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# Do formulas prevent students from understanding concepts?

1. Jan 11, 2007

### Ja4Coltrane

I recently made several comments about the sole use of formulas when a logical approach would be as effective.

I am a high school senior taking AP physics C. Last year I took a more basic course, and in both of these classes (more so in the more basic course) I couldnt help but feel that students abuse formulas and feel that the key to succeeding in physics is memorizing these formulas.
Earlier this year I watched a girl move through a huge pile of flash cards with every formula that she had learned all year. Formulas like the acceleration of an atwood machine and the acceleration of a box car with a pendulum inside making an angle with the verticle. I know that if she instead understood the concepts well enough to realize the reason that the formula is what it is, she would not only feel like there is much less to memorize, but she would also be able to manipulate the formula in order to solve different types of problems.

Physics is not about memorization--it's not a language, history, or geography class. A physics course should be about thought and the pure mathematics should always follow that thought.
I'd like any feed back. What do the other members of PF think on this issue?
Do you all think that formulas prevent students from understanding and enjoying physics?

2. Jan 11, 2007

### sunbunny

I completely agree with you! I'm a university student taking introductory physics and I've never been really strong with the conceptionalization and the thinking process that goes along with this course. I find memorizing formulas as comforting because I think that I've leaned something when in reality I haven't learned much.

3. Jan 11, 2007

### Kurdt

Staff Emeritus
I agree that you need to know the concepts behind a formula but my point from the earlier thread was that mathematics is the embodiment of logical thought. With a little mathematical knowledge you can manioulate almost any basic equation presented without knowing too much about it and I agree this is counter-productive. But so is learning concepts and not modelling them mathematically. If you want to learn physics the both go hand in hand.

Another point is that sometimes mathematical methods can glean valuable insights that are otherwise obscured if one is limited to solely thinking about problems with no maths. This is exceptionally evident in the thread where this started which was about the equivalence principal which is something counter-intuitive and has puzzled many of the greatest minds throughout history.

4. Jan 11, 2007

### Claude Bile

The New South Wales (in Australia) high school Physics syllabus was changed six years ago to remove the emphasis on the equation/calculation aspect of Physics. The current syllabus as it stands requires NO mathematical prerequisities (other than the compulsory maths taught at junior level).

Is this the right balance though? From a personal point of view, I do not think so. (I think this change was implemented more due to the lack of science teachers in high schools that have studied physics at a tertiary level). Students ought to have some exposure to equations and the role they play, and not every concept in physics can be tackled using qualitative logic alone.

So on that note I would have to answer this question with a no (I'm assuming that understanding and enjoyment go hand in hand ). It is imperative though that students see past the symbols that comprise an equation and understand what they represent, and under what conditions the equation is valid, because without this understanding, the equation is means nothing more than squiggles on a page.

The best approach to teaching I feel is a synergistic approach using both qualitative logic and equations. You can't do good physics without one or the other. Equations are important because they - in a sense - encapsualte the logic of the problem (though it may sometimes be difficult to decipher), while qualitative logic is important as what is commonly reffered to as 'sanity checks' because no equation is guaranteed to give a sensible result.

I'll make one final comment as to why equations are important - when it comes to not simply learning physics, but actually USING this knowledge - whether it be in engineering, applied physics or theory - does require the use of equations, if for nothing more than being able to calculate a quantitative result.

Claude.

5. Jan 11, 2007

### Gokul43201

Staff Emeritus
I disagree. Any thought that you apply to physics that lacks a mathematical structure is simply intuition. You can only take intuition so far, and it is not the right way to learn physics.

This doesn't mean you merely memorize formulas either - that approach is even more flawed, for it results in no learning.

6. Jan 18, 2007

### Ja4Coltrane

Thanks for all of your replies--I have always found this to be an interesting argument in physics. Personally I feel that this goes down to the fact that every person thinks in a slightly different way--thus, the different types of IQ. Just a little note, Claude Bile mentioned that the New South Wales has this zero math idea. Last year, when I took my first physics class, I started with Paul G. Hewitt's Conceptual Physics--that book, although it has some very basic mathematics (like calculating the magnitude of the acceleration of a 1 kg block when a 1 newton net force acts on it) the book made physics very interesting and very enjoyable.
Thank you for your response Kurdt. I like your point--I do feel that by using mathematics one can often find very interesting results that he would never actually notice just by thinking. Of course, once that result is seen in the equation, one can understand the conceptual aspect of it. I think this proves one thing that I said wrong--that thought should always come before math. Perhaps math often stimulates thought and thus has a place before it. (not always--hehe)

7. Jan 18, 2007

### Kurdt

Staff Emeritus
Well I think we can safely say that the two go hand in hand and to replace one exclusively with the other would be a mistake.

8. Jan 18, 2007

### tylerp

I don't think history is about memorization either, I think it's similar to Physics in the aspect that you have to know the whole story to understand the most important question, why?

9. Jan 18, 2007

### Ja4Coltrane

absolutely! Sorry about that impulsive and not-very-well thought out claim.

10. Jan 18, 2007

### Staff: Mentor

From my perspective (which is a mix of real-world EE R&D work and a love for general physics), you should memorize formulas and equations that you use often, so that you don't have to slow down to look them up, and beyond that memorize enough fundamentals so that you can derive other formulas and equations within a reasonable time when you need them (instead of looking them up). Like, I absolutely need to have memorized and ready the values for $$\mu_o$$ and $$\epsilon_o$$ and equations for capacitance and characteristic impedance and many other daily things. And I need to have several E&M fundamental equations readily memorized (like the Biot-Savar Law (sp?) ), and I prefer to be able to re-derive things like the s-parameter equations instead of looking them up again.

But especially with the easy access to the Internet and wiki and google, there are lots of formulas and equations that I don't bother memorizing, like Snell's Law (which I had to wiki a few minutes ago to provide some PF homework question help).

My advice would be to be sure that you can re-derive important relations when necessary (because you understand how to, and what underlying relations to use), and then decide what you want to have memorized for immediate daily/weekly use, what you want to be able to derive with a few minutes work, and what you use rarely and can wiki for.

11. Jan 18, 2007

### Ja4Coltrane

berkeman--I'm sure that in your 3600 posts you have at some point felt that the sometimes people look at formulas without really thinking about them and just struggle to plug any numbers in that sort of work (aka have the right units or something) take a look at post number five in this thread: https://www.physicsforums.com/showthread.php?t=152028

that sort of shows what I mean.

12. Jan 19, 2007

### samh

I'm taking an introductory physics course at my university right now.

Let me say something about myself: I *hate* memorizing things. I do anything I can to avoid it. I've held on strong to this viewpoint for years but this physics class is starting to change my mind.

I've spent the semester so far trying very hard to understand everything. I can derive the formulas from thin air now and haven't memorized a thing. But it took me so long to get this understanding that I'm falling way behind in this class and very far behind in all my other classes. At this rate I can't keep up. Everyone else is memorizing and they're doing better than I am. That's BS. These super crunched, 10-week semesters aren't meant for people like me.

You might say, "well it doesn't take that long to acquire a real understanding." But I would say that it doesn't take YOU that long. I'm not good at physics and neither are most people (i.e. 98% of the population). For us physics takes a long time. Memorizing a few formulas saves a lot of effort. As of right now I do nothing but homework constantly and I'm sick of it. Next chapter I'm just going to memorize the formulas so I have some free time for once and maybe I'll be able to enjoy my life a little bit.

Also, don't forget that most people taking physics courses are not aspiring to be physicists. For a lot of people, physics is just another class they're forced to take and they just want to get it out of the way.

13. Jan 19, 2007

### DougD720

i agree 100% with the poster.

My teacher emphasizes this and makes sure we understand the concepts and then uses a General formula that works for All cases instead of the many special ones that work for each. We don't use a formula in class unless we can derive it somehow.

My teacher is excellent i must say.

14. Jan 19, 2007

### dontdisturbmycircles

Do formulas prevent students from understanding concepts?

No. There are two ways to learn high school physics, in my mind. One is to understand the formulas and what they mean and do, and where they came from. There is another way that I see some people using to try to learn physics, they write out the equation and say "Ok, well I need to get v so I need to find an equation with v in it. Ok I see one, but what is t? temperature?" In other words they just learn how to use equations, but aren't really learning physics.

In order to be successful in high school physics (which is the only level of physics I am qualified to speak about) I think it is neccesary to understand the underlying concepts. But without the formulas, physics is just a conversational course, and I don't think that is physics.

Last edited: Jan 19, 2007
15. Jan 19, 2007

### dontdisturbmycircles

This is a good point. If the student can use one or two equations to derive the 6 or so kinematics equations (for example) then I think it is safe to say that they understand the equations.

16. Jan 19, 2007

### Kurdt

Staff Emeritus
The problem with this sometimes is that at the level where the kinematic equations are introduced, the students don't yet have the skill to derive them from first principles. I'm not sure if this is the case in all countries though but i've certainly encountered it.

17. Jan 19, 2007

### Staff: Mentor

Absolutely. I even fell into that trap a couple times myself way back in undergrad. Like in my first solid state physics class, I didn't really have a good intuitive feel for what was going on in the early parts of the class, and spent time "looking for equations" that might fit the problem. But I managed to start figuring stuff out after not too long, and then I got a lot better at deriving what I needed from the fundamental equations, and then applying the derived equation to a particular problem.

18. Jan 19, 2007

### 3trQN

What about thinking geometrically, i often find myself understanding a concept and abstract concepts relations to each other geometrically (as pictures and geometrical objects in my mind, intersections, tangents etc) yet not being able to fully express those algebraically or in a symbolic form very easily (or rapidly).

I can say to myself "I understand this, and i know it is related to this other concept because of the way these pieces fit together", yet i cant show that very easily.

This comes from early development of conceptualising things without mathematical equations (through a late development of mathematical skill), symbolic, and developing a picture/model approach (which basically achieves largely the same thing in terms of understanding and abstract relation).

The problem is then one of communication, learning the language to express these concepts unambiguously, not one of understanding. If you see my dilemma.

19. Jan 19, 2007

### CPL.Luke

if your talking about an intro college physics class, then everybody should have the skill to derive the kinematic equations from first principles, heck even more than that those concepts should be so intuitive to a student that they don't even need to derive it. It should come naturally.

20. Jan 19, 2007

### Kurdt

Staff Emeritus
Yeah I see what you're getting at. I think this thread has firmly established the value of both conceptual and mathematical understanding but has raised the uglier question of when and how to introduce both successfully to physics lessons. Its a tough question because the tutors and lecturers I've encountered have always preferred either one way or the other.

21. Jan 19, 2007

### Kurdt

Staff Emeritus
I was talking about high school and pre calculus for the kinematic equations.

22. Jan 19, 2007

### complexPHILOSOPHY

What the hell is pre-calculus exactly, I never took the class. I taught myself trigonometry after high school (because my highest math was algebra) and then my college let me take Calculus 1.

What do you learn in Pre calc? Is there anything I need to go back over and review? I hate missing math courses.

23. Jan 19, 2007

### dontdisturbmycircles

In pre-calc the teacher basically just wants to make sure that you are ready for calculus, and if you are not, attempt to get you ready. It is mainly trigonometry. But also introduces the concept of an open and closed interval and set notation. Composition of functions and stuff is also covered. They also make sure you know all of your algebra rules like adding fractions since it is very important for the study of derivatives, but thats about it. Oh, and exponent laws and log laws.

