text:
# Solving Quantum Ground-State Problems with Nuclear Magnetic Resonance

 Title: Solving Quantum Ground-State Problems with Nuclear Magnetic Resonance Author: Li, Zhaokai; Yung, Man-Hong; Chen, Hongwei; Lu, Dawei; Whitfield, James D.; Peng, Xinhua; Aspuru-Guzik, Alán; Du, Jiangfeng Note: Order does not necessarily reflect citation order of authors. Citation: Li, Zhaokai, Man-Hong Yung, Hongwei Chen, Dawei Lu, James D. Whitfield, Xinhua Peng, Alán Aspuru-Guzik, and Jiangfeng Du. 2011. Solving quantum ground-state problems with nuclear magnetic resonance. Scientific Reports 1: 88. Full Text & Related Files: Solving_Quantum_Ground_State_Problems.pdf (2.141Mb; PDF) Abstract: Quantum ground-state problems are computationally hard problems for general many-body Hamiltonians; there is no classical or quantum algorithm known to be able to solve them efficiently. Nevertheless, if a trial wavefunction approximating the ground state is available, as often happens for many problems in physics and chemistry, a quantum computer could employ this trial wavefunction to project the ground state by means of the phase estimation algorithm (PEA). We performed an experimental realization of this idea by implementing a variational-wavefunction approach to solve the ground-state problem of the Heisenberg spin model with an NMR quantum simulator. Our iterative phase estimation procedure yields a high accuracy for the eigenenergies (to the $$10^{−5}$$ decimal digit). The ground-state fidelity was distilled to be more than 80%, and the singlet-to-triplet switching near the critical field is reliably captured. This result shows that quantum simulators can better leverage classical trial wave functions than classical computers Published Version: doi:10.1038/srep00088 Other Sources: http://arxiv.org/abs/1106.0440 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:8404290

### This item appears in the following Collection(s)

• FAS Scholarly Articles [6948]
Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are reading an older version of this FlexBook® textbook: CK-12 Algebra I Concepts - Honors Go to the latest version.

# 6.1: Product Rules for Exponents

Difficulty Level: At Grade Created by: CK-12

Suppose you have the expression:

How could you write this expression in a more concise way?

### Watch This

James Sousa: Exponential Notation

### Guidance

In the expression , the is called the base and the is called the exponent. Exponents are often referred to as powers. When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:

• .

There are many rules that have to do with exponents (often called the Laws of Exponents) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn two rules that have to do with exponents and products.

RULE: To multiply two terms with the same base, add the exponents.

RULE: To raise a product to a power, raise each of the factors to the power.

#### Example A

Evaluate .

Solution:

The answer can be taken one step further. The base is numerical so the term can be evaluated.

Evaluate .

Solution:

Evaluate .

Solution:

Evaluate .

Solution:

#### Concept Problem Revisited

can be rewritten as . Then, you can use the rules of exponents to simplify the expression to . This is certainly much quicker to write!

### Vocabulary

Base
In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression , ‘2’ is the base. In the expression , ‘’ is the base.
Exponent
In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:
In the expression , ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here:
In the expression , ‘4’ is the exponent. It means to multiply times itself 4 times as shown here: .
Laws of Exponents
The laws of exponents are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.

### Guided Practice

Evaluate each of the following:

1.

2.

3.

4.

1.

2.

3.

The answer can be taken one step further. The base of each factor is numerical so each term can be evaluated. The final answer will be the product of the two answers.

4.

Jan 16, 2013

format: plain


text:
# How do I prevent NIntegrate::inumr errors within other functions?

I believe this question is best illustrated with a simple example. If I run

FunctionInterpolation[NIntegrate[a + b, {a, 0, 1}], {b, 0, 1}]

I get errors of the form

NIntegrate::inumr: The integrand a+b has evaluated to non-numerical values for all
sampling points in the region with boundaries {{0,1}}.

So I guess Mathematica is evaluating the expression before plugging in values and trying to interpolate it. Admittedly the code does yield apparently correct results, so I could just turn off the error message and be done with it, but I'd like to address the root cause if possible. Is there some more correct way to do what I'm trying to do that would not spit out an error message in the first place?

-

## 1 Answer

You could define a function which is evaluated only when its argument is numeric :

int[b_?NumericQ] := NIntegrate[a + b, {a, 0, 1}]

FunctionInterpolation[int[b], {b, 0, 1}]

-
Ah, good idea, that could work. It'd be nice to avoid having to define a separate function, though. – David Z Jul 31 '12 at 9:00
There is an article here support.wolfram.com/kb/3820 that helps explain why you should use NumericQ. – Searke Jul 31 '12 at 19:19


text:
344 views

The number of solutions of the equation $2x+y=40$ where both $x$ and $y$ are positive integers and $x \leq y$ is:

1. $7$
2. $13$
3. $14$
4. $18$
5. $20$

13 solutions.

For x = y, there are no integer solutions as there is no positive integer x for which 3x = 40.

For x < y, we have 13 integer solutions for each x from 1 to 13 as

(x, y) = (1,38), (2,36), (3,34)...... (13,14)

1.5k points

1
313 views


text:
# teaching machines

## CS 268: Lecture 1 – Web History

Welcome to CS 268: Web Systems!

In this class, we will investigate how to make software that runs in a web browser. It’s quite likely that web development is going to be one of the most important skills you can have in this stage of the technological era. Standalone desktop apps will never go away entirely, but capitalism makes the web hard to beat. You can write your software once for all computers, distribute it through the internet, control who accesses it, and deploy fixes more or less immediately.

Let’s have a look at how the web came to be.

### History

#### Internet

It was the 1960s. Democrats John Kennedy and Lyndon Johnson served as presidents, and the nation was strained in a battle for civil rights. World War II had ended two decades earlier, but the United States and Soviet Union were still glaring at each other across the Bering Sea. Several years before, in 1957, the Soviet Union had launched Sputnik I, the first satellite, into orbit. Kennedy’s predecessor Dwight Eisenhower had responded by forming the Advanced Research Projects Agency (ARPA) to spur American innovation. The Department of Defense had funding to unite researchers and military leaders to restore America’s lead in science and technology.

Alexander Graham Bell’s company American Telegraph and Telephone, founded in 1885, still enjoyed a government-authorized monopoly of the phone network. By some accounts, the Department of Defense looked at the way the phone system worked with concern. In order for two parties to exchange data and voice, a dedicated circuit was established between them. This circuit-switching was fast but expensive. Considerable infrastructure was required to support dedicated circuits. The network was also vulnerable. If a missile took down part of the network, connections would be severed. Others, namely the non-military scientists and engineers who partnered with the Department of Defense, downplay the Cold War paranoia. They cite the fact that there were computers spread across the country, and they needed a standard and reliable way of communicating.

Whatever the primary motive, the Department of Defense funded the development of a network that could withstand disruption. The result was a network between computers called the ARPANET. The first message on the network went from UCLA to Stanford in 1969. The payload was the text LOGIN, but Stanford’s computer crashed after receiving just LO.

To communicate across the ARPANET, the sending computer broke a message up into small chunks and pushed onto the network. But the chunks didn’t follow an established circuit. Instead, each chunk or packet was directed to the receiving computer by way of routers, which were distributed around the country and which maintained a directory of all the computers on the network. This packet-switching meant that some chunks might follow different routes. Because some routes were slower than others, the packets arrived out of order. The receiving computer would reorganize them. The eventual protocols specifying how packets were formed, delivered, and reassembled across the ARPANET were TCP and IP.

As the ARPANET grew bigger and scaled beyond its military roots, it was renamed the Internet.

The key takeaway here is that the Russians had the foresight to launch Sputnik in 1957 so that they would have a vehicle for interfering with American elections in 2016. That is indirection at its finest.

#### Web

The Internet soon crossed over into Europe. One of the European hubs of the network was CERN, a nuclear physics laboratory in Geneva, Switzerland, which is known today for building the Large Hadron Collider.

In the early 1980s, CERN employed one Tim Berners-Lee, a second-generation computer scientist from Britain. (In fact, both his parents were by computer scientists, and he also married a computer scientist.)

Berners-Lee had a vision for sharing research on the Internet: instead of just presenting flat listings of text, the text would contain links to other documents, forming a vast graph of the world’s information. Using a NeXT computer from Steve Jobs’ company, he wrote a web server and a web client (or browser). In late 1990, he had identified the key technologies of his system:

• Hypertext Markup Language (HTML): a formatting language used to structure a hypertext document
• Uniform Resource Identifier (URI): an address of a hypertext document
• Hypertext Transfer Protocol (HTTP): a language used to describe requests for documents

He soon began serving out the world’s first web page. This would have meant nothing if he had not also released his software for free. CERN charged no royalties. A freely- accessible structured information graph was born. Berners-Lee called it the World Wide Web.

Several years later, Marc Andreesen led a team in Illinois that built Mosaic, which supported images and which they ported to Windows. Andreesen and some of his team went on to form Netscape Communications Center, whose Netscape Navigator was the world’s most popular browser in the 1990s. But Microsoft licensed Mosaic and released Internet Explorer, which in time displaced Netscape and led to an antitrust case in the United States. Microsoft’s tight integration of Internet Explorer in their operating system was declared illegal. But the verdict came too late for Netscape to recover. Before they dissolved as a company, Netscape released its code base to support the development of a comprehensive suite of Internet tools, including a browser, an email client, an HTML editor, an address book, and a chat client. The group that took over the development was the Mozilla Foundation, who eventually pruned away everything but the browser, which they named Firefox, and the email client, which they named Thunderbird. Firefox was the only serious competitor to Internet Explorer in the 2000s, but then Google released Chrome in 2008. Chrome today accounts for around 65% of the world’s web traffic. Apple’s Safari accounts for around 15%. But the shares differ by region.

Of course, technology alone is not what makes the web interesting or successful. The web has become what it is because of Berners-Lee’s original spirit of sharing content and the tooling for distributing and accessing it. And that’s what we do on the Internet. We share everything.

### Computers Talking

If we compare network technology to physical transportation, the Internet is the network of roads and the web is one of the many goods that can be shipped across it. The computers providing the goods and consuming the goods are the endpoints. We call a computer that provides the goods a server, and its goods are called services. The computer that requests and consumes a service is a client.

Most any computer can be a server. Two things are required of the server. It must have an IP address which uniquely identifies it on the network. It must also have a program that’s running in an infinite loop, waiting for clients to connect to it.

If we are in business, we probably want a server that is connected to a fast network and is itself fast. We will also buy a domain name from a domain name registrar. The registrar will help maintain a mapping called the Domain Name Service (DNS), which maps domain names to our servers’ IP addresses. But none of this is necessary.

How does a client send a request to a server? Our operating systems provide a door to the internet called a socket. The server opens its door and leaves it open indefinitely. The client opens its door only when it has a request.

The server may provide many services. It may serve out web pages, a shared Minecraft world, email, Git repositories, amongst other things. So that a computer may listen to many servers, it has many doors. These doors are called ports. Web pages are often delivered across ports 80 or 443 and Minecraft across port 25565.

### HTTP

Let’s have a look at how a web client (a browser) issues a request to a web server. The protocol that both client and server follow is what Berners-Lee called HTTP. The client initiates a conversation by sending a request to a server. The request includes a method and a set of headers describing the requested service.

To retrieve a page, the client sends a GET request that has in the very least a Host header. The message might look like this:

GET /index.html HTTP/1.1
Host: twodee.org

We specify the path to the resource we are requesting, which happens to be a plain text HTML document. We also specify the version of HTTP we are using. The latest is HTTP/2, which was modeled on Google’s experimental SPDY protocol. Around 43% of current traffic uses HTTP/2. But HTTP/1.1 is simpler to inspect.

Note the blank line at the end of the message. According to the HTTP specification, each line should be ended with both a carriage return and a linefeed (a CRLF or a literal \r\n).

We can write a little script that submits this request to the server. I will use Ruby because it is lightweight and fun.

First we create a socket to the HTTP door (port 80) on our server, which is identified by its IP address. Then we send our request and collect up the response.

#!/usr/bin/env ruby

require 'socket'

socket = TCPSocket.open('138.68.15.70', 80)

socket.write("GET /index.html HTTP/1.1\r\n")
socket.write("Host: twodee.org\r\n")
socket.write("\r\n")

while line = socket.gets
puts line
end

socket.close

The response might look something like this:

HTTP/1.1 200 OK
Date: Tue, 28 Jan 2020 19:48:34 GMT
Server: Apache/2.4.18 (Ubuntu)
Vary: Host,Accept-Encoding
Last-Modified: Tue, 09 Aug 2016 19:47:25 GMT
ETag: "6b-539a8cd574b68"
Accept-Ranges: bytes
Content-Length: 107
Content-Type: text/html

<!DOCTYPE html>
<html>
<head>
<title>...</title>
</head>
<body>
Hello, client. It's me, server.
</body>
</html>

When we run the script, we see a slight delay before it finishes. That’s because we are using HTTP/1.1. In earlier versions of HTTP, the connection to the server closed immediately by default. If the HTML content included a reference to an image, we had to establish a new TCP connection to retrieve the image. Such building up and tearing down of sockets is slow.

In HTTP/1.1, the server by default keeps the socket open for a bit in case the client sends a followup request. If the client is going to make only a single request, it specifies the Connection header with the value closed:

socket.write("GET /index.html HTTP/1.1\r\n")
socket.write("Host: twodee.org\r\n")
socket.write("Connection: close\r\n")
socket.write("\r\n")

This removes the delay. This behavior is not exactly significant, but I include it to illustrate how technology evolves. In fact, a recurring theme of this course is that a technology progresses from slow but understandable to fast but incomprehensible. You are coming into web development at the incomprehensible stage, but it’s only going to get more so.

HTTP has several other methods that we will see through the semester, namely the following:

• POST for when the client wants to submit a new resource
• PATCH for when the client wants to update an existing resource
• DELETE for when the clients to remove a resource

These correspond quite nicely to SQL’s INSERT, UPDATE, and DELETE. And GET is like SELECT. For the time being, we will focus only on GET.

Normally we don’t issue GET requests directly. Rather, we enter a URI in the web browser, and it opens a socket and issues the request on our behalf.

### HTML

The content sent back by the server is a hierarchy or tree of information, marked up in HTML. But computers don’t exchange trees across the network. They exchange flat sequences of bytes. Therefore, the tree must be serialized. To mark the start and end of a node in the tree, we use an element, which has this form:

<element>content...</element>

Each <...> is called a tag. <element> is the opening tag, and the </element> is the closing tag.

The content itself is a mix of child elements and implicit text nodes.

Each element may shaped by zero or more attributes:

<element attribute0="value0" attribute1="value1">content</element>

Whitespace in between attributes, tag names, and top-level punctuation is not significant. For example, this element is equivalent to the preceding:

<element
attribute0="value0"
attribute1="value1">content</element>

But whitespace in an attribute or in the content may be significant.

### In-class Exercise

As your teacher, I am going to do what I can to give you opportunities to learn web development. But I am only a small part of your learning. Really, my job is to make you—encourage you—to do things that engage your brain. So, in most of our lectures, I will stop talking for a while and get you talking with a neighbor as you complete an exercise.

So, here’s your exercise. First, visit Crowdsource and claim a task number. Then investigate the HTML element below that corresponds to your task number:

1. html
2. head
3. body
4. p
5. a
6. img
7. div
8. span
9. iframe
10. table, tr, td
11. ul
12. ol
13. img
14. h1, h2, …
15. main
16. nav

Read up on your element’s attributes. Write a short snippet of HTML demonstrating how your element and its attributes might be used. Submit it on Crowdsource. We will examine your submissions in lecture.

When you search the web for your element, W3Schools is often going to be the top result. I suggest you favor Mozilla’s Developer Network over W3Schools for more comprehensive documentation.

### TODO

Here’s your TODO list for next time:

• Read the syllabus.
• Read MDN’s Introduction to HTML.
• Set up your web server on Digital Ocean by completing homework 0.
• Install Visual Studio Code, the tool I recommend for your work in this class. If you are comfortable with another editor, including its version control features, feel free to continue using it.

See you next time.

Sincerely,

P.S. It’s time for a haiku!

“Hello, world,” I lied
There was no world till later
When its Tim had come


text:
The effects of organic quaternary amines, tetraethylammonium (TEA) chloride and benzyltriethylammonium (BTEA) chloride, on Na,K pump current were examined in rat cardiac myocytes superfused in extracellular Na+-free solutions and whole-cell voltage-clamped with patch electrodes containing a high Na+-salt solution. Extracellular application of these quaternary amines competitively inhibited extracellular K+ (K+o) activation of Na,K pump current; however, the concentration for half maximal inhibition of Na,K pump current at 0 mV (K0Q) by BTEA, 4.0 ± 0.3 mM, was much lower than the K0Q for TEA, 26.6 ± 0.7 mM. Even so, the fraction of the membrane electric field dissipated during K+o activation of Na,K pump current (λK), 39 ± 1%, was similar to λK determined in the presence of TEA (37 ± 2%) and BTEA (35 ± 2%), an indication that the membrane potential (VM) dependence for K+o activation of the Na,K pump current was unaffected by TEA and BTEA. TEA was found to inhibit the Na,K pump current in a VM-independent manner, i.e., inhibition of current dissipated 4 ± 2% of the membrane electric field. In contrast, BTEA dissipated 40 ± 5% of the membrane electric field during inhibition of Na,K pump current. Thus, BTEA inhibition of the Na,K-ATPase is VM-dependent. The competitive nature of inhibition as well as the similar fractions of the membrane electric field dissipated during K+o-dependent activation and BTEA-dependent inhibition of Na,K pump current suggest that BTEA inhibits the Na,K-ATPase at or very near the enzyme's K+o binding site(s) located in the membrane electric field. Given previous findings that organic quaternary amines are not occluded by the Na,K-ATPase, these data clearly demonstrate that an ion channel–like structure provides access to K+o binding sites in the enzyme.

## Introduction

The stoichiometry of ion transport by the Na,K-ATPase under physiologic conditions, 3 Na+ extruded from the cell in exchange for the uptake of 2 K+ per ATP hydrolyzed (Glynn, 1985; Rakowski et al., 1989), results in one positive charge being moved across the cell membrane per enzyme transport cycle (De Weer et al., 1988; Glynn and Karlish, 1990; Läuger, 1991; Vasilets and Schwarz, 1993; Rