6x+10\big) \\ For example, (if a and b are real, then) the complex conjugate of a + b i {\displaystyle a+bi} is a − b i. \ _\square19+7i5+14i⋅19−7i19−7i=410193−410231i. Observe that if α=p+qi (p,q∈R)\alpha=p+qi \ (p, q \in \mathbb{R})α=p+qi (p,q∈R) and α‾=p−qi,\overline{\alpha}=p-qi ,α=p−qi, then αα‾=p2+q2≥0.\alpha \overline{\alpha}=p^2+q^2 \geq 0.αα=p2+q2≥0. The complex conjugate is particularly useful for simplifying the division of complex numbers. $b = -6, \ c = 14, \ d = -24, \ e = 40$. a^2-b^2+a &= 0 \qquad (1) \\ Complex conjugate for a complex number is defined as the number obtained by changing the sign of the complex part and keeping the real part the same. This function is used to find the conjugate of the complex number z. ', performs a transpose without conjugation. \ _\square The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. Operations on zzz and z‾:\overline {z}:z: Based on these operations, we can add some more properties of conjugate: \hspace{1mm} 9. z+z‾=2Re(z)\hspace{1mm} z+\overline{z}=2\text{Re}(z)z+z=2Re(z), twice the real element of z.z.z. $\left \{ - 3i,\ 3i, \ -1, \ 3 \right \}$ Since b>0,b > 0,b>0, we obtain a=12a=\frac{1}{2}a=21 from (2),(2),(2), and by substituting this into (1)(1)(1) we have b2=34b^2=\frac{3}{4}b2=43 or b=32b=\frac{\sqrt{3}}{2}b=23 since b>0.b > 0.b>0. We can divide f(x)f(x)f(x) by this factor to obtain. f(x) &= (x-5)\big(x-(3+i)\big)\big(x-(3-i)\big) \\ The nonconjugate transpose operator, A. \frac { 4+3i }{ 5+2i } $f(x) = \begin{pmatrix}x + 2 \end{pmatrix}.\begin{pmatrix}x - (1-i) \end{pmatrix}.\begin{pmatrix}x - (1+i) \end{pmatrix}$, Given $$2+3i$$ is a root of $$f(x) = -2x^3 + 10x^2 -34x+26$$, so is $$2 - 3i$$. The complex conjugates of these complex numbers are written in the form a-bi: their imaginary parts have their signs flipped. Examples: Properties of Complex Conjugates. □x. When a complex number is added to its complex conjugate, the result is a real number. Find the sum of real values of xxx and yyy for which the following equation is satisfied: (1+i)x−2i3+i+(2−3i)y+i3−i=i.\frac { \left( 1+i \right) x-2i }{ 3+i } + \frac { \left( 2-3i \right) y+i }{ 3-i } =i.3+i(1+i)x−2i+3−i(2−3i)y+i=i. (α−α‾)+(1α−1α‾)=0(α−α‾)(1−1αα‾)=0.\begin{aligned} Hence, $$z_1 = 3$$, $$z_2 = i$$ and $$z_3 = 2-3i$$ are roots of the equation: &=\frac { 2+3i }{ 4-5i } .\frac { 4+i }{ 1+3i } \\\\ &=\frac { (4+3i)(5-2i) }{ { 5 }^{ 2 }+{ 2 }^{ 2 } } \\ The conjugate of a complex number a + i ⋅ b, where a and b are reals, is the complex number a − i ⋅ b. $2x^3 + bx^2 + cx + d = 0$, $$z_1 = 2i$$ and $$z_2 = 3+i$$ are both roots of the equation: Click here to learn the concepts of Modulus and Conjugate of a Complex Number from Maths } We learn the theorem and illustrate how it can be used for finding a polynomial's zeros. Note that a + b i is also the complex conjugate of a - b i. expanding the right hand side, simplifying as much as possible, and equating the coefficients to those on the left hand side we find: \frac { -3x }{ 1-5xi } +\frac { 3i }{ 3+i } $x^4 + bx^3 + cx^2 + dx + e = 0$, $$z_1 = 1+\sqrt{2}i$$ and $$z_2 = 2-3i$$ are roots of the equation: Conjugate complex numbers. While this may not look like a complex number in the form a+bi, it actually is! Hence, let f(x)f(x)f(x) be the cubic polynomial with roots 3+i,3+i,3+i, 3−i,3-i,3−i, and 5,5,5, then, f(x)=(x−5)(x−(3+i))(x−(3−i))=(x−5)((x−3)−i)((x−3)+i)=(x−5)(x2−6x+9−i2)=(x−5)(x2−6x+10)=x3−11x2+40x−50. $f(z) = 0$ Given $$2- i$$ is a root of $$f(x) = 2x^4 - 14x^3 + 38x^2 - 46x +20$$, find this polynomial function's remaining roots and write $$f(x)$$ in its root-factored form. This consists of changing the sign of the imaginary part of a complex number. (a2−b2+pa+q)+(2ab+pb)i=0.\big(a^2-b^2+pa+q\big)+(2ab+pb)i=0.(a2−b2+pa+q)+(2ab+pb)i=0. Assuming i is the imaginary unit | Use i as a variable instead. &=\frac { 5+14i }{ 19+7i } . Complex numbers tutorial. Example. Rationalization of Complex Numbers. You can rate examples to help us improve the quality of examples. Thus the complex conjugate of 1−3i is 1+3i. Example To ﬁnd the complex conjugate of −4−3i we change the sign of the imaginary part. □\frac { 5+14i }{ 19+7i } \cdot \frac { 19-7i }{ 19-7i } =\frac { 193 }{ 410 } - \frac { 231 }{ 410 } i. Read formulas, definitions, laws from Modulus and Conjugate of a Complex Number here. Posted 4 years ago. if it has a complex root (a zero that is a complex number), $$z$$: Given $$2i$$ is one of the roots of $$f(x) = x^3 - 3x^2 + 4x - 12$$, so is $$-2i$$. The conjugate of a complex number (real,imag) is (real,-imag). □. |z|^2=a^2+b^2. For example, . Experienced IB & IGCSE Mathematics Teacher Computes the conjugate of a complex number and returns the result. Im folgenden Beispiel wird die konjugierte Zahl zweier komplexer Zahlen angezeigt.The following example displays the conjugate of two complex numbers. Return value: This function returns the conjugate of the complex number z. A location into which the result is stored. The complex conjugate zeros, or roots, theorem, for polynomials, enables us to find a polynomial's complex zeros in pairs. \ _\squarex. Since a+bia+bia+bi is a root of the quadratic equation, it must be true that. So, a Complex Number has a real part and an imaginary part. Complex Conjugates Every complex number has a complex conjugate. Let's divide the following 2 complex numbers $\frac{5 + 2i}{7 + 4i}$ Step 1. Complex Numbers; conj; On this page; Syntax; Description; Examples. Using the fact that: z^6 &= \big(z^3\big)^2=1 \\ In below example for std::conj. using System; using System.Numerics; public class Example { public static void Main() { Complex[] values = { new Complex(12.4, 6.3), new Complex… &=\frac { \overline { 2-3i } }{ \overline { 4+5i } } \cdot \frac { \overline { 4-i } }{ \overline { 1-3i } }\\\\ What are this equation's remaining roots? out ndarray, None, or tuple of ndarray and None, optional. the complex number whose imaginary part is the negative of that of a given complex number, the real parts of both numbers being equal a –i b is the complex conjugate of a +i b 2 A complex number is written in the form a+bi. $\left \{ - i,\ i,\ -3, \ - 1, \ 2 \right \}$ How does that help? The conjugate can be very useful because ..... when we multiply something by its conjugate we get squares like this:. We get squares like this: Step 1 they are basically the same real component,... Number z as ( real, img ), Now, if represent!, imag ) is ( real, img ), Now, if we represent a complex conjugate of in. = 1-2i # # to its conjugate is ( real, for example, any non-real root will have shape... Means they are basically the same in the complex conjugate of complex Values in ;... Thus, a freshly-allocated array is returned be used for finding a polynomial 's complex zeros in.! We deno... ” work through an exercise complex conjugate examples in which we use.! Number \ ( 3 + 4i\ ) is the complex number z write this polynomial in root-factored! Part can be very useful because..... when we multiply a complex number by its complex of... ( x-5+i ) ( x−5−i ) ( 5−2i ) =2920−8i+15i−6i2=2926+297i=2926, b=297 namely... Is ( real, -imag ) conjugate bundle of just 2 sreeteja641 's “... Through an exercise, in which we use it } \ ) explain the complex conjugate of the for. 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That is typically used in this light we can rewrite above equations as follows: tanx=1 tan2x=1.\tan... Polynomial, some solutions may be arrived at in conjugate pairs: 4 7. Will discuss the Modulus and conjugate of \ ( 3 − 4i\ ) imaginary. That will not involve complex numbers there are 33 positive integers less then 100 that make znz^nzn integer... This factor to obtain the complex numbers \$ \frac { 5 + 2i } { \alpha } },! Is the complex conjugate can be used for finding a polynomial are all real, -img ) of −4−3i change. Multiply the numerator and denominator by that conjugate and simplify however, you 're to..., -1+1/2i, and engineering topics following tutorial we further explain the complex conjugate of a complex has... Numbers, both are indistinguishable and tan2x=1 { 5 + 2i } \alpha!, any non-real root will have a conjugate pair complex conjugateof a complex number \ ( 3 + ). + iy is denoted by z ˉ = x – iy } ^ 2... 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Realize a MAC using a Pseudo-random function?

Given a pseudo-random function and assuming that we do not have any other tools, How can we construct a MAC?

I believe this can be done. Would like to know if there is more than one way of doing this.

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If you have a PRF (with larger input than output), you can use it as compression function in a Merkle-Damgård structure, yielding a hash function which you can subsequently turn into a MAC with HMAC. Indeed, the security proof of HMAC relies on indistinguishability of the compression function from a PRF.

There are still an awful lot of details, though. And the PRF assumption is quite strong.

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Just to quibble over minor points, I think NMAC makes more sense than HMAC here (since you're starting with a compression function, rather than a hash function), and I believe the requirement is that the output has to be at least as long as the PRF key (since you'd be chaining through the key input). –  Seth Feb 19 '12 at 10:03

If you have a keyed PRF, where the key is sufficient to turn the function into an independent PRF, you can turn it into a keyed PRP using a Luby Rackoff construct. If you got a keyed PRP, you can use it in CBC-MAC mode.

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I am not sure if $\mathsf{PRF}$ alone can be used to construct; however there is a natural way to construct $\mathsf{MAC}$ using $\mathsf{PRF}$ and $\mathsf{UHF}$. You can see the following papers for more detail:

1. LFSR-based hashing and authentication. In CRYPTO 1994.
2. Bucket hashing and its application to fast message authentication. In CRYPTO 1995.
3. On fast and provably secure message authentication based on universal hashing. In CRYPTO 1996.

I just found a very easy way to construct $\mathsf{MAC}$ from $\mathsf{PRF}$ in the lecture notes of Barak. More concisely, it says that if $\{ f_k \}_{k \in \mathcal{K}}$ is a family of $\mathsf{PRF}$, then the following is a $\mathsf{MAC}$:

$$\mathsf{SIGN}_k(x) = f_k(x) \qquad \mathsf{VERIFY}_k(x,s)=1 \text{ iff } f_k(x)=s.$$

I think this is pretty neat.

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I think the exact details depend from the used definition of pseudo-random function. If you have one that allows input of any size (like a hash function and the $f_k$ used here), it is a lot easier than if you have one with fixed input size. –  Paŭlo Ebermann Feb 20 '12 at 20:17

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# Why this loop carries an integral if there is no undetermined momenta?

Consider the following Feynman diagram:

I've read that it will have associated with it one integral over the loop. The issue is, in Schwartz book the Feynman rules for momentum space are:

1. Internal lines get propagators;
2. Vertices get factors of $i\lambda$;
3. Lines connected to external points get nothing;
4. Momentum is conserved at each vertex;
5. Integrate over undetermined 4-momenta;

Now in this diagram I let $p_1$ be the incoming momentum on the left and $p_2$ the outgoing momentum on the right.

If in the loop I assign a counterclockwise momentum $k$, we must have at the vertex $p_1 - p_2 - k = 0$ so that $k = p_1-p_2$. In that case, $k$ is not undetermined.

I would guess by these rules that the amplitude is

$$i\mathcal{M}= (i\lambda)\dfrac{i}{k^2-m^2+i\epsilon}=(i\lambda) \dfrac{i}{(p_1-p_2)^2-m^2+i\epsilon}$$

but searching the internet it seems that the correct form would be

$$i\mathcal{M}=(i\lambda)\int \dfrac{i}{k^2-m^2+i\epsilon}d^4k,$$

but $k$ is not undetermined momentum, so why integrate over it following this set of rules?

What is actually the right viewpoint on this diagram?

The momentum conservation at the vertex is NOT $p_1-p_2-k=0$ but rather $p_1-p_2-k+k=0$ which leaves $k$ completely free. The vertex has two half-edges carrying the momentum $k$ but in opposite directions.
• I get your point now. The issue isn't with the line, that gets a single propagator really, but the fact that in the vertex it "sees" the one line going with momentum $k$ and another coming with momentum $-k$, without any sensitivity to whether they are the same line or not, right? – user1620696 May 17 '17 at 14:55


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# Limits

• May 13th 2010, 06:37 PM
riemann
Limits

We know that since the right hand limit and left hand limits are not same for the following limit, therefore

$
\lim_{x \to 0} 1/x\ does\ not\ exist
$

Now rewriting the above limit in another way,

$
\lim_{x \to \infty} x
$

does it exist? It should be infinity, thus possessing a limit. But how can rewriting a non-existing limit give an existing limit?
I am somewhere wrong here, can anyone point where? (thanks)

We are totally clear about the non-existance of

$
\lim_{x \to 0} 1/x
$

yet I came to know that

$
\lim_{x \to 0} sin(1/x)= -1\ to\ 1\
$

Therefore the above limit exists. But how can sine of an undefined function give a definite value? (The word 'definite value' does not mean in its exact sense, i.e., I tried to mean (i).how sine of an undefined function gave some value between -1 and 1 and (ii).though we are not sure of any definite value, yet we say it's limit exists?) My natural instincts tell that the above limit should not exist, yet it exists. Can someone please explain?

Your replies would be highly appreciated. Thank you.(Happy)
• May 13th 2010, 06:58 PM
pickslides
Why $\lim_{x \to 0 }\frac{1}{x} = \lim_{x \to \infty }x$ ?
• May 13th 2010, 07:19 PM
Bruno J.
Quote:

Originally Posted by riemann

We know that since the right hand limit and left hand limits are not same for the following limit, therefore

$
\lim_{x \to 0} 1/x\ does\ not\ exist
$

Now rewriting the above limit in another way,

$
\lim_{x \to \infty} x
$

does it exist? It should be infinity, thus possessing a limit. But how can rewriting a non-existing limit give an existing limit?
I am somewhere wrong here, can anyone point where? (thanks)

We are totally clear about the non-existance of

$
\lim_{x \to 0} 1/x
$

yet I came to know that

$
\lim_{x \to 0} sin(1/x)= -1\ to\ 1\
$

Therefore the above limit exists. But how can sine of an undefined function give a definite value? (The word 'definite value' does not mean in its exact sense, i.e., I tried to mean (i).how sine of an undefined function gave some value between -1 and 1 and (ii).though we are not sure of any definite value, yet we say it's limit exists?) My natural instincts tell that the above limit should not exist, yet it exists. Can someone please explain?

Your replies would be highly appreciated. Thank you.(Happy)

Good question! In a sense, the limit of $x^{-1}$ as $x \rightarrow 0$ does exist; it's equal to $\infty$, but you have to be clear about what $\infty$ is. There is made no distinction between $+\infty$ and $-\infty$ here. There is only one point at infinity; read this!

The limit $\lim_{x \rightarrow 0}\sin(1/x)$, on the other hand, does not exist; the function oscillates between $-1$ and $1$ infinitely many times in any neighbourhood of $0$.


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### Discussion

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### Video Transcript

we're told that we have two plane mirrors at right angles to one another and asked to look at figure p 22.10 as reference. So the light rays into a mere one angle, fatal with the vertical. So use the law of reflections and geometry to show that this rays reflected off both mirrors and the outgoing reflected rays parallel to the incident reflected. Right. Okay, So the angle of the incident on the Mir one estate, as we're told and by the law of reflection, the angle of incidence is England equal to the angle of reflection. Hence the angle of reflection at mirror one. We're gonna call this data. One is equal to Fada. Okay, Pretty simple. So far, so in the triangle than ABC on the image that some of the interior angles, of course, is going to be equal to 180 degrees. So from that figure, we have data one Plus, they did too. Plus, since it's a right, a right angle here, 90 degrees, all of this has to add up to 180 degrees. Okay, Okay. And since data one equal stated to excuse me since data one equal state of what we said at the very top. Right here, we can go ahead and sulfur data to In terms of data, we find that data to is equal to 90 degrees, minus data. So hence the angle of incident on near to which is stated to is 90 degrees minus data. So now again, by the law of reflection, the angle of incidence is equal to the angle of reflection. So the angle of reflection that mirror to data to is going to be equal to data three. So now we have 82 is equal to data three, which means that they had a three is equal to 90 degrees minus data. Right. So the angle the incident Ray makes with Mir one looking there at the image we're told is fine. And start a new page for this five is equal to 90 degrees minus data. And we're told that that is the angle of incident at mere one. And because if you look back on the equation, we just found for Data three. We found that it was also 90 degrees minus data. That means that the angle of incident on Mir one is equal to fate of three and data three is the outgoing reflected Ray from ear to. So we just proved that we were asked to show and we'll go ahead and just type it out here we can say. Thus outgoing reflected Ray from Mir two is parallel to the incident. Ray on mirror one. See if we can get all this in the No, There we go and we can box. I did as our solution to the question period in box it it

University of Kansas


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## Introduction

Potassium channels are integral membrane proteins present in prokaryotic and eukaryotic organisms where they contribute to the control of potassium flow, cell volume, release of hormones and neurotransmitters, resting potential, excitability, and behavior. Under physiological conditions these molecules are highly selective, allowing the permeation of K+ at near diffusion-limited rates, whereas Na+ is effectively excluded from passing through1. So far, the molecular basis of selectivity and permeation is still a matter of debate. KcsA is a proton-activated, voltage-modulated channel cloned from S. lividans and used as a prototypical protein on the biophysical studies of the K+ channels due to its simple structure: four identical subunits around a central pore, each one comprising an N-terminal domain, two transmembrane segments and the cytosolic C-terminal section. The two transmembrane segments (TM1 and TM2) are connected by a pore region that contains a tilted short-helix (pore helix), two loops and an ion selectivity filter (SF) (Fig. 1A). The SF contains four putative K+ binding sites delineated by the signature sequence TVGYG (Fig. 1B,C), clearly homologous to the more complex eukaryotic potassium channels and its conformational dynamics is mainly responsible for the permeation and the selectivity features of these membrane proteins2,3. Based on the conformation of the intracellular bundle (intracellular gate or gate I) and the SF (gate II), the gating cycle of KcsA is defined by at least four kinetic states: closed/conductive (pH7, high K+), open/conductive (transient state found in high K+ after the pH4 gating), open/inactivated (pH4, high K+, at steady state) and closed/inactivated (pH7, high K+, immediately after increasing the pH from 4 to 7)4. The opening movement at gate I is allosterically coupled to changes in SF conformation, inducing the slow inactivation process, and consequently leading to a steady-state, very low open probability of the wild-type (WT) channel5,6. The inactivation phenomenon after several milliseconds of activity occurs in a very similar way to eukaryotic channels7,8. Additionally, X-ray crystallography data from the closed state revealed two different conformations of the SF: a conductive conformation (Fig. 1B,C) in the presence of sufficient amounts of permeant cations (e.g. K+, Rb+ or Cs+), and a non-conductive or collapsed structure when in low concentration of permeant species or high quantities of blocking species such as Na+, with only the most external binding sites (S1 and S4) accessible to ion-protein interaction (Fig. 1C)3,9,10.

The SF conformation is also modulated by the amino acid residues that lie behind the signature sequence, especially those in the pore helix region, constituting a key scaffold component of the potassium channel3,10. The inactivation triad corresponds to the interaction between Glu71, Asp80 and Trp67 residues, forming a hydrogen-bond network that modulates the conductive/inactivated states by affecting the conformation of the K+-binding sites (Fig. 1B)6. In fact, mutation of Glu71 to alanine lead to a SF “frozen” in a conductive conformation in the presence of K+ and also Na+11,12. Macroscopic currents measured in this mutant channel revealed an almost abolished C-type inactivation process and consequently a high open probability. Some authors relate the non-conductive conformation of the SF seen in the X-ray experiments (PDB entry 1K4D) with the inactivated conformation13, since this phenomenon, is favored at low K+ concentrations in eukaryotic potassium channels11,14,15,16, whereas some other experimental information suggests only modest conformational changes at the G77 residue during inactivation17.

Here, we have used polarized steady-state and time-resolved fluorescence measurements to study tryptophan-tryptophan energy migration/Förster resonance energy transfer (FRET) in detergent-solubilized KcsA mutant channels to characterize the interplay between the pore helix and the SF conformation and its consequences over the different functional states of the protein. These measurements have the advantage of being performed at room temperature and using a low concentration of unlabeled protein, in contrast to X-ray crystallography, electron paramagnetic resonance (EPR) or nuclear magnetic resonance (NMR). Time-resolved fluorescence anisotropy is a powerful tool that has allowed to study homo-FRET in different systems and to determine distances and orientations between the fluorophores18,19,20,21,22. The extent of FRET depends on the reciprocal of the sixth power of the distance between donor and acceptor, making this technique a suitable “spectroscopic ruler” to measure distances within and between proteins23. When FRET occurs between identical fluorophores (homo-FRET or energy migration), the experimental observable is the fluorescence anisotropy of the sample because the successive reversible FRET steps lead to a strong depolarization of the fluorescence emitted by the molecule without any concomitant variation of its fluorescence intensity or lifetime24,25. In order to apply advanced FRET methodologies, a system with a high symmetry is required, so a KcsA mutant containing a single tryptophan residue per subunit is essential. The KcsA channel contains five tryptophan residues per subunit located at both ends of the transmembrane helical segments of the protein (Fig. 1A). W26 and W113 are located at the membrane-cytosol interface at the intracellular ends of the transmembrane segments; W87 is in contact with the lipid bilayer and W67 and W68 are buried in the channel core, all the latter at the extracellular side. The triple mutant channel W26,87,113F has already been shown to present ion conduction properties and spectroscopic characteristics very similar to the wild-type (WT) protein26. Moreover, the fluorescence signal from W67 and W68 residues are able to detect SF conformations (conductive or non-conductive) according to the type and concentration of cations27. These two residues are located at the pore helix, with their indole side chains practically in contact with the polypeptide backbone of the selectivity filter2. W67 is highly conserved among prokaryotic and eukaryotic potassium channels28 and, as mentioned above, is involved in C-type inactivation regulation29. Thus, this fluorescent residue was kept while W68 was mutated to phenylalanine in order to obtain a quadruple KcsA mutant with a single tryptophan residue per subunit (W26,68,87,113F or “W67” KcsA). We have also taken advantage of the non-inactivating E71A mutation to study the KcsA channel predominantly kept in a conductive conformation (non-inactivated state)11.

Our results indicate that W67 residues undergo an efficient homo-FRET process among the four channel subunits, and that their fluorescence anisotropy is an excellent reporter of the changes in the pore helix conformation induced by the progressive increase in the SF ion occupancy and by the gating transitions. Moreover, the analytical solution describing homo-FRET among a homo-tetramer in a square geometry allowed retrieving W67-W67 lateral distances from the W67 KcsA fluorescence anisotropy decays, shedding new light on the KcsA inactivated state conformation and its relationship to S2 SF binding site occupation.

## Results

### The W67 KcsA mutant behaves as a wild-type like protein and shows unique photophysical properties

The presence of a predominant tetrameric structure of the detergent-solubilized W67 KcsA mutant channel was confirmed in the presence of 200 mM KCl, RbCl, CsCl and NaCl, at both pH 7 (closed state) and 4 (open state) (Fig. 2A). The tetrameric structure was also preserved in the presence of 5 mM tetrabutylammonium (TBA+) chloride. This blocker salt binds with nM affinity to the channel cavity, leaving the SF almost “empty” of cations30. The W26,68,87,113F substitutions had a minimal impact on the structural stability of the detergent-solubilized protein as shown by the thermal denaturation assays performed in the presence of either 100 mM Na+ or K+ at pH 7 (Fig. 2B). In addition, ion permeation properties tested by patch-clamp methods confirmed that W67 KcsA gates at pH 4 and behaved as the WT channel, showing a low open probability in the presence of K+ (Fig. 2C), thus validating this mutant channel as a wild-type like protein.

W67 KcsA remains a highly fluorescent protein, retaining 40–45% of the WT overall florescence intensity, due to the shielded position of its single tryptophan residue (W67) per subunit (protected from the aqueous solution) (inset to Fig. 2D). The fluorescence properties of W67 (blue-shifted emission spectra (maximum emission wavelength ca. 325 nm) and high quantum yield (Φ~ 0.3, using N-acetyl-L-tryptophanamide (NATA) in water as reference31) yielded a Förster radius of R0 = 12 Å, close to the highest values described in the literature for the tryptophan-tryptophan Förster pairs20. The steady-state (Fig. 2D, Table S1) and time-resolved (Table S2) spectral properties of W67 KcsA were slightly sensitive to the monovalent cation present in the buffer.

### Efficient inter-subunit homo-FRET occurs between the tryptophan residues of W67 KcsA channel depending on the ion occupancy of the SF

Direct inspection of several high resolution crystal structures of KcsA crystals in the absence of their C-terminal domains and with an antibody Fab fragment bound to the extracellular loop revealed W67-W67 lateral separation distances, R ~ 17–18 Å. In addition, R ~ 15.2–15.9 Å was found when working with the full-length channel bound to an intracellular Fab fragment (Table 1). These distances are close to the Trp-Trp Förster radius of R0 = 12 Å calculated from our spectroscopic data for the detergent-solubilized W67 KcsA and therefore intramolecular homo-FRET phenomena among the tryptophanyl residues within each channel is highly probable. To test this hypothesis, both steady-state and time-resolved fluorescence anisotropy measurements were carried out for the mutant channel under different ionic conditions and pH. The samples were excited at 300 nm as the fundamental anisotropy of tryptophan reaches its highest value (r0 ~ 0.3) at this wavelength32, allowing to maximize the dynamic range of the time-resolved measurements.

At pH 7, the steady-state anisotropy of W67 KcsA showed a strong dependence with the cation present in the bathing solution: the more permeant the monovalent cation is, the lower the steady-state anisotropy presented by the protein (<r>K+ ~ 0.135 ± 0.004 (n = 5) <<<r>Rb+ <<r>Cs+ <<<r>Na+ ~ 0.166 ± 0.004 (n = 5)). The highest steady-state anisotropy was obtained in the presence of 5 mM TBA+, i.e. with an “empty” SF ($$< r{ > }_{{{\rm{TBA}}}^{+}}$$ ~ 0.180 ± 0.002 (n = 3)) (Fig. 3A). Altogether, these data strongly suggest that the more permeant cations induce local conformational changes in the channel at the SF and pore helix level, hence increasing the efficiency homo-FRET among the tryptophanyl residues and consequently producing a stronger depolarization of the protein fluorescence (lower <r> values). This hypothesis was corroborated by the time-resolved anisotropy data obtained for the mutant channel. These measurements are fundamental to retrieve quantitative structural information (distances and orientations) from systems undergoing homo-FRET18,19,20,21. The anisotropy decay of W67 KcsA became much faster in the presence of K+, the more permeant cation, as compared to the blocking agents TBA+ and Na+, a result that is a spectroscopic signature for the occurrence of an efficient homo-FRET process (Fig. 3C)25.

The four-fold axis of symmetry of the KcsA channel and consequently the square geometry delimited by the W67 residues (Fig. 4), allowed deriving a formalism describing the fluorescence anisotropy decay of the W67 KcsA mutant-DDM complex. As detailed in the Theory section (SI), the depolarization of the fluorescence emitted by the W67 residues was considered to be due to two main processes: (i) the overall rotational tumbling of the tetrameric KcsA-DDM complex in buffer solution, and (ii) energy transfer between the subunits within each tetrameric channel. This model has three fitting parameters, namely the initial anisotropy of each W67 residue, r(0), the FRET rate constant between neighbouring Trp residues, k1, (Fig. 4), and the global rotational correlation time, ϕg, of the KcsA-DDM complex (Eq. 1). This rotational correlation time was first determined independently by measuring the time-resolved anisotropy decay of KcsA L90C labeled with a long-lived fluorescent probe, N-1-pyrene-maleimide (see SI for a detailed description of the process); a ϕg = 43 ± 3 ns was obtained and fixed during the subsequent fitting procedures. Then, we investigated whether changes in ionic conditions (use of different blocking/permeant cations at saturating concentrations) influenced the tryptophan-tryptophan lateral distance R within each channel. The derived homo-FRET formalism (Eq. 1) was fitted to the fluorescence anisotropy decays obtained for W67 KcsA at pH 7 (closed state at the inner gate). As exemplified in Fig. 3D, the relative2 surfaces showed well-defined minima and therefore the fittings were adequate to determine the lateral rate constant k1. Using Eq. 2, the intramolecular lateral inter-tryptophan distances, R, could now be retrieved from the fitted rate constants k1. As tryptophan is an intrinsic fluorescent residue of the proteins, the distances calculated by the homo-FRET approach can be directly compared to those obtained from the WT KcsA crystal structures. The results indicate that at pH 7.0, the calculated distances gradually decreased (RNa+ > RCs+ > RRb+ > RK+ (Fig. 3E and Table 1)) upon increasing the ion occupancy of the SF, and therefore with the permeability sequence already described for the WT channel27. Remarkably, the larger and the shorter average W67-W67 Cδ2-Cε2 lateral distance within the protein were estimated to be ~19 Å in the presence of 5 mM TBA+, when the SF is essentially empty23, and ~15 Å in the presence of 200 mM K+, when the average ion occupancy of the SF is ~2, respectively, illustrating the conformational plasticity of this domain when the potassium channel is in solution at room temperature (ΔRmax ~ 4 Å (Table 1)). In contrast, W67-W67 Cδ2-Cε2 distances calculated from the X-ray data available from the truncated channel (Δ125) (bound to an Fab segment tagging the extracellular loop) show only small differences in the presence of different conducting and blocking cations (R~ 17–18 Å and therefore ΔRmax < 1 Å (Table 1)). Interestingly, the homo-FRET derived W67-W67 lateral distance in the presence of K+ was found to be almost identical to the distance retrieved from the X-ray data obtained only for the full length channel bound to a stabilizing “C-terminal domain” antibody (PDB entry: 3EFF, Table 1). This is clear evidence that the conditions required for X-ray studies critically affect the structure of the channel, imposing restrictions to the conformational changes at the SF level.

To further test the conformational dynamics of the short pore helices, we took advantage of the well-known conformational equilibria of the SF, which is able to interconvert between a collapsed, non-conducting conformation, and a conductive structure, where all the four binding sites for K+ are detected. At pH 7 this equilibrium is regulated by the type and concentration of the cation present in the filter, since only sufficient amounts of permeant cations are capable to shift the equilibrium towards the SF conductive state. Moreover, these conformations have been described to have different affinities for K+ or Na+26,27. Here, we were able to monitor the transition between these conformational states by performing both steady-state and time-resolved anisotropy measurements of W67 KcsA as a function of either K+ or Na+ concentration, at pH 7. Both the steady-state fluorescence anisotropy (Fig. 5A) and the calculated inter-subunit W67-W67 lateral distances obtained from analyzing the corresponding time-resolved data with the derived homo-FRET formalism (Fig. 5B) varied pronouncedly in a dose-response manner with either the K+ or Na+ concentration used. These results reveal how powerful the homo-FRET approach here described is, since it is able to verify previous thermodynamic studies performed with the WT KcsA channel solubilized in DDM, where K+ binding to the protein has been described by two consecutive binding events, the first one with an apparent 1000-fold higher affinity than the second one26. The first binding event, which corresponds to the stabilization of the non-conducting structure, in which K+ is able to bind either to S1 or S4 SF binding sites (with an average occupancy of just one K+ distributed between these two sites)3, resulted here in a ~1 Å decrease in the calculated average inter-tryptophan lateral distances (Fig. 5B). Further increasing the concentration of the permeant K+ cation leads to a higher occupancy of the pore, as a second ion goes into the middle of the filter (site S2 or S3), self-inducing the formation of the conductive structure. This conformation shows a lower affinity for the cation in order to allow for the high-efficiency K+ permeation process to occur26. As it is shown in Fig. 5B, this second step further narrowed down the average inter-tryptophan neighboring distances between the four subunits of W67 KcsA by ~3 Å. A similar titration experiment was also conducted with Na+, where the concentration-dependent response of both <r> and R was now monophasic, resulting in an overall ΔR ~ 1.5–2 Å, also in agreement with the previously described binding of Na+ to the most external site (S1)33.

### Ion occupancy at S2 binding site controls the modest conformational changes linked to pH-induced gating of W67 KcsA

We next sought to gain some insight into the conformational changes at the SF and pore helix derived from the pH-induced gating of the channel and its relationship to the C-type inactivation process that takes place in the presence of K+ once the intracellular gate is open. With this objective in mind, fluorescence anisotropy measurements made for the W67 KcsA mutant channel at pH 7 (closed state) and 4 (open state) were compared. Figure 3 (panels A and E) show that both the steady-state anisotropy and average lateral inter-tryptophan distances presented the same general trend at both pH values as the two parameters gradually decreased upon replacing the blocking agents TBA+ or Na+ with the progressively more permeant cations Cs+, Rb+ and K+, respectively. More importantly, triggering the opening of the inner bundle gate generated only a modest change on the anisotropy decays (Fig. S1), and consequently on the average inter-subunit W67-W67 lateral distances between the open and closed states (ΔR ~ 0.6–0.8 Å). The difference between the two distances was found to be statistically significant (P < 0.001 (pH 7 vs pH 4) as determined from two-way ANOVA (with Bonferroni multiple comparison posttest), only in the presence of saturating concentrations of Na+, Cs+ and Rb+ (Fig. 3E). Analysis of full length KcsA X-ray data also revealed that in the presence of high amounts of K+, there is no significant variation in W67-W67 lateral distances when comparing the closed and open states at the inner gate (Table 1). It should be stressed that there is no structural information available for any other cation with this experimental approach.

At saturating cation concentration, the main difference between the KcsA complexes with Na+, Cs+, Rb+ versus K+, is the ability of the latter to occupy the S2 position at the SF10, so we decided to test the hypothesis that S2 binding site occupancy would lead to shorter W67-W67 lateral distances and insensitivity to acid pH gating. To this end, barium binding to the mutant channel was also evaluated. Ba2+ is a divalent cation which acts as universal blocker of K+ channels by binding to the selectivity filter with high affinity (nM range). KcsA X-ray crystallographic data (PDB entry: 2ITD) revealed that WT KcsA can complex barium cations not only at S4 but also at S2 binding site, leaving both positions S1 and S3 empty. As shown in Fig. S1, W67 KcsA presented similar steady-state and time-resolved fluorescence anisotropies in the presence of saturating concentrations of Ba2+, at both pH values. The change in the W67 inter-subunit distance between the open/inactivated and closed states of the mutant channel was now non-significant (P > 0.05 (pH 7 vs pH 4) as determined from two-way ANOVA with Bonferroni multiple comparison posttest), suggesting that occupation of the S2 SF binding site precludes a pH-induced conformational change at the W67-W67 lateral distance level.

### E71A mutation locks W67 E71A KcsA in a conductive conformation

The second approach used here to explore the structural changes linked to channel inactivation was to study the behavior of W67 E71A KcsA. The E71A mutation exhibits a non-inactivating phenotype since it abrogates the C-type inactivation process, presumably because its SF is stabilized in a conductive conformation, even in the presence of low K+ concentrations or high Na+”11,12. The E71A W67 KcsA mutant behaved in a similar way to the E71A counterpart (see Fig. S2), so we expected the anisotropy decays of the W67 E71A mutant channel to resemble the one obtained for the conductive conformation of W67 KcsA in 200 mM KCl buffer. As exemplified in Fig. S2, this prediction was experimentally confirmed as the inter-subunit W67 distances calculated for W67 E71A KcsA were ~15 Å under all conditions tested (in the presence of 200 mM K+, Rb+, Cs+ and 1 M Na+ at both pH (Fig. 3F)), close to the values previously obtained for W67 KcsA in 200 mM KCl, when homo-FRET among the tryptophan residues was most efficient (Fig. 3E).

Crystal structures of the closed state of the E71A channel at high K+ concentration show that D80 and W67 residues could adopt two different conformations: one that resembles the WT structure (“non-flipped”) and a “flipped” structure where there is disruption of the H-bond between these two residues leading to a rotation of the W67 side chain “away” from the SF (Fig. S3A)11. It should be noticed that all distances here determined were not compatible with the presence of the “flipped” conformer of the potassium channel. The data obtained here are also in agreement with previous thermal stabilization studies carried out with E71A KcsA34.

## Discussion

### The conformation of the short pore helices is strongly correlated with SF ion occupancy of W67 KcsA in the closed state

The location of W67 in the short pore helix allowed studying the interplay between the SF occupancy and pore helix conformation. The W67-W67 inter-subunit lateral distances were found to be sensitive to cation-type and cation concentration within the pore. At saturating salt concentration, the larger W67-W67 distances were found when the blocker TBA+ was used, as expected for a cation that only binds to the channel cavity, barely interacting with an “empty” SF at the T75 level. As the SF ion occupancy increases (Na+ < Cs+~Rb+ < K+), the tryptophan residues present in each subunit of the W67 KcsA channel become closer to each other, the inter-subunit FRET rate constants increased and the excitation energy was more effectively distributed among the four tryptophan residues of the channel which, in turn, led to a stronger depolarization of the emitted fluorescence. The KcsA X-ray data (PDB entry 1K4C and 2ITC) show a tendency of the protein subunits to get closer to each other at the SF level in the presence of high K+ compared to high Na+ (even though to a lesser extent than the channel in solution), and previous EPR experiments also revealed that channel subunits gather around the central pore during the C-type inactivation process35. Thermal denaturation assays performed in the detergent-solubilized WT channel also describe that cations can bridge together the protein subunits by interacting through multiple contacts with the internal cavity and the SF26. Based on these evidences, we claim that changes in W67-W67 lateral distances seen in these homo-FRET experiments are mainly due to the movement of the pore helices approaching each other towards the pore axis, even though modifications on Trp side chain angle (away/closer to the SF, in a similar way to the flipped conformation of E71A mutant) cannot not be dismissed.

Regarding the physiologically relevant K+ ion, our data provide additional experimental evidence for the nonconductive-to-conductive conformational change undergone by the KcsA channel upon increasing the total number of ions in the filter, as previously described in the Introduction section. This equilibrium between the two dominant channel structures is driven by the association of the second ion to the central binding sites (sites S2 or S3) in the singly occupied filter. The homo-FRET dose-response experiments detected two consecutive K+ binding events, the first one bridging closer together the channel subunits by ~1 Å and the second one by ~3 Å respectively. As previously seen in the WT channel26, these two consecutive transitions correspond to the stabilization of the non-conductive state and to the induction of the conductive conformation of the SF, respectively. The greatest change in the intersubunit lateral distances observed during the second binding event of K+ correlates well with the vast thermal stabilization observed during this process (35–40 °C) in the WT channel26. These experiments also reveal that the steady-state fluorescence anisotropy of our systems, a much more accessible experimental parameter than the determination of the rate constant k1 via time-resolved anisotropy measurements, can be used directly as a spectroscopic ruler of the average conformation adopted by W67 KcsA in solution.

Finally, it is important to highlight that the overall ΔR that was obtained from the homo-FRET measurements upon varying the cation type/concentration in solution was much larger than the one calculated from the available crystallographic data for the truncated channel stabilized externally with a bound Fab antibody fragment (Fig. 6 and Table 1). Several factors may contribute to these discrepancies: first, our measurements were performed with a very low concentration of the detergent-solubilized, full-length mutant W67 KcsA at room temperature, when the channel is expected to display a much larger conformational freedom than in its crystal form. In fact, crystal packing forces near the SF might contribute to the lack of significant structural alterations in this channel region. Secondly, the auxiliary antibody Fab fragments included in the crystallographic studies were absent from our experiments. Depending on different authors, the Fab fragment is either directed towards the channel extracellular domain, involving interaction with the pore helix R64 and nearby residues11, or towards the intracellular C-terminal36. Both co-adjuvants have already been shown to reduce the rate of channel entry into the inactivated state coupled to pH-induced gating (more noticeable in the “extracellular” Fab segment). The homo-FRET derived distances obtained in the presence of saturating concentrations of K+ are indeed more similar to the ones reported for the full length KcsA crystallographic data (PDB entry: 3EFF (Table 1)), further hinting that the Fab fragment bound to the channel ectodomain is restraining its conformational freedom. In addition, comparison of the X-ray data from the full length channel attached to a Fab segment (PDB entry: 3EFF, R = 15.9 Å, Table 1) and the truncated KcsA channel without any antibody (PDB entry: 1BL8, R = 17.3 Å, Table 1) also hints to a possible allosteric effect induced by chymotrypsin cleavage of the C-terminal domain as an additional factor responsible for the discrepancies detected between the homo-FRET and X-ray calculated distances. Hence, the high-resolution structures provided by protein crystallography may not always reflect the predominant structure displayed by the protein in less extreme experimental conditions (i.e., at low concentration in aqueous solution at room temperature).

### The “conductive” E71A mutant freezes the W67 position in the “non-flipped” conformation

As previously described, the substitution of a glutamate residue by the smaller alanine residue at position 71 completely abolished the C-type inactivation process, leaving the SF “frozen” in a conductive conformation, not only in the presence of K+ but also Na+11,12. According to molecular dynamics simulations, the interaction between E71 and D80 residues contributes to the stabilization of the pore helix and the SF, and the strength of this interaction could be associated to movements of this helix. On the other hand, the H bonding between D80 and W67 would be less stabilizing in terms of channel structure, but more relevant since is more conserved among the K+ channel family37. The lack of SF flexibility in this mutant was further corroborated by our analysis, where the calculated inter-subunit lateral distances were now essentially the same whether Na+, K+, Rb+ or Cs+ ions were present at both pH7 and 4.0. Previous thermal denaturation experiments performed on this mutant also described a very similar behavior in the presence of these permeant cations38, and radioactive flux assays also demonstrated a decrease in cation selectivity induced by the E71A mutation in KcsA12. Apparently, upon disruption of the hydrogen bond between E71 and D80 from the inactivation triad, the SF of this mutant channel maintains a permanently conductive conformation, which is able to accommodate monovalent ions with very different sizes while displaying very similar lateral W67-W67 distances at the level of the short pore helices of the mutant channel. Furthermore, all the calculated distances now resemble those obtained from the W67 channel in the presence of K+, confirming the similarity between all the “conductive” states in terms of SF and pore-helix conformation. Cuello et al. recently explored the open conformation of a truncated form of the E71A mutant39. The reported X-ray crystallography data in the presence of K+ showed a very similar conformation of the SF compared to the resting state, supporting our homo-FRET results at pH 4, and pointing out to a modification in the strength of the interaction between the SF carbonyl groups and the K+ cations induced by channel gating in this mutant.

As previously mentioned, X-ray crystallography experiments also revealed two possible W67 conformers in the closed E71A mutant: a “non-flipped” conformation, very similar to that seen in the WT channel, and a “flipped” position of the D80 residue, oriented extracellularly, and leaving the W67 free to move away from the channel core. However, the distances calculated from our experiments support only the presence of the “non-flipped” channel structure, since a strong reduction in the homo-FRET process would be expected for the “flipped” conformer (i.e., a larger W67-W67 lateral distance should be found). Cordero-Morales and collaborators found that the stabilization of D80 in the flipped conformation, and consequently the loss of the interaction with W67, is probably due to an electrostatic interaction with Arg93 in the Fab segment, helping to stabilize that particular conformation of the filter11. Taking all these data into account, our results indicate that the W67-D80 H bond interaction is stable at room temperature in the detergent solubilized system in the absence of any antibody segment, and the mutation of E71 residue to alanine is not sufficient to disrupt this bond under the described conditions.

### The X-ray collapsed state does not account for the conformation of the inactivated state

As a proton activated channel, buffer exchange to decrease the pH from 7.0 to 4.0 triggers the opening of the KcsA channel at the intracellular gate and its subsequent inactivation. During the last years, attempts have been made to characterize the inactivated state of KcsA. Whereas some authors propose that the “collapsed” state, initially seen by X-ray crystallography at low K+ concentration (PDB entry: 1K4D), accounts for the conformation of the SF in the inactivated state39,40,41, new pieces of information indicate that structural modifications linked to the inactivated state are not equivalent to that described for the collapsed state at low K+, including different binding affinities for this cation and modifications on the ion occupancy at the most extracellular S1 binding site17,42,43,44 Comparison between the distances determined in this work for the closed/conductive (pH 7, 200 mM K+, R = 15.4 Å), closed/non-conductive (pH7, 0.1 mM K+, R = 16.5 Å) and open/inactivated (pH4, 200 mM K+, R = 15.3 Å) states clearly rule out the hypothesis that the inactivated state is structurally similar to its collapsed state, in agreement with recent studies.

It should also be stressed that once again, W67-W67 lateral distances recovered from the X-ray crystallography data show significant differences when comparing the open channel in the presence of a Fab segment bound to the extracellular face (PDB entry: 3F5W, R = 17.7 Å (Table 1)) or to the intracellular C-terminal domain (PDB entry: 3PJS, R = 15.2 Å (Table 1)). The distances for the W67 mutant channel calculated from the anisotropy decays in the presence of K+ at pH4 are in accordance with those observed in the crystals of the full length channel bound to the intracellular antibody. Moreover, comparison between the W67 inter-subunit crystallographic distances from the full-length KcsA channel in the closed and open states (PDB entries 3EFF and 3PJS, respectively) shows no significant changes, indicating that there is a fine and subtle tuning of KcsA conformation which modulates the inactivation of K+ channels.

Finally, a more detailed analysis of our results reveals that the relative changes in the homo-FRET distances (ΔR pH7 versus pH4) depends on the type of cation present in the solution: whereas Na+, Rb+ and Cs+ showed a statistically significant small increase in R, no significant modification was found in the presence of K+ or Ba2+. As previously described throughout the manuscript, K+ and Ba+ share the ability to bind to the S2 binding site of the SF (among others). This maintenance in the recovered distances using the W67 from the pore helix as a reporter suggests that the occupation of the S2 site lead to a very similar conformation of the SF in the open and closed states. Additionally, the ion occupancy at S2 also seems to be relevant in terms of the C-type inactivation process: Cs+ and Rb+ are permeant species that cannot bind to S2 site in the SF, a result that has been correlated to a 1000-time slower C-type inactivation process and 20-fold increase in mean open times (compared to K+) in the WT channel17,38. Here, Rb+ and Cs+ are the only permeant species that showed a small but significant increase in the W67-W67 lateral distances when the channel is in the open state. Interestingly, the increase in the W67 inter-subunit distances from pH 7 to pH 4 suggests that the open-conductive (pH 4) structure of the channel is not exactly equal to the closed-conductive (pH 7) conformation when working with these two cations. These results advocate that in the presence of Rb+ and Cs+ the interaction of the inactivation triad is somehow “less stable” and gives the W67 more conformational freedom, whereas the binding of K+ to two out of the four putative binding sites in the SF stabilizes the inactivation triad both in the closed (pH 7) and open (pH4) states. Matulef et al.42,45 also suggested that occupation of S2 SF binding site is a key element for the C-type inactivation process to occur in K+ channels.

In sum, the homo-FRET analysis has provided a quantitative picture of the changes in inter-subunit distances (lateral inter-tryptophan distances) at the level of the pore helices of the W67 KcsA mutant channel upon ion binding and pH gating, with an uncertainty of about +/−0.2 Å. Furthermore, comparison with X-ray crystallographic distances highlights how the presence of a bound Fab fragment and/or the hydrolysis of the C-terminal domain modify the extracellular loop and the SF conformation. In the short term, we plan to extend the homo-FRET studies to evaluate how the lipid environment modulates the structural plasticity of KcsA, a topic that it is difficult to explore by the X-ray crystallography approach.

## Methods

For a detailed description of materials and methods used, please see S.I. section.

### Protein expression, purification and samples preparation

Briefly, KcsA W26,68,87,113F (W67); W26,68,87,113F E71A (W67 E71A) and L90C mutant channels were purified in E.coli M15 (pRep4) heterologous expression system and purified by Ni2+/His-tag affinity chromatography according to previous reports46. Purified channels were then dialyzed against the pH 7 (20 mM HEPES, 5 mM n-Dodecyl-β-D-maltoside (DDM), 5 mM N-methyl-D-glucamine) or pH 4 (10 mM succinic acid, 5 mM DDM, 5 mM N-methyl-D-glucamine) buffers supplemented with the adequate concentration of KCl, RbCl, CsCl, NaCl, BaCl2 or tetrabutylammonium chloride (TBA.Cl) salts. Unless otherwise indicated, the typical protein concentration used in spectroscopic assays was 6 µM (monomer based). For patch-clamp functional exploration, KcsA was reconstituted into asolectin liposomes at 100:1 lipid to protein ratio (w/w) using Bio-Beads SM-247.

### Fluorescence measurements and analysis

Intrinsic fluorescence emission spectra and steady-state fluorescence anisotropy were measured on a Horiba Jobin Yvon Fluorolog-3-21 spectrofluorometer. Time-resolved fluorescence and anisotropy measurements with picosecond resolution were obtained using the time-correlated single-photon timing (SPT) technique. The fluorescence decays of the 6 μM detergent-solubilized W67 KcsA samples (λexc = 300 nm) and pyrene-labeled L90C KcsA channels (λexc = 335 nm) were measured at 345 nm and 400 nm, respectively, using an emission polarizer set at the magic angle (54.7°) relative to the vertically polarized excitation beam produced by a frequency doubled Rhodamine 6 G laser, as previously described48.

Analysis of the fluorescence intensity decays were performed with the TRFA software (Scientific Software Technologies Center, Minsk, Belarus). The anisotropy decays of the detergent-solubilized W67 and W67 E71A KcsA samples were analyzed as (see S.I. for details):

$$r(t)=\frac{r(0)}{4}[1+\exp (-4{k}_{1}t)+2\exp (-\frac{9}{4}{k}_{1}t)]\cdot \exp (-t/{\varphi }_{g})$$
(1)

where r (0) is the initial anisotropy, k1 is the rate constant for FRET (between neighboring tryptophan residues (Fig. 3B) and ϕg is the rotational correlation time of the KcsA-DDM complex (independently determined using pyrene-labeled L90C KcsA). The inter-tryptophan lateral distance R can be directly calculated via k1

$${k}_{1}=\frac{1}{\tau }{(\frac{{R}_{0}}{R})}^{6}$$
(2)

where τ is the intensity-weighted mean fluorescence lifetime, and R0 is the critical radius computed with an orientational factor κ2 = 2/3.

The time-resolved anisotropy measurements of the pyrene-labeled L90C KcsA channels solubilized in the detergent micelles were globally analyzed using a two-step procedure using the TRFA software as previously described49. The usual statistical criteria, namely a reduced χ2 < 1.2 and a random distribution of weighted residuals and autocorrelation plots, were used to evaluate the goodness of the fits31.

### Statistics

Statistical analysis was performed using Graphpad Prism version 5 (Graphpad Software). The statistical significance of observed differences was assessed by one-way or two-way ANOVA with posttest as indicated. Results are expressed as mean ± standard deviation, S.D.