Edit: Here are the contents from my old precalculus book

1. Numbers and their disguises (introduction to the set of real numbers, what is a rational number, what makes a number irrational, etc.)
2.Completing the Square
3.Solving equations
4.Functions and their graphs (graphical addition, subtraction, inverses, compositions, all that fun stuff)
5.Cyclic phenomenon : The Six basic trig ratios
6.Exponential functions
7.Logarithmic Functions
8.Inverse trigonometric functions
9.Changing the form of a function (factoring, canceling, long division, synthetic division, rationalizing)
10.Simplifying Algebraic Expressions (Difference quotients, cancelling common factors, more rationalizing of expressions)
11.Decomposition of functions (sort of an intro to the Chain Rule)
12.Trig identities

Hope that helps you.

Last edited: Jan 19, 2007
24. Jan 19, 2007

### complexPHILOSOPHY

I haven't run into any problems studying Calculus so far, but do you think it would benefit me to take pre-calc anyways? I enjoy doing maths, I still go back and work through algebra problems to stay fresh (is this a waste of time? I have a fear I will forget my algebra and then all of a sudden need to relearn it in some higher math class). I just feel behind because I hated math until a year ago (which was a year after I graduated high school) so I don't have a formal math education prior to calculus.

Sorry for all of the questions, I just despise not having a precise, informed perspective about something.

25. Jan 19, 2007

### dontdisturbmycircles

I am basically in the exact same boat. I sluffed off until just last year. It took me quite a while to build up my base, about 1-2 months of studying algebra very hard. But I really truly did start from scratch. I didn't even know how to add fractions :yuck:. But once that was over Ihad a fairly easy time of basic diff and integral calculus. Pick up http://www.amazon.ca/Just-Algebra-T...ef=sr_1_1/701-3176379-4913139?ie=UTF8&s=books if you have any worries about your precalculus skills. And no it is not a waste of time to practice algebra skills. A big mistake some people make is to skip over the algebra and head on over to the harder stuff. I really do believe that when my old teachers used to say that getting good at math is like building a house, you need a good foundation, they were right. My personnal oppinion is that you probably don't need to take a formal pre-calculus course. Just pick up that book, it is actually designed to be a supplement for your calculus course and help you along throughout it.

Last edited by a moderator: Apr 22, 2017


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# Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$$y = e^{\sqrt{x}}$

## $=\frac{e^{\sqrt{x}}}{2 \sqrt{x}}$

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here we have a composite function and we're going to identify the inside function and the outside function before we differentiate. So the inside function would be the square root of X that is inside the E to the X function. And I would like to write the square root of X as extra the 1/2 when I differentiate it. So then the outside function is f of X equals e to the X okay, to find the derivative Do I d. X? What we want to dio according to the chain rule is start by taking the derivative of the outside function and we have learned that the derivative of each of the X is e to the X, So the derivative of each of the square root X we're going to have to start with the to the square root X. Then we multiply by the derivative of the inside function. So now we're taking the derivative of X to the 1/2 power and that would be 1/2 times X to the negative 1/2 power just using the power rule. Okay, so we have our derivative and now we're going to simplify So one of the things we can do is eliminate the negative exponents. So we have e to the square root X times 1/2 x to the 1/2 because X to the negative 1/2 is equivalent to one over X to the 1/2 and now we can just write it as a single fraction. So we have e to the square root X over two X to the 1/2. But why don't we take it one step further and change back the X to the 1/2 into radical notation and we have each of the square root x over two square root X that's are derivative.

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# How do you FOIL (x+3)(x+3)(x+3)?

Jun 16, 2015

FOIL will only get you part of the way, since once you multiply one pair of binomials you will have the product of a binomial and a trinomial, but

$\left(x + 3\right) \left(x + 3\right) \left(x + 3\right) = {x}^{3} + 9 {x}^{2} + 27 x + 27$

#### Explanation:

Using FOIL, first multiply $\left(x + 3\right) \left(x + 3\right)$

First: $x \cdot x = {x}^{2}$
Outside: $x \cdot 3 = 3 x$
Inside: $3 \cdot x = 3 x$
Last: $3 \cdot 3 = 9$

$F + O + I + L = {x}^{2} + 3 x + 3 x + 9 = {x}^{2} + 6 x + 9$

To multiply $\left({x}^{2} + 6 x + 9\right) \left(x + 3\right)$ use distributivity:

$\left({x}^{2} + 6 x + 9\right) \left(x + 3\right)$

$= \left({x}^{2} + 6 x + 9\right) \cdot x + \left({x}^{2} + 6 x + 9\right) \cdot 3$

$= \left({x}^{2} \cdot x\right) + \left(6 x \cdot x\right) + \left(9 \cdot x\right) + \left({x}^{2} \cdot 3\right) + \left(6 x \cdot 3\right) + \left(9 \cdot 3\right)$

$= {x}^{3} + 6 {x}^{2} + 9 x + 3 {x}^{2} + 18 x + 27$

$= {x}^{3} + \left(6 {x}^{2} + 3 {x}^{2}\right) + \left(9 x + 18 x\right) + 27$

$= {x}^{3} + \left(6 + 3\right) {x}^{2} + \left(9 + 18\right) x + 27$

$= {x}^{3} + 9 {x}^{2} + 27 x + 27$

Alternatively, pick the 4th row of Pascal's triangle to get:

$1 , 3 , 3 , 1$

List the first four ascending powers of three:

$1 , 3 , 9 , 27$

Multiply these two sequences together to get:

$1 , 9 , 27 , 27$

These are the coefficients of the descending powers of $x$:

${x}^{3} + 9 {x}^{2} + 27 x + 27$


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# Local Search: Hill Climbing Search

## A Short Look at Hill Climbing Search and Variations

December 11, 2013

## Introduction

Local Search is a method that is used to solve optimization problems, where we are after a solution's state, and the path to the goal is not the solution it self. This contrast from shortest-path problems we can solve with A* and Dijkstra's Algorithm, in that the solution it is the path from a start to a goal node. We only care about a final configuration.

Local search is a way to solve problems like vehicle routing, portfolio management, scheduling, and positioning problems without having to explore all possible solutions which increases in complexity with the number of variables.

In this post I will be quickly covering what Hill Climbing, Simulated Annealing, Local Beam, and Genetic Search are and what their algorithms look like.

The important thing to note about Local Search is that we must be aware that the search space has a sort of landscape that has slope with global and local maxima and minima at which our algorithms can get stuck, and since these algorithms do not explore all areas of a search space it is also possible to arrive at a solution which is good enough to solve a problem, such as in the case of scheduling.

The majority of the algorithms are derivations of those presented in AIMA by Peter Norvig.

## Hill Climbing

Hill Climbing is an algorithm that move in a direction of increasing value, and terminates once it reaches a peak. At every step of the way it uses a heuristic evaluation function, which we can call $$h(n)$$ to evaluate its state and the state of all its neighboring solutions.

After using $$h(n)$$ to evaluate the neighboring solutions it makes a move to the next best state, in other words, we can consider Hill Climbing to be greedy. Since this is a greedy algorithm we know it is likely that we will reach a local maxima sooner than later, but weather or not this is a solution good enough is dependent on the problem definition.

The problem with hill climbing is that it is possible to get stuck at a local maxima which does not solve our problem. Similarly, it is also possible to reach a plateau where we do not have a next best move and we can wonder infinitely back and forth if we allow sideways moves.

### Best First Hill Climbing Algorithm

// Pseudo Code

function h(State s) {
// Heuristic Evaluation Function
}

// Pseudo Code
function HillClimbing(State s) {
State best = s;
State current;
while (true) {
current = best;
for (State next : s.nextStates()) // foreach...in..
// Move to the next best possible state
best = h(best) < h(next) ? next : best;

// If current & best are STILL the same, then we reached a peak.
if (best == current)
return best;
}
}

### Hill Climbing Variations

To get around some of the limitations on Hill Climbing like reaching a plateau or a local maxima we can go about allowing a limited number of sideways moves before failing, or we can go about making choices in a stochastic manner or simply doing a randomized restart when the Hill Climbing gets stuck.

### Stochastic Hill Climbing

We can try choosing a randomized uphill move, rather than a greedy one. This would still move us towards a solution, but would lead us to explore different areas of the search space, and depending on the problem it could lead to better solutions.

// Pseudo Code

function h(State s) {
// Heuristic Evaluation Function
}

function List::ChooseRandom() {
// return move with probability proportional to the improvement.
}

function HillClimbing(State s) {
State best = s;
State current;
List betterMoves = List();

while (true) {
current = best;

// Look for better moves
for (State next : s.nextStates()) // foreach...in..
if (h(best) < h(next))

// Choose randomly, from better moves -- if any
if (betterMoves.length() > 0)
best = betterMoves.chooseRandom();

// If current & best are STILL the same, then we reached a peak.
if (best == current)
return best;
}
}

### Fist Choice Hill Climbing

First Choice Hill Climbing is all about generating successors randomly until a better one than our current state is generated. The reason you would want to do this is that in certain problems the number of successors states can be too big to compare directly to the current state, so generating randomly and moving to a better state may yield a better outcome in less time.

// Pseudo Code

function h(State s) {
// Heuristic Evaluation Function
}

function generateRandomState() {
// return a new randomly generated state.
}

function HillClimbing(State s) {
State best = s;
State current;

while (true) {
current = best;

// Look for better moves.
for(i = 0; i < THRESH_HOLD; i++) // foreach...in..
{
generated = generateRandomState();
if (h(best) < h(generated)) {
best = generated
break;
}
}

// If current & best are STILL the same, then we reached a peak.
if (best == current)
return best;
}
}

### Random Restart Hill Climbing

Random Restart Hill Climbing is all about doing a series of Hill Climbing searches from randomly generated initial states, until a goal is found. So every time the algorithm gets stuck at a local maxima, or plateau the algorithm simply does a random restart and tries again and again.

// Pseudo Code
function main() {
State current = new State(...); // Initial state
while (true) {
solution = HillClimbing(current)
if (isSolution(solution))
break;
// Solution good enough has not been found do a randomized restart.
current = generateRandomState()
}
}

function generateRandomState() {
// Generate a Random State to start from.
}

function h(State s) {
// Heuristic Evaluation Function
}

// Best-First Hill Climbing, but may be any type of hill climbing.
function HillClimbing(State s) {
State best = s;
State current;
while (true) {
current = best;
for (State next : s.nextStates()) // foreach...in..
// Move to the next best possible state
best = h(best) < h(next) ? next : best;

// If current & best are STILL the same, then we reached a peak.
if (best == current)
return best;
}
}

## Conclusion

Hill Climbing appears to be a powerful method for local search problems where we are looking to get to a solution configuration. Some version of the algorithm like the Greedy Best First will be incomplete, and fail to find a solution where there is one if it gets stuck in a local maxima, or plateau. We can also see that that can be resolved by adding in a randomized restart which will allow the hill climbing to start in another location of the search space. The goal was to more or less introduce the general idea of hill climbing and present some pseudo code to possible implementations.


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# Alternating multisymmetric functions

I am looking for a reference on certain modules of invariants. I think that the question is quite natural so that I believe there should be some results already, but I am not able to find anything.

The situation is the following: let $\mathfrak{S}_n$ be the symmetric group that acts on $\mathbb{C}[x_1,y_1,x_2,y_2,\dots,x_n,y_n]$ by permuting simultaneously the $x_i,y_i$. Then the ring of invariants $S=\mathbb{C}[x_1,y_1,x_2,y_2,\dots,x_n,y_n]^{\mathfrak{S}_n}$ is sometimes called the ring of multisymmetric functions.

The generators of $S$ have been computed by Vaccarino in https://arxiv.org/pdf/math/0205233.pdf, but I am interested in the module of alternating covariants $Alt = \mathbb{C}[x_1,y_1,x_2,y_2,\dots,x_n,y_n]^{alt_n}$, where $alt_n$ denotes the alternating representation of $\mathfrak{S}_n$. More precisely,

$$Alt = \{ f \in \mathbb{C}[x_1,y_1,\dots,x_n,y_n] \,|\, \sigma\cdot f = \operatorname{sgn}(\sigma)f \quad \text{ for all } \sigma\in \mathfrak{S}_n \}$$ where $\operatorname{sgn}$ denotes the sign of a permutation.

So my question is

• Is there any reference about the generators of $Alt$ as an $S$-module? I would already be interested in the cases $n\leq 4$.

Any help would be greatly appreciated.

• In the highly cited paper "Conjectures on the Quotient Ring by Diagonal Invariants", ams.org/mathscinet-getitem?mr=1256101, your $S$ is called diagonal invariants. I have seen it in many places unter this name. – Christian Stump May 16 '17 at 16:51
• I think that she wants the polynomials $f(x,y)$ satisfying $w\cdot f(x,y) = \mathrm{sgn}(w)f(x,y)$ for all $w\in\mathfrak{S}_n$. If it were just $f(x)$ then the module is a free module of rank one over the symmetric functions, generated by $\prod_{i<j}(x_i-x_j)$. – Richard Stanley May 16 '17 at 17:57
• Yes, thank you, that's precisely what I am looking for. I am sorry if I was not clear before, I added more details now. – Daniele A May 17 '17 at 9:09
• @ChristianStump: thank you for the pointers to the diagonal invariants and the paper of Haiman. I'll have a look at it and see whether it says something on the alternating case. – Daniele A May 17 '17 at 9:12

If I am not mistaken, the space $Alt$ has a basis given by $\{ \Delta_M(\mathbf{x},\mathbf{y}) \mid M \subset \mathbb{N}\times\mathbb{N} \text{ finite}\}$. Here, $\Delta_M(\mathbf{x},\mathbf{y}) = \det[x_i^{p_j}y_i^{q_j}]_{1\leq i,j\leq n}$ for $M = \{ (p_1,q_1),\ldots,(p_n,q_n) \}$.


text:
An analytic cancellation theorem and exotic algebraic structures on ${C}^{n},n\ge 3$
Colloque d'analyse complexe et géométrie - Marseille, janvier 1992, Astérisque no. 217  (1993), p. 251-282
@incollection{AST_1993__217__251_0,
author = {Zaidenberg, M. G.},
title = {An analytic cancellation theorem and exotic algebraic structures on $C^n , n \geq 3$},
booktitle = {Colloque d'analyse complexe et g\'eom\'etrie - Marseille, janvier 1992},
author = {Collectif},
series = {Ast\'erisque},
publisher = {Soci\'et\'e math\'ematique de France},
number = {217},
year = {1993},
pages = {251-282},
language = {en},
url = {http://www.numdam.org/item/AST_1993__217__251_0}
}

Zaidenberg, M. G. An analytic cancellation theorem and exotic algebraic structures on $C^n , n \geq 3$, in Colloque d'analyse complexe et géométrie - Marseille, janvier 1992, Astérisque, no. 217 (1993), pp. 251-282. http://www.numdam.org/item/AST_1993__217__251_0/

[Ab Ha Ea] Sh. S. Abhyankar, W. Heinzer, P. Eakin: On the uniqueness of the coefficient ring in a polynomial ring. J. Algebra 23 (1972), 310-342

[Da] W. Danielewski: On a cancellation problem and automorphism group of affine algebraic varieties. Preprint.

[Fu 1] T Fujita: On the topology of non-complete algebraic surfaces. J. Fac. Sci. Univ. Tokyo (Ser. 1A) 29 (1982), 503-566

[Fu 2] T. Fujita: On Zariski problem. Proc Japan Acad. (Ser. A) 55 (1979), 106-110

[Gi] M. H. Gizatullin: Invariants of incomplete algebraic surfaces that can be obtained by means of completions (Russian). Izvestiya Akad. Nauk SSSR, Ser. Mat. 35 (1971), 485-497

[Gu 1] R. V. Gurjar: Affine varieties dominated by ${C}^{2}$. Comment. Math. Helvetici 55 (1980), 378-389.

[Gu 2] R. V. Gurjar: Topology of affine varieties dominated by an affine space. Invent. Math. 59 (1980), 221-225

[Gu Mi] R. V. Gurjar, M. Miyanishi: Affine surfaces with $\overline{k}\ge 1$. In: Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, 1987, pp. 99-124.

[Gu Sha] R. V. Gurjar, A. R. Shastri: On the rationality of complex homology $2$-cells. I. J. Math. Soc. Japan 41-1 (1989), 37-56

R. V. Gurjar, A. R. Shastri: On the rationality of complex homology $2$-cells. II. J. Math. Soc. Japan 41-2 (1989), 175-212

[Ii 1] Sh. Iitaka: On logarithmic Kodaira dimensions of algebraic varieties. In: Complex analysis and Algebraic Geometry. Tokyo: Iwanami 1977, pp. 175-189.

[Ii 2] Sh. Iitaka: Some applications of logarithmic Kodaira dimensions. In: Algebraic Geometry (Proc. Intern. Symp. Kyoto 1978), Tokyo: Kinokuniya 1978, pp. 185-206

[Ii Fu] Sh. Iitaka, T Fujita: Cancellation theorem for algebraic varieties. J. Fac. Sci. Univ. Tokyo (Sec. 1A) 24 (1977), 123-127

[Ka] Sh. I. Kaliman: Polynomials on ${C}^{2}$ with isomorphic generic fibers. Soviet Math. Doklady 33-1 (1986), 600-603.

[Km] T. Kambayashi: On Fujita's strong cancellation theorem for the affine plane. J. Fac. Sci. Univ. Tokyo (Sec.1A) 27 (1980), 535-548

[Kaw] Y. Kawamata: On the classification of non-complete algebraic surfaces. In: Algebraic Geometry (Proc. Copenhagen Summer Meeting). Lecture Notes in Math. 732, Berlin e. a.: Springer-Verlag 1971, pp. 215-232.

[Mil] Lectures on the h-cobordism theorem. Princeton: Princeton Univ. Press 1965.

[Mi Tsu] M. Miyanishi, S. Tsunoda: Open algebraic surfaces with Kodaira dimension $-\infty$. In: Proc. Symposia in Pure Math. 46-1, 1987, pp. 435-450

[Mo] J. A. Morrow: Compactifications of C2. Bull. AMS 78 (1972), 813-816.

[Ni Suz] T. Nishino, M. Suzuki: Sur les singularites essentielles et isolees des applications holomorphes a valeuers dans une surface complexe. Publ. RIMS Kyoto Univ. 16 (1980), 461-497.

[Ra] C. P. Ramanujam: A topological characterization of the affine plane as an algebraic variety. Ann. Math. 94 (1971), 69-88.

[Sa] F. Sakai: Kodaira dimensions of complements of divisors. In: Complex analysis and Algebraic Geometry. Tokyo: Iwanami 1977, pp. 238-257.

[Su] T. Sugie: On T. Petrie's problem concerning homology planes. J. Math. Kyoto Univ. 30-2 (1990), 317-342.

[tDi] T. Tom Dieck: Homology planes without cancellation property. Preprint (1990).

[tDi Pe] T. Tom Dieck, T. Petrie: Homology planes. An announcement and survey. In: Topological Methods in Algebraic Transformation Groups. Progress in Math. Vol. 80. Boston: Birkhauser 1989, pp. 27-48

[Ts] R. Tsushima: Rational maps to varieties of hyperbolic type. Proc Japan Acad. (Ser.A) 22 (1979), 95-100

[Ur] T. Urata: Holomorphic automorphisms and cancellation theorems. Nagoya Math. J. 81 (1981), 91-103.

[VdV] A. Van De Ven: Analytic compactifications of complex homology cells. Math. Ann. 147 (1962), 189-204.

[Za 1] M. G. Zaidenberg: On Ramanujam surfaces, ${C}^{**}$-families and exotic algebraic structures on ${ℂ}^{n},n\ge 3$. Trudy Moscow Math. Soc. (to appear).

[Za 2] M. G. Zaidenberg: Ramanujam's surfaces and exotic algebraic structures on ${ℂ}^{n},n\ge 3$. Soviet Math. Doklady; transl. from: Dokl. Akad. Nauk SSSR 314-6 (1990), 1303-1307.

[Za 3] M. G. Zaidenberg: Isotrivial families of curves on affine surfaces and characterization of the affine plane. Math. USSR Izvestiya 30-3 (1988), 503-532. Additions and corrections to the paper...

M. G. Zaidenberg: Isotrivial families of curves on affine surfaces and characterization of the affine plane transl. from: Izvestiya AN SSSR 25-1 (1991), 221-223.

[Za 4] M. G. Zaidenberg: Rational actions of the group ${C}^{*}$ on ${C}^{2}$, their quasi-invariants, and algebraic curves in ${C}^{2}$ with Euler characteristic $1$. Soviet Math. Dokl. 31-1 (1985), 57-60.


text:
# Thought provoking problems on heat and thermodynamics

These problems are from pre-test conducted in teacher's workshops by Prof HC Verma and his team.

## Test 1

Source: RESPECT KV MRP Workshop 15th-24th May 2017
1. Which of these is a function of the state of a gas?
1. Heat $Q$
2. Work $W$
3. Internal energy $U$
4. $U+PV$
2. The figure shows adiabatic processes for samples of equal moles of three gases. Write the name of one possible gas for each of the three processes. Process-1 gas:__________. Process-2 gas:_________. Process-3 gas:_______
3. Consider isothermal expansion of an ideal gas.
1. If the gas does a work W, how much heat Q is given to the gas?
2. Is the heat Q completely converted into work in this process?
3. Is this a violation of the 2nd law of thermodynamics?
4. Consider the following gas systems all at room temperature. In which case $|pV-nRT|$ will be smallest and in which case it is largest?
1. gas in an LPG cylinder
2. hydrogen gas filled in a balloon which tries to go up in atmosphere
3. air in your water bottle
4. air in the tubes of a car
5. A suit of a hotel has two identical rooms connected by a door. The living room is air conditioned and is set at 24 deg C while the outer room has no air conditioning. Suppose the door remains open for quite some time. Which of the rooms will have more air if it is
1. a hot summer day (outside temperature 45 deg C),
2. a chill winter night (outside temperature 5 deg C)?
6. Consider an air molecule in a closed plastic box. What is the order of distance moved by this molecule in an hour?
1. 1 km
2. 10 km
3. 100 km
4. 1000 km
5. 10000 km or more
7. Two processes $AB$ and $A^\prime B^\prime$ on a sample of ideal gas are shown on volume vs temperature diagram. In which of the two processes, the work done by the gas is larger? Give a one-sentence justification.
8. A gas is suddenly compressed. During the process what kind of relation between the pressure and the volume should be used?
9. Most of the solids expand on increasing temperature. This is because
1. Some of the atoms leave the lattice site at higher temperature
2. The vibration amplitudes of the atoms increase at higher temperature
3. The inter-atomic potential energy is not symmetric about the minimum energy
4. The electrons go to larger orbits.


text:
The tetrameric prokaryotic potassium channel KcsA is activated by protons acting on the intracellular aspect of the protein and inactivated through conformational changes in the selectivity filter. Inactivation is modulated by a network of interactions within each protomer between the pore helix and residues at the external entrance of the channel. Inactivation is suppressed by the E71A mutation, which perturbs the stability of this network. Here, cell-free protein synthesis followed by protein purification by sodium dodecyl sulfate–polyacrylamide gel electrophoresis was used to produce heterotetramers of KcsA that contain different combinations of wild-type and E71A subunits. Single-channel recordings from these heterotetramers reveal how the network of interactions in individual protomers affects ionic conduction and channel inactivation, suggesting that the latter is a cooperative process.

## INTRODUCTION

The potassium channel KcsA from Streptomyces lividans has been used as a model for understanding the structure–function relationships of ion channels (Doyle et al., 1998; MacKinnon, 2004; Gouaux and Mackinnon, 2005). It is a small tetrameric protein with a pore region that resembles that of larger potassium channels (Doyle et al., 1998; MacKinnon, 2004; Gouaux and Mackinnon, 2005). The channel is activated by protons acting on the intracellular aspect of the protein (Cuello et al., 1998; Heginbotham et al., 1999; Thompson et al., 2008) and inactivated through conformational changes in the selectivity filter (Blunck et al., 2006; Cordero-Morales et al., 2006a,b, 2007). Macroscopic current measurements of KcsA channels have shown that after exposure to low pH, the channels are rapidly activated and subsequently enter an inactivated state from which recovery is slow (Gao et al., 2005; Cordero-Morales et al., 2006b; Chakrapani et al., 2007a). As a result, under steady-state conditions, the channel has a low open probability (Cordero-Morales et al., 2006b; Chakrapani et al., 2007b). In single-channel recordings, this activity is characterized by short bursts of activity that are separated by long periods of inactivity (Cordero-Morales et al., 2006b; Chakrapani et al., 2007b). In addition, the steady-state gating of KcsA at low pH is affected by the applied potential (Cuello et al., 1998; Heginbotham et al., 1999; Cordero-Morales et al., 2006a), which influences the rate of entry and exit into the inactivated state as reflected in macroscopic currents (Chakrapani et al., 2007a).

The inactivation of KcsA is modulated by intraprotomeric interactions between Glu71 and Trp67 in the pore helix and Asp80 in the external entrance of the channel (Fig. 1). Therefore, the inactivation of KcsA channels can be described as an allosteric process, where the conformation of the regulatory site (the network of interactions between the pore helix and the external entrance) affects ion permeability through the selectivity filter. The mutation E71A, which disrupts this network, greatly reduces the rate of inactivation as observed in macroscopic currents (Cordero-Morales et al., 2006a,b, 2007 ,Chakrapani et al., 2007a,b) and produces a remarkably high steady-state open probability in single-channel recordings at low pH (Cordero-Morales et al., 2006a,b, 2007; Chakrapani et al., 2007b). Further, it has been proposed that crystal structures of wild-type (WT) KcsA in the presence of high K+ concentrations, where the selectivity filter contains two ions distributed over the four binding sites (Zhou et al., 2001), represent the permeable (active) configuration of the KcsA selectivity filter (Fig. 1). On the other hand, the structures of WT KcsA at low K+ concentrations (Zhou et al., 2001) and the crystal structure of the M96V mutant (Lockless et al., 2007), where the selectivity filter contains one ion distributed over the two remaining binding sites, are thought to represent the configuration of the selectivity filter during inactivation.

C-type inactivation of eukaryotic voltage-dependent K+ (Kv) channels, which occurs in response to prolonged depolarization, has also been associated with conformational changes in the selectivity filter (Yellen, 1998). C-type inactivation of Kv channels is a cooperative process that involves concerted conformational changes of all four subunits (Ogielska et al., 1995; Panyi et al., 1995). The KcsA inactivation process shares features with C-type inactivation: (a) mutation of Glu71 and Tyr82 in KcsA (Cordero-Morales et al., 2006a,b), and the corresponding residues in Shaker Kv channels, Val438 (Yifrach and MacKinnon, 2002) and Thr449 (López-Barneo et al., 1993; Ogielska et al., 1995), respectively, all affect the inactivation of the channels; (b) the rate of inactivation of KcsA and Shaker Kv channels has a similar dependence on the extracellular potassium ion concentration (López-Barneo et al., 1993; Cordero-Morales et al., 2006b; Chakrapani et al., 2007a). These results, together with the sequence and structural similarities of KcsA and eukaryotic K+ channels in the region of the selectivity filter (Long et al., 2005), suggest that the inactivation processes of the channels are mechanistically similar.

Single-channel recordings are widely used to clarify the mechanisms of action of ion channels. The majority of single-channel activity measurements on ion channels that have been performed in planar lipid bilayers, including those on KcsA channels, have been of proteins overexpressed and purified from bacteria. The ion channels are typically reconstituted into lipid vesicles, which are subsequently fused with bilayers (e.g., Schrempf et al., 1995; Heginbotham et al., 1999). Although this approach has produced valuable data and expanded our understanding of the functional properties of ion channels, time-consuming overexpression and purification steps are involved. It has previously been shown that some β barrel pores, such as staphylococcal α-hemolysin, and the small viral K+ channel Kcv can be expressed in a cell-free system and purified by SDS-PAGE before direct insertion into planar lipid bilayers (Braha et al., 1997; Shim et al., 2007; Mason, 2008). Cell-free protein synthesis has advantages over conventional protein expression techniques, including its rapidity and the capability to synthesize proteins with unnatural or labeled amino acids (Spirin, 2002; Wang et al., 2006). Cell-free synthesis has been used to produce many membrane proteins, including KcsA channels (van Dalen et al., 2000). In addition, it has been shown that KcsA, like the α-hemolysin pore and the Kcv channel, maintains its oligomeric structure during SDS-PAGE (Heginbotham et al., 1997). These results suggested that KcsA tetramers might also be purified directly from polyacrylamide gels in a functional form, which is the approach taken here.

Cell-free protein synthesis followed by direct protein purification by SDS-PAGE was previously exploited to form heteromers of WT and chemically modified mutant subunits of α-hemolysin (Braha et al., 1997). More recently, when this technique was used, the mutant α-hemolysin subunits contained oligoaspartate extensions at their C termini (Howorka et al., 2001). Accordingly, it was possible to separate heteromers with different combinations of WT and mutated subunits based on their differing electrophoretic mobilities. Electrophoresis of KcsA heteromers formed by the coexpression of subunits with different peptide extensions at their C termini was used previously to demonstrate that KcsA forms tetramers (Heginbotham et al., 1997). However, these channels were not purified from the gels and used for functional studies. In the present work, heteromers that contained WT and E71A KcsA protomers with different peptide extensions at their N termini were produced and separated by SDS-PAGE. The extensions were then cleaved by proteolysis, and single-channel recordings were made with heteromers in which the protomers differed from each other only at the desired site of mutation, position 71.

By this approach, we reveal how regulatory sites in the KcsA channel affect inactivation. These regulatory sites are the networks of interaction within individual protomers between the pore helix and the external entrance of the channel. Based on our results, we propose an inactivation mechanism that features cooperativity between the four subunits. In addition, this study shows that the E71A mutation causes the formation of a ring of charges at the extracellular entrance of the channel that increases inward ionic conduction.

## MATERIALS AND METHODS

Unless otherwise stated, chemicals and oligonucleotides were purchased from Sigma-Aldrich. Enzymes and prestained gel markers were purchased from New England Biolabs, Inc. and lipids from Avanti Polar Lipids, Inc.

### Genetic constructs

Constructs were assembled in the pT7-SC1 expression vector (Cheley et al., 1997) and verified by sequencing of the entire genes. The KcsA gene encoding the point mutant E71A was generated by PCR mutagenesis and in vivo recombination as described previously (Jones, 1995; Howorka and Bayley, 1998). The same technique was used to generate the genes for KcsA with N-terminal polypeptide extensions by using overlapping primers to form the coding sequences for the extensions. Two extensions were generated: “long,” with the amino acid sequence MHHHHHHNNNNNNNNRLEQKLISEEDLL…, and “short,” with the sequence MHHHHHHL.... In both cases, the sequences were followed by a tobacco etch virus (TEV) protease recognition site (ENLYFQG) and eight asparagine residues.

### Cell-free coupled transcription and translation

Polypeptides were generated by coupled in vitro transcription and translation (IVTT) by using an Escherichia coli T7-S30 extract system for circular DNA (Promega). The complete 1-mM amino acid mixture minus cysteine and the complete 1-mM amino acid mixture minus methionine were mixed in equal volumes to obtain the working amino acid solution required to generate high concentrations of the proteins. The amino acids (10 µl) were mixed with premix solution (40 µl), 4 µl [35S]methionine (1,175 Ci mmol−1, 10 mCi ml−1; MP Biomedicals) and 30 µl T7 S30 extract supplemented with 20 µg ml−1 (final concentration) rifampicin (Cheley et al., 1997). To form heteromers, different ratios of the plasmid DNA encoding WT KcsA with the long polypeptide extension and E71A-KcsA with the short polypeptide extension were mixed (total volume 16 µl, from 400–ng µl−1 stock solutions of each DNA). Synthesis was performed for 60 min at 37°C to generate 100 µl IVTT polypeptide solution. Laemmli sample buffer (2×, 100 µl; Bio-Rad Laboratories) was then added and the sample was loaded, without heating, into two 5-mm wells of a 15-cm-long, 1.5-mm-thick 10% SDS polyacrylamide gel. Both the stacking and the separation gels were supplemented with 10 mM KCl. The gel was run overnight at 45 V, and then for a few more hours at 100 V to achieve the maximal separation between the different heteromers. The unfixed gel was vacuum dried without heating onto 3MM chromatography paper (GE Healthcare) and visualized by exposure to film (BioMax MR-1; Kodak). Each of the heteromer-containing bands was cut from the gel by using the autoradiogram as a guide. The excised pieces were rehydrated with buffer (200–300 µl) containing 10 mM HEPES, pH 7.0, and 10 mM KCl for 1 h. After removing the paper, each gel strip was thoroughly crushed in the solution, and the protein was separated from the gel by centrifugation through a 0.2-µm cellulose acetate filter (Rainin). The protein was stored frozen at −80°C.

### Proteolysis with TEV protease

The polypeptide extensions of the heteromers were removed by proteolysis with TEV protease. Before proteolysis, the storage buffer of the protein was exchanged for 50 mM Tris HCl, pH 8.0, 0.5 mM EDTA, 0.02% (wt/vol) n-dodecyl-β-d-maltoside, and 10 mM KCl by dilution and concentration by centrifugation, three times at 14,000 g for 18 min, with a filter (YM-10 Microcon; Millipore). 1 mM (final concentration) dithiothreitol and AcTEV protease (1 µl, 10 U µl−1; Invitrogen) were added to a protein sample (50 µl) (Parks et al., 1994; Suh-Lailam and Hevel, 2009). The sample was incubated overnight at 25°C and stored at −80°C. This material was used directly for electrical recordings.

### Single-channel electrical recording and analysis

Single-channel recordings were performed by using the folded planar lipid bilayer method (Montal and Mueller, 1972). The apparatus consisted of a 0.025-mm-thick Teflon septum glued between two Delrin chambers. The septum contained an aperture of 100-µm diameter, which was treated with 10% (vol/vol) hexadecane in pentane. Each chamber was filled with 1 ml of filter-sterilized buffered salt solution. Unless otherwise stated, the salt solutions were 200 mM KCl and 10 mM HEPES, pH 7.0, in the cis chamber and 200 mM KCl and 10 mM succinic acid, pH 4.0, in the trans chamber (solutions were titrated with KOH or HCl).

1-Palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE) and 1-palmitoyl-2-oleoyl-sn-glycero-3-[phospho-rac-(1-glycerol)] (POPG) (8 µl; 7.5 and 2.5 mg ml−1, respectively, in pentane) or 1,2-diphytanoyl-sn-glycero-3-phosphocholine (DPhPC; 8 µl; 10 mg ml−1 in pentane) were added to the surface of each chamber, and the pentane was allowed to evaporate to leave behind a lipid monolayer. A lipid bilayer was formed by lowering and raising the liquid levels in the chambers below and above the aperture.

The electrical current was detected with two Ag/AgCl electrodes, amplified with a patch clamp amplifier (Axopatch 200B; MDS Analytical Technologies), filtered with a low-pass Bessel filter (80 dB per decade) with a corner frequency of 5 kHz, and digitized with an A/D converter (DigiData 1320; MDS Analytical Technologies) at a sampling frequency of 50 kHz. In all experiments, the cis chamber was at ground. A positive current represents the movement of cations from trans to cis. The bilayer recording platform, chambers, amplifying headstage, and micromanipulator were enclosed in a Faraday cage to shield against ambient electrical noise. KcsA channels were directly inserted into the bilayer from the cis (ground) chamber either by adding the protein to the bath solution and stirring or by using an agarose probe, as described previously (Holden and Bayley, 2005).

Traces of 2-min duration (beginning in the open state) were further filtered with a digital Gaussian filter at 1 kHz in Clampfit 10 (MDS Analytical Technologies). Amplitude histograms were fitted to Gaussian functions, and the quoted unitary conductance values were obtained from the peak (mean) values of the fits. Open probabilities were determined with the Single-Channel Search module from Clampfit. When the activity of more than one channel was observed in a trace, the single-channel open probability was determined by dividing the total open probability (NPo) by the total number of active channels (N). NPo is calculated by Clampfit:

$NPo=∑1NntnT,$

where n is the number of channels in level n (1, 2, … N), tn is the total amount of time spent in level n during the entire trace, and T is the total recording time. A value for N was obtained from the highest current level observed during a recording.

## RESULTS

### In vitro expression of functional KcsA channels

KcsA was expressed and labeled with [35S]methionine in a cell-free IVTT system. Analysis by SDS-PAGE revealed two major bands with Mr values of ∼15 and ∼45 kD (Fig. 2 A). The molecular mass of KcsA, predicted from the polypeptide sequence, is 18 kD for the monomer and hence 72 kD for the tetramer. Tetrameric KcsA has been shown previously to migrate in SDS polyacrylamide gels with Mr < 72 kD (Heginbotham et al., 1997), and we surmised that the ∼45-kD band is the tetramer.

For single-channel recording, the KcsA tetramers formed by IVTT were isolated from an SDS polyacrylamide gel by elution into 10 mM HEPES, pH 7.0. The protein was added to the cis chamber (ground) of a bilayer apparatus, and the solution was stirred until single-channel activity was seen. Alternatively, the protein was introduced directly into the bilayer by using an agarose probe (Holden and Bayley, 2005). During this study, folded lipid bilayers were used (Montal and Mueller, 1972), composed initially of either DPhPC or a 3:1 ratio of POPE and POPG.

Previous studies have suggested a requirement for a negatively charged lipid for channel activity (Heginbotham et al., 1998; Valiyaveetil et al., 2002). Typically then, for single-channel recordings, KcsA is reconstituted into vesicles and fused with bilayers of 3:1 POPE/POPG (Heginbotham et al., 1999; LeMasurier et al., 2001; Marius et al., 2008) or soybean asolectin lipids (Cortes et al., 2001). However, we found that KcsA functions in a lipid bilayer composed solely of DPhPC with similar activity, in terms of channel gating and ion conductance, compared with that observed in POPE/POPG bilayers (Fig. 2, B and C). As the S30 cell extract contains a high level of E. coli lipids (van Dalen et al., 2002; Conlan, 2003), it is possible that the tetramer formed by IVTT contains tightly bound PG, which might be retained throughout the purification process and thereby allow the channel to function in a DPhPC bilayer. All subsequent experiments were performed with DPhPC bilayers.

To be active in reconstituted systems, the intracellular domain of KcsA should be at pH <5 (Cuello et al., 1998). Therefore, the chamber solutions were buffered with 10 mM HEPES, pH 7.0, in the cis chamber, and with 10 mM succinic acid, pH 4.0, in the trans chamber. This ensured that the active channels were of known orientation with respect to the bilayer (Heginbotham et al., 1999). The cis chamber therefore represented the extracellular side of a cell and the trans chamber the intracellular side. The channels displayed the well-documented activity of KcsA: burst-like behavior, with a higher open probability at positive potentials compared with negative potentials (Fig. 2 C) (Cordero-Morales et al., 2006b; Chakrapani et al., 2007b). The openings at negative potentials were significantly noisier than at positive potentials (not depicted) and the I-V curve displayed partial outward rectification (Fig. 3). WT KcsA had a conductance of 108 ± 8 pS at +100 mV and 50 ± 10 pS at −100 mV in 200 mM KCl. These values are in reasonable agreement with previously reported values of 56 pS at +100 mV and 31 pS at −100 mV recorded in 100 mM KCl (Heginbotham et al., 1999). This anticipated single-channel behavior demonstrates that it is possible to examine the functional properties of a K+ channel expressed in vitro and purified directly from a gel, thereby eliminating the time-consuming labor of traditional protocols.

### Formation of heteromeric channels

It had been shown that KcsA heteromers with different combinations of subunits, containing short and long peptide extensions, can be separated by SDS-PAGE (Heginbotham et al., 1997). Therefore, we attempted to form heteromeric KcsA channels by co-translation of two different constructs. We added DNAs encoding WT and E71A KcsA polypeptides, which contained, respectively, long and short peptide extensions at the N terminus (see Materials and methods), to an IVTT mix. When the products were examined by SDS-PAGE, the E71A tetramer was not observed, and it was supposed that it either did not form or dissociated into monomers in SDS. Cations known to permeate or strongly block K+ channels (such as K+, Rb+, and Cs+) confer stability to the WT tetramer (Krishnan et al., 2005). Indeed, the addition of 10 mM KCl to the SDS polyacrylamide gels increased the stability of the E71A tetramer sufficiently for purification for single-channel recording. Therefore, 10 mM KCl was routinely added to gels and to the storage solutions of the purified protein.

When the WT/E71A tetramers were analyzed by SDS-PAGE, a ladder of five evenly spaced bands was evident (Fig. 4). The different heteromeric combinations had been separated according to their molecular masses, conferred by the N-terminal extensions. However, by using this method it was not possible to distinguish between the two potential WT2E71A2 tetramers, where the mutant protomers are in either symmetric or asymmetric configurations (Fig. 4). The tetramers with different ratios of WT and E71A subunits were isolated from the gel, and the protein was eluted into 10 mM KCl and 10 mM HEPES, pH 7.0. Most of the purified protein retained its tetrameric structure, as shown by the rerunning of samples in SDS polyacrylamide gels (Fig. 5). The ratios of the two subunits in the dissociated tetramers were as anticipated based on the expected “parent” tetramer structures (Fig. 5).

For further work, the peptide extensions at the N termini of the KcsA subunits in the tetramers were removed by proteolysis with TEV protease to form subunits that differed from each other only at the mutated position 71 (Fig. 5). After the proteolysis step, all the subunits in a tetramer (WT or E71A) retain a short extension at the N terminus comprising one Gly residue followed by eight Asn residues (G8N). Interestingly, in the absence of the G8N linker between the N terminus of the protomers and the TEV recognition site, protease cleavage was very inefficient (not depicted). It has been determined by site-directed spin labeling (Cortes et al., 2001) and nuclear magnetic resonance spectroscopy (Baker et al., 2007) that the N terminus of the KcsA channel forms an α helix. It is possible that this well-ordered secondary structure prevents the binding of the protease or inhibits protease activity when it is adjacent to the recognition site. After the proteolysis step, the KcsA tetramers with different ratios of WT and E71A subunits all formed active channels in planar lipid bilayers (Fig. 6). Comparisons of the properties of WT and E71A KcsA channels with or without the G8N extension demonstrated that the presence of the extension did not affect the activity of channels in terms of their unitary conductance values (Fig. 3) and gating activities (compare Figs. 2 and 6).

### The effect of the E71A mutation on the KcsA inward conductance

The I-V curve of WT KcsA in 200 mM KCl reflects significant outward rectification (Fig. 3). By comparison, under the same conditions, E71A homotetramers show mild inward rectification. Although the mutation has little effect on the magnitude of the current in the outward direction, it has a major effect on the inward current, e.g., at −100 mV the conductance of WT channels is 50 ± 10 pS (n = 4), and the conductance of E71A channels is about three times higher, 148 ± 5 pS (n = 4). Similar results were previously published (Thompson et al., 2008). The E71V homotetramer in planar lipid bilayers showed a similar phenomenon, with an even larger inward conductance (Choi and Heginbotham, 2004). The E71A homotetramers exhibited significant noise in single-channel recordings (Fig. 2). The origin of the noise is unclear, but it is of a similar magnitude in currents recorded from WT tetramers.

Comparison of the crystal structures of WT (Zhou et al., 2001) and E71A (Cordero-Morales et al., 2006b) KcsA channels suggests a possible explanation for the differences in the conductance values. In the WT KcsA crystal structure, the side chain of Asp80 faces into the protein core (Zhou et al., 2001) (Fig. 1). In one of the two structures of E71A (called “flipped”), the Asp80 side chain moves 8 Å and is situated in the extracellular entrance of the channel, exposed to the bulk solvent (Fig. 1) (Cordero-Morales et al., 2006b). The authors suggest that eliminating the carboxyl–carboxylate interaction between Asp80 and Glu71 increases the flexibility of Asp80 side chain and allows the observed movement (Cordero-Morales et al., 2006b). Based on the conformational change observed in the flipped structure, the exposed carboxyl side chains of Asp80 creates an external ring of negative charge (Fig. 1). This ring would increase the inward conductance of the channel by decreasing the height of the energy barrier for ion entry through the external entrance into the selectivity filter.

The effect of charged rings on conduction has been studied in various ion channels, including BK potassium channels (Qian et al., 2002; Brelidze et al., 2003; Carvacho et al., 2008), acetylcholine receptors (Imoto et al., 1988), and mutated KcsA (Nimigean et al., 2003). For all of these channels, it was demonstrated that the presence of negatively charged amino acids in one of the entrances into the channel increases the rate of transport of cations from this entrance to the other side of the membrane.

To further explore the effect of a negatively charged ring at the extracellular entrance of E71A KcsA, we determined the effect of pH on the conductance of the channel. If the source of the increase of inward conductance in E71A KcsA channels is a ring of charge at the extracellular entrance formed by the exposed carboxyl side chains of Asp80, it is predicted that a reduction in the pH on the extracellular side of the channel would cause a decrease in the inward current as a result of protonation of the Asp side chains. When the pH in the cis chamber (the extracellular side of the channel) is decreased from pH 7.0 to 3.0, only the inward conductance of E71A KcsA is reduced, whereas for WT KcsA, both the inward and the outward conductances are unaffected (Fig. 7). Therefore, the results support the idea that the rise in the inward conductance of the mutated channel is caused, at least in part, by the exposure of the Asp80 carboxyl side chains at the extracellular entrance of the channel, as shown in the flipped E71A KcsA crystal structure. On the other hand, the WT KcsA inward conductance was not changed at low extracellular pH because the carboxyl side chain of Asp80 is buried inside the protein, as shown in the WT crystal structure (Fig. 1), and does not form a ring of charge at the extracellular entrance.

Measurements on the WT/E71A heteromers gave a further opportunity to determine the effect of Ala substitution at position 71 on the channel conductance (Fig. 8 A). The I-V curves of the WT/E71A heteromers show that as the number of E71A subunits in the tetramers is increased, the inward current increases (Fig. 8 B). These results suggest that each of the E71A subunits in the tetramer makes a cumulative contribution to the elevation of the inward current.

### Inactivation of KcsA

Proton-dependent activation of KcsA is followed by an inactivation process similar to C-type inactivation. As a result, at low pH, homotetrameric WT KcsA channels are characterized by short bursts of activity that are separated by long periods of inactivity (Cordero-Morales et al., 2006b; Chakrapani et al., 2007b). Under our conditions, WT4 KcsA channels behave as described in the earlier reports (Fig. 6). WT4 channels indeed have a low open probability, Po < 0.1, and as the bilayer is depolarized, the open probability of the channel increases (Fig. 9 A and Table I) (Cordero-Morales et al., 2006a; Chakrapani et al., 2007b). The inactivation gate is located in the selectivity filter of the channel, and inactivation is suppressed by the E71A mutation (Cordero-Morales et al., 2006b). In contrast with WT4, previous work has shown that the open probability of E71A4 at low pH is close to one, Po = 1 (Cordero-Morales et al., 2006b; Chakrapani et al., 2007b; Thompson et al., 2008). In the experiments presented here, E71A4 channels exhibited variable and lower open probabilities compared with those reported previously (e.g., Po = 0.58 ± 0.18 at +50 mV; n = 12) (Fig. 9 A and Table I). The basis of the differences between this study and previous work is not clear. Nevertheless, we do find that the open probability of E71A4 is much higher than that of WT4 (Fig. 9 A and Table I) and is practically voltage independent (Cordero-Morales et al., 2007).

Measurements of the single-channel activity of the WT/E71A heteromers gave a closer view of the mechanism behind the transition of the selectivity filter between active and inactive states. The open probability of WT3E71A1 channels, where one WT subunit is substituted with one E71A subunit, was higher than that of WT4 channels at all the potentials examined (Fig. 9 A and Table I). However, like that of WT4 channels, the open probability of WT3E71A1 channels was voltage dependent, and it increased as the membrane was depolarized. At high depolarizing potentials (above +50 mV), the open probability of WT3E71A1 channels approached the same level as that of the E71A4 channels (Fig. 9 A and Table I). Because of the high variability in the open probability values, it was impossible to determine the exact values of V1/2 and the gating charge (z) for WT4 and WT3E71A1. However, the plots of normalized open probability as a function of membrane potential of both channels can be fit to Boltzmann distributions with similar slopes (i.e., similar gating charges), in which the V1/2 of WT3E71A1 is shifted to a more negative potential (Fig. 9 B). These results suggest that the presence of even one E71A subunit destabilizes the inactive state of the selectivity filter in the tetrameric channel.

Heteromers with additional E71A subunits (i.e., WT2E71A2 and WT1E71A3 channels) had open probability values of the same order of magnitude as E71A4 channels (Fig. 9 A and Table I). Although it is difficult to make a precise comparison of the open probability values of WT2E71A2, WT1E71A3, and E71A4 tetramers because of their high variability, it is clear that in all of these channels, the open probability is no longer voltage dependent (Fig. 9 A and Table I). These results suggest that the presence of two or more E71A subunits in a KcsA tetramer provides maximal suppression of the transition of the selectivity filter into the inactivated state.

## DISCUSSION

The potassium channel KcsA has been used extensively as a model for the understanding of structure–function relationships in ion channels. Here, it has been demonstrated that functional KcsA can be expressed in vitro by cell-free coupled IVTT by using an E. coli extract. The tetrameric proteins formed in the IVTT mix are stable in SDS and therefore can be purified by SDS-PAGE. The extracted protein inserts spontaneously into planar lipid bilayers. By using single-channel recording, it was shown that the activity of KcsA extracted from gels is similar to the activity of KcsA purified by conventional means.

### Cell-free expression and purification of heteromeric KcsA channels

Cell-free protein expression and purification by SDS-PAGE eliminates the time-consuming steps of traditional membrane protein purification protocols. In addition, in vitro synthesis provides numerous useful options, including the labeling of amino acids and the incorporation of unnatural amino acids (Spirin, 2002; Wang et al., 2006). Here, we have exploited one such application, namely, the production of heteromers of an oligomeric protein. Natural KcsA is a homotetramer. Therefore, it is difficult to analyze the contributions to protein function of individual subunits and the interactions between them. To this end, heterooligomers formed from WT and mutated subunits can be used. Two major techniques have been used previously for the production of heteromeric ion channels. The first is in vivo coexpression of WT and mutated polypeptides. This technique, combined with data-fitting analyses of macroscopic currents, was useful in determining the subunit stoichiometry of channels such as Shaker Kv (MacKinnon, 1991). The second technique uses tandemly fused polypeptides connected by flexible linkers. The genes are built from WT and mutant genes fused tail to head. This technique has been used in analyses of subunit cooperativity in processes including the activation of Kv channels (Zandany et al., 2008). One of the limitations of the first approach is the stochastic (or otherwise) sorting of the WT and mutant subunits in the heteromers. The second method addresses this problem. However, the fact that the subunits are fused tail to head can affect their ability to function normally or to insert correctly into membranes (McCormack et al., 1992; Sack et al., 2008).

In the study presented here, an IVTT system was used to produce heteromers of WT and mutant KcsA subunits with peptide extensions at their N termini. These extensions allowed heteromers with different ratios of WT and mutant subunits to be separated on the basis of size by electrophoresis. In this way, our technique overcomes the limitations of the other approaches; heteromers are produced in which the subunits are not covalently connected and the stoichiometry of the subunits is known. In recent years, this method has been used extensively to form heteroheptamers of α-hemolysin pores (Bayley and Jayasinghe, 2004). In the study presented here, the peptide extensions were removed by proteolysis, so that the subunits in the tetramers differed only by the desired mutation at position 71. This was an important step because the extensions were up to about one quarter of the mass of each monomer and situated close to the channel entrance. They might therefore have influenced channel activity. Both KcsA and α-hemolysin form stable oligomers in SDS polyacrylamide gels and therefore can be purified by electrophoresis. “Native” gel electrophoresis might be suitable for the separation and purification of oligomeric proteins that are not stable to SDS-PAGE.

### The E71A mutation affects KcsA channel stability and ion conduction

E71A, the mutation used throughout this work, was shown to affect the stability of KcsA channels and their gating and conductance properties. The source of these effects is likely to be the elimination of the carboxyl–carboxylate interaction between Asp80 and Glu71 (Cordero-Morales et al., 2006b). As a result, the side chain of Asp80 becomes more flexible, which disrupts its interaction with Trp67. Consequently, Trp67 is able to switch from one rotameric state to another (Cordero-Morales et al., 2006b). In WT channels, the side chain of Trp67 is turned toward the adjacent protomer and lies in close proximity to several residues, including Arg89, which has been implicated in the stability of the tetramers (Irizarry et al., 2002). In E71A4, one of the rotamers of Trp67 is orientated away from the contact plane with the adjacent protomer (whereas the other is located as found in the WT channel). Therefore, the flexibility of Trp67 in E71A channels weakens the stability of the tetramer. Inorganic cations known to permeate or strongly block K+ channels confer stability to the KcsA channel (Krishnan et al., 2005), and indeed we found that the addition of 10 mM KCl to SDS polyacrylamide gels increased the stability of the E71A homotetramer.

Our results also show that the inward conductance of E71A KcsA channels is much higher than that of WT channels and is reduced at low pH. In the crystal structure of the E71A homotetramers (in the flipped conformation; Cordero-Morales et al., 2006b), Asp80 residues are exposed at the extracellular entrance. Therefore, our findings suggest that the exposed side chains create a ring of negative charge that decreases the height of the energy barrier for ion entry into the selectivity filter and therefore increases the inward current. At low pH, the Asp80 side chains become protonated, and their effect on the electrostatic potential at the entrance is reduced. By using the WT-E71A heteromers, it was shown that the addition of each E71A subunit to the tetrameric channel makes a cumulative contribution to the elevation of the inward conductance. In the flipped crystal structure of E71A tetramers, the distance between the carboxyl groups of the Asp80 side chains is ∼9 Å. Because the Debye length in 200 mM KCl is ∼7 Å, the independence of the contributions of each carboxyl group makes sense. Therefore, the stability of the intraprotomeric network of interactions between the pore helix and the external entrance of the channel is a major factor in the outward rectification of the WT KcsA channel.

### The inactivation of KcsA as an allosteric process

Based on electrophysiological measurements of mutated channels and molecular dynamics simulations, it has been proposed that an intraprotomeric network of interactions between Glu71 and Trp67 in the pore helix and Asp80 in the external entrance of KcsA (Fig. 1) promotes a distortion of the filter into a nonconductive configuration, leading to inactivation (Cordero-Morales et al., 2006b). In addition, because the open probability of the channel is increased at high positive applied potentials, it has been suggested that membrane depolarization relieves the inactivation of the channel by pulling Glu71 toward the intracellular side of the bilayer, thereby disrupting the network (Cordero-Morales et al., 2006a). The open probability of E71A4 is much higher than that of WT4, and it is voltage independent. Based on these results, it was proposed that the E71A mutation essentially abolishes inactivation because the presence of Ala in position 71 cannot support the network of interactions (Cordero-Morales et al., 2006b, 2007).

As mentioned earlier, our measurements of the unitary conductance of WT/E71A heteromers suggest that the identity of the residue at position 71 (Glu or Ala) in a protomer determines whether Asp80 in the same subunit is buried in the protein or exposed at the external entrance. Therefore, the network of interactions that is described above only exists in WT subunits and not in E71A subunits, whether they are present in homotetramers or heterotetramers. From our examination of inactivation, we surmise that the tetrameric selectivity filter can be in only one of two configurations: active (conductive) or inactive (nonconductive). For example, we did not observe intermediate current steps between the closed and the open levels (although we cannot reject the possibility that very short-lived states exist that were not acquired in our measurements). In other words, the transition of the four selectivity filter sequences (TVGYG) between their active and inactive conformations is concerted. Therefore, we can describe the inactivation of KcsA as an allosteric process, where the permeability of the selectivity filter (the active site) is affected by the state of the four independent networks of interactions between the pore helices and the external entrance (together: the regulatory sites). Each WT subunit in a tetramer contains a functional regulatory site that stabilizes the inactive conformation of the selectivity filter. In an E71A protomer, this regulatory site is defective.

Our results show that under strongly depolarizing potentials (above +50 mV), the open probabilities of WT/E71A heteromers with one or more E71A subunits are similar to the open probability of E71A homotetramers (Fig. 9 A and Table I). Therefore, we suggest that under these conditions, the transition of the selectivity filter sequences into the inactive state in all E71A-containing heteromers is defective. On the other hand, channels with four WT subunits have a much lower open probability in this voltage range, reflected by their greater tendency to inactivate. The network of interactions that stabilizes the inactivated state is relatively weak under a strong depolarizing potential, which pulls Glu71 toward the intracellular side of the bilayer, disrupting the network of interactions between Glu71, Trp67, and Asp80. We propose that, under these conditions, the concerted transition of the selectivity filter sequences of all four subunits into their inactive conformation will happen only when all of the subunits are stabilized in that conformation by the regulatory network (Fig. 10).

A decrease in the membrane potential does not affect the open probability of channels that contain two or more E71A subunits. In contrast, the open probabilities of E71A1WT3 and WT4 are reduced as the membrane potential becomes more negative (Fig. 9, A and B, and Table I). It has been suggested that at more negative potentials, the network of interactions in WT subunits that stabilizes inactivation becomes stronger (Cordero-Morales et al., 2006a). We propose that, under these conditions, the concerted transition of the selectivity filter sequences of all the subunits of a tetramer into the inactive conformation occurs even when the regulatory network is broken in one of the subunits (Fig. 10). However, because the open probability of channels with one E71A protomer is higher than WT4 across the entire measured voltage range (Fig. 9 A), we also propose that at hyperpolarized potentials (when the network of interactions is relatively strong), the destabilization of the regulatory network in one protomer still exerts a negative effect on the concerted transition of the selectivity filter sequences into the inactive conformation.

### Conclusion

These studies of WT/E71A KcsA heteromers have offered a glimpse into the mechanism of inactivation of the KcsA channel. It was previously shown that the C-type inactivation of Kv channels is a cooperative process that involves a concerted conformational change in all four protomers (Ogielska et al., 1995; Panyi et al., 1995). Superimposition of the KcsA crystal structure (PDB entry 1K4C) with the crystal structures of the Kv1.2 channel from the Shaker family (PDB entries 2A79 and 2R9R) and the KvAP channel (PDB entry 1ORQ) suggests that in these channels there also exist networks of interactions between the pore helix and the external entrance of the channel. In addition, mutation of the residue corresponding to Glu71 of KcsA in the Shaker Kv (Val438) (Yifrach and MacKinnon, 2002) and Kv2.1 (Ile369) channels (De Biasi et al., 1993) affects the C-type inactivation of these channels. Therefore, we can speculate that C-type inactivation of voltage-gated channels, like KcsA, is also regulated by an allosteric site. In contrast with the KcsA channel, it has been shown that C-type inactivation in Shaker Kv channels is voltage independent, at least in the range of −25 to +50 mV (Hoshi et al., 1994). As proposed earlier (Chakrapani et al., 2007a), the absence of a charged residue in “position 71” of these channels might be the basis for the difference in the voltage dependency. Further, it was shown that when the conserved Val in position 370 (equivalent to position 71 in KcsA) of the Shaker channel Kv1.2 was mutated to Glu, the inactivation of channel was accelerated (Cordero-Morales et al., 2007).

In conclusion, comparison of the single-channel activity of homomeric and heteromeric channels formed from WT and E71A KcsA subunits shows that the stability of the intraprotomeric network of interactions between the pore helix and the external entrance of the channel makes a major contribution to both the outward rectification of the channel and to its transition into an inactive state. The effect of the stability of the network on the ion conductance is graded: the more protomers with a stabilized network, the greater the reduction in the inward current. On the other hand, channel inactivation, which is a concerted process, occurs only when, depending on the membrane polarization, the network of interactions is stabilized in three or four protomers.

## Acknowledgments

This work was supported by a grant from the Medical Research Council. H. Bayley is the holder of a Royal Society Wolfson Research Merit Award.

Christopher Miller served as editor.

## References

References
Baker
K.A.
,
Tzitzilonis
C.
,
Kwiatkowski
W.
,
Choe
S.
,
Riek
R.
.
2007
.
Conformational dynamics of the KcsA potassium channel governs gating properties
.
Nat. Struct. Mol. Biol.
14
:
1089
1095
. doi:
Bayley
H.
,
Jayasinghe
L.
.
2004
.
Functional engineered channels and pores (Review)
.
Mol. Membr. Biol.
21
:
209
220
. doi:
Blunck
R.
,
Cordero-Morales
J.F.
,
Cuello
L.G.
,
Perozo
E.
,
Bezanilla
F.
.
2006
.
Detection of the opening of the bundle crossing in KcsA with fluorescence lifetime spectroscopy reveals the existence of two gates for ion conduction
.
J. Gen. Physiol.
128
:
569
581
. doi:
Braha
O.
,
Walker
B.
,
Cheley
S.
,
Kasianowicz
J.J.
,
Song
L.
,
Gouaux
J.E.
,
Bayley
H.
.
1997
.
Designed protein pores as components for biosensors
.
Chem. Biol.
4
:
497
505
. doi:
Brelidze
T.I.
,
Niu
X.
,
Magleby
K.L.
.
2003
.
A ring of eight conserved negatively charged amino acids doubles the conductance of BK channels and prevents inward rectification
.
100
:
9017
9022
. doi:
Carvacho
I.
,
Gonzalez
W.
,
Torres
Y.P.
,
Brauchi
S.
,
Alvarez
O.
,
Gonzalez-Nilo
F.D.
,
Latorre
R.
.
2008
.
Intrinsic electrostatic potential in the BK channel pore: role in determining single channel conductance and block
.
J. Gen. Physiol.
131
:
147
161
. doi:
Chakrapani
S.
,
Cordero-Morales
J.F.
,
Perozo
E.
.
2007a
.
A quantitative description of KcsA gating I: macroscopic currents
.
J. Gen. Physiol.
130
:
465
478
. doi:
Chakrapani
S.
,
Cordero-Morales
J.F.
,
Perozo
E.
.
2007b
.
A quantitative description of KcsA gating II: single-channel currents
.
J. Gen. Physiol.
130
:
479
496
. doi:
Cheley
S.
,
Malghani
M.S.
,
Song
L.
,
Hobaugh
M.
,
Gouaux
J.E.
,
Yang
J.
,
Bayley
H.
.
1997
.
Spontaneous oligomerization of a staphylococcal alpha-hemolysin conformationally constrained by removal of residues that form the transmembrane beta-barrel
.
Protein Eng.
10
:
1433
1443
. doi:
Choi
H.
,
Heginbotham
L.
.
2004
.
Functional influence of the pore helix glutamate in the KcsA K+ channel
.
Biophys. J.
86
:
2137
2144
. doi:
Conlan
S.P.
2003
.
Biochemical and biophysical characterization of a monomeric porin, OmpG
.
PhD thesis. Texas A&M University
,
College Station, TX
.
180 pp
.
Cordero-Morales
J.F.
,
Cuello
L.G.
,
Perozo
E.
.
2006a
.
Voltage-dependent gating at the KcsA selectivity filter
.
Nat. Struct. Mol. Biol.
13
:
319
322
. doi:
Cordero-Morales
J.F.
,
Cuello
L.G.
,
Zhao
Y.
,
Jogini
V.
,
Cortes
D.M.
,
Roux
B.
,
Perozo
E.
.
2006b
.
Molecular determinants of gating at the potassium-channel selectivity filter
.
Nat. Struct. Mol. Biol.
13
:
311
318
. doi:
Cordero-Morales
J.F.
,
Jogini
V.
,
Lewis
A.
,
Vásquez
V.
,
Cortes
D.M.
,
Roux
B.
,
Perozo
E.
.
2007
.
Molecular driving forces determining potassium channel slow inactivation
.
Nat. Struct. Mol. Biol.
14
:
1062
1069
. doi:
Cortes
D.M.
,
Cuello
L.G.
,
Perozo
E.
.
2001
.
Molecular architecture of full-length KcsA: role of cytoplasmic domains in ion permeation and activation gating
.
J. Gen. Physiol.
117
:
165
180
. doi:
Cuello
L.G.
,
Romero
J.G.
,
Cortes
D.M.
,
Perozo
E.
.
1998
.
pH-dependent gating in the Streptomyces lividans K+ channel
.
Biochemistry.
37
:
3229
3236
. doi:
De Biasi
M.
,
Hartmann
H.A.
,
Drewe
J.A.
,
Taglialatela
M.
,
Brown
A.M.
,
Kirsch
G.E.
.
1993
.
Inactivation determined by a single site in K+ pores
.
Pflugers Arch.
422
:
354
363
. doi:
Doyle
D.A.
,
Morais Cabral
J.
,
Pfuetzner
R.A.
,
Kuo
A.
,
Gulbis
J.M.
,
Cohen
S.L.
,
Chait
B.T.
,
MacKinnon
R.
.
1998
.
The structure of the potassium channel: molecular basis of K+ conduction and selectivity
.
Science.
280
:
69
77
. doi:
Gao
L.
,
Mi
X.
,
Paajanen
V.
,
Wang
K.
,
Fan
Z.
.
2005
.
Activation-coupled inactivation in the bacterial potassium channel KcsA
.
102
:
17630
17635
. doi:
Gouaux
E.
,
Mackinnon
R.
.
2005
.
Principles of selective ion transport in channels and pumps
.
Science.
310
:
1461
1465
. doi:
Heginbotham
L.
,
Odessey
E.
,
Miller
C.
.
1997
.
Tetrameric stoichiometry of a prokaryotic K+ channel
.
Biochemistry.
36
:
10335
10342
. doi:
Heginbotham
L.
,
Kolmakova-Partensky
L.
,
Miller
C.
.
1998
.
Functional reconstitution of a prokaryotic K+ channel
.
J. Gen. Physiol.
111
:
741
749
. doi:
Heginbotham
L.
,
LeMasurier
M.
,
Kolmakova-Partensky
L.
,
Miller
C.
.
1999
.
Single streptomyces lividans K+ channels: functional asymmetries and sidedness of proton activation
.
J. Gen. Physiol.
114
:
551
560
. doi:
Holden
M.A.
,
Bayley
H.
.
2005
.
Direct introduction of single protein channels and pores into lipid bilayers
.
J. Am. Chem. Soc.
127
:
6502
6503
. doi:
Hoshi
T.
,
Zagotta
W.N.
,
Aldrich
R.W.
.
1994
.
Shaker potassium channel gating. I: transitions near the open state
.
J. Gen. Physiol.
103
:
249
278
. doi:
Howorka
S.
,
Bayley
H.
.
1998
.
Improved protocol for high-throughput cysteine scanning mutagenesis
.
Biotechniques.
25
:
764
766
,
768, 770 passim
.
Howorka
S.
,
Cheley
S.
,
Bayley
H.
.
2001
.
Sequence-specific detection of individual DNA strands using engineered nanopores
.
Nat. Biotechnol.
19
:
636
639
. doi:
Imoto
K.
,
Busch
C.
,
Sakmann
B.
,
Mishina
M.
,
Konno
T.
,
Nakai
J.
,
Bujo
H.
,
Mori
Y.
,
Fukuda
K.
,
Numa
S.
.
1988
.
Rings of negatively charged amino acids determine the acetylcholine receptor channel conductance
.
Nature.
335
:
645
648
. doi:
Irizarry
S.N.
,
Kutluay
E.
,
Drews
G.
,
Hart
S.J.
,
Heginbotham
L.
.
2002
.
Opening the KcsA K+ channel: tryptophan scanning and complementation analysis lead to mutants with altered gating
.
Biochemistry.
41
:
13653
13662
. doi:
Jones
D.H.
,
1995
.
PCR mutagenesis and recombination in vivo
.
In
PCR Primer: A Laboratory Manual
.
Dieffenbach
C.W.
,
Dveksler
G.S.
,
editors
.
Cold Spring Harbor Laboratory Press
,
Cold Spring Harbor, NY
.
591
601
.
Krishnan
M.N.
,
Bingham
J.P.
,
Lee
S.H.
,
Trombley
P.
,
Moczydlowski
E.
.
2005
.
Functional role and affinity of inorganic cations in stabilizing the tetrameric structure of the KcsA K+ channel
.
J. Gen. Physiol.
126
:
271
283
. doi:
LeMasurier
M.
,
Heginbotham
L.
,
Miller
C.
.
2001
.
KcsA: it's a potassium channel
.
J. Gen. Physiol.
118
:
303
314
. doi:
Lockless
S.W.
,
Zhou
M.
,
MacKinnon
R.
.
2007
.
Structural and thermodynamic properties of selective ion binding in a K+ channel
.
PLoS Biol.
5
:
e121
. doi:
Long
S.B.
,
Campbell
E.B.
,
Mackinnon
R.
.
2005
.
Crystal structure of a mammalian voltage-dependent Shaker family K+ channel
.
Science.
309
:
897
903
. doi:
López-Barneo
J.
,
Hoshi
T.
,
Heinemann
S.H.
,
Aldrich
R.W.
.
1993
.
Effects of external cations and mutations in the pore region on C-type inactivation of Shaker potassium channels
.
Receptors Channels.
1
:
61
71
.
MacKinnon
R.
1991
.
Determination of the subunit stoichiometry of a voltage-activated potassium channel
.
Nature.
350
:
232
235
. doi:
MacKinnon
R.
2004
.
Potassium channels and the atomic basis of selective ion conduction (Nobel Lecture)
.
Angew. Chem. Int. Ed. Engl.
43
:
4265
4277
. doi:
Marius
P.
,
Zagnoni
M.
,
Sandison
M.E.
,
East
J.M.
,
Morgan
H.
,
Lee
A.G.
.
2008
.
Binding of anionic lipids to at least three nonannular sites on the potassium channel KcsA is required for channel opening
.
Biophys. J.
94
:
1689
1698
. doi:
Mason
A.E.
2008
.
Single-channel characterisation of potassium channels with high temperature studies
.
PhD thesis. University of Oxford
,
England, UK
.
200 pp
.
McCormack
K.
,
Lin
L.
,
Iverson
L.E.
,
Tanouye
M.A.
,
Sigworth
F.J.
.
1992
.
Tandem linkage of Shaker K+ channel subunits does not ensure the stoichiometry of expressed channels
.
Biophys. J.
63
:
1406
1411
. doi:
Montal
M.
,
Mueller
P.
.
1972
.
Formation of bimolecular membranes from lipid monolayers and a study of their electrical properties
.
69
:
3561
3566
. doi:
Nimigean
C.M.
,
Chappie
J.S.
,
Miller
C.
.
2003
.
Electrostatic tuning of ion conductance in potassium channels
.
Biochemistry.
42
:
9263
9268
. doi:
Ogielska
E.M.
,
Zagotta
W.N.
,
Hoshi
T.
,
Heinemann
S.H.
,
Haab
J.
,
Aldrich
R.W.
.
1995
.
Cooperative subunit interactions in C-type inactivation of K channels
.
Biophys. J.
69
:
2449
2457
. doi:
Panyi
G.
,
Sheng
Z.
,
Deutsch
C.
.
1995
.
C-type inactivation of a voltage-gated K+ channel occurs by a cooperative mechanism
.
Biophys. J.
69
:
896
903
. doi:
Parks
T.D.
,
Leuther
K.K.
,
Howard
E.D.
,
Johnston
S.A.
,
Dougherty
W.G.
.
1994
.
Release of proteins and peptides from fusion proteins using a recombinant plant virus proteinase
.
Anal. Biochem.
216
:
413
417
. doi:
Qian
X.
,
Nimigean
C.M.
,
Niu
X.
,
Moss
B.L.
,
Magleby
K.L.
.
2002
.
Slo1 tail domains, but not the Ca2+ bowl, are required for the β1 subunit to increase the apparent Ca2+ sensitivity of BK channels
.
J. Gen. Physiol.
120
:
829
843
. doi:
Sack
J.T.
,
Shamotienko
O.
,
Dolly
J.O.
.
2008
.
How to validate a heteromeric ion channel drug target: assessing proper expression of concatenated subunits
.
J. Gen. Physiol.
131
:
415
420
. doi:
Schrempf
H.
,
Schmidt
O.
,
Kümmerlen
R.
,
Hinnah
S.
,
Müller
D.
,
Betzler
M.
,
Steinkamp
T.
,
Wagner
R.
.
1995
.
A prokaryotic potassium ion channel with two predicted transmembrane segments from Streptomyces lividans
.
EMBO J.
14
:
5170
5178
.
Shim
J.W.
,
Yang
M.
,
Gu
L.-Q.
.
2007
.
In vitro synthesis, tetramerization and single channel characterization of virus-encoded potassium channel Kcv
.
FEBS Lett.
581
:
1027
1034
. doi:
Spirin
A.S.
,
2002
.
Cell-free protein synthesis
.
In
Cell-Free Translation Systems
.
Spirin
A.S.
,
editor
.
Springer
,
New York
.
3
20
.
Suh-Lailam
B.B.
,
Hevel
J.M.
.
2009
.
Efficient cleavage of problematic tobacco etch virus (TEV)-protein arginine methyltransferase constructs
.
Anal. Biochem.
387
:
130
132
. doi:
Thompson
A.N.
,
Posson
D.J.
,
Parsa
P.V.
,
Nimigean
C.M.
.
2008
.
Molecular mechanism of pH sensing in KcsA potassium channels
.
105
:
6900
6905
. doi:
Valiyaveetil
F.I.
,
Zhou
Y.
,
MacKinnon
R.
.
2002
.
Lipids in the structure, folding, and function of the KcsA K+ channel
.
Biochemistry.
41
:
10771
10777
. doi:
van Dalen
A.
,
Schrempf
H.
,
Killian
J.A.
,
de Kruijff
B.
.
2000
.
Efficient membrane assembly of the KcsA potassium channel in Escherichia coli requires the protonmotive force
.
EMBO Rep.
1
:
340
346
. doi:
van Dalen
A.
,
Hegger
S.
,
Killian
J.A.
,
de Kruijff
B.
.
2002
.
Influence of lipids on membrane assembly and stability of the potassium channel KcsA
.
FEBS Lett.
525
:
33
38
. doi:
Wang
L.
,
Xie
J.
,
Schultz
P.G.
.
2006
.
Expanding the genetic code
.
Annu. Rev. Biophys. Biomol. Struct.
35
:
225
249
. doi:
Yellen
G.
1998
.
The moving parts of voltage-gated ion channels
.
Q. Rev. Biophys.
31
:
239
295
. doi:
Yifrach
O.
,
MacKinnon
R.
.
2002
.
Energetics of pore opening in a voltage-gated K(+) channel
.
Cell.
111
:
231
239
. doi:
Zandany
N.
,
M.
,
Orr
I.
,
Yifrach
O.
.
2008
.
Direct analysis of cooperativity in multisubunit allosteric proteins
.
105
:
11697
11702
. doi:
Zhou
Y.
,
Morais-Cabral
J.H.
,
Kaufman
A.
,
MacKinnon
R.
.
2001
.
Chemistry of ion coordination and hydration revealed by a K+ channel-Fab complex at 2.0 Å resolution
.
Nature.
414
:
43
48
. doi:

Abbreviations used in this paper:

• DPhPC

1,2-diphytanoyl-sn-glycero-3-phosphocholine

•
• IVTT

in vitro transcription and translation

•
• POPE

1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine

•
• POPG

1-palmitoyl-2-oleoyl-sn-glycero-3-[phospho-rac-(1-glycerol)]

•
• TEV

tobacco etch virus

•
• WT

wild-type


text:
# [OS X TeX] Using align with "or"

Ross Moore ross.moore at mq.edu.au
Mon Apr 18 00:47:22 CEST 2011

Hi David,

On 18/04/2011, at 8:35 AM, David Arnold wrote:

> Bruno et al,
>
> This is precisely what I needed. But now I think I also need an explanation. We have:
>
> \begin{alignat}{5}
>
> But the first row has 9 ampersands. I've stumbled into solutions like this on my own, but I just don't understand the use of the ampersands.

Lee Witt said:

>>>  If you want to align n columns use 2n-1 &s

The point is that the alignment at each ampersand alternates:
right alignment, then left alignment.

So if you have columns each with an alignment point "inside" the content,
then you need "&&" between the columns,

viz.

A&=B   &&   C&=D   && ...

> Is this use of the ampersands explained in any documentation?

Try  The LaTeX Companion    1st or 2nd editions.

>
> D.
>
> P.S. Also, is there any way to spread out the output a bit?

try the variants:

\begin{xalignat}{5}
....

or

\begin{xxalignat}{5}
....

and use  {alignat*}  {xalignat*}  if you don't want equation numbers.
{xxalignat} suppresses them anyway.

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>
>
> D.
>
>
> On Apr 17, 2011, at 12:56 PM, Bruno Voisin wrote:
>
>> Le 17 avr. 2011 ? 21:48, Lee D. Witt a ?crit :
>>> Have you tried "alignat"? You specify the number of "columns" to align. If you want to align n columns use 2n-1 &s
>>>
>> \begin{alignat}{5}
>> u      &=0 &&&                       u      &=-4 &&&                       u      &=4 \\
>> \log x &=0 &&&                       \log x &=-4 &&&                       \log x &=4
>> \end{alignat}
>>
>> Bruno

Hope this helps,

Ross

------------------------------------------------------------------------
Ross Moore                                       ross.moore at mq.edu.au
Mathematics Department                           office: E7A-419
Macquarie University                             tel: +61 (0)2 9850 8955
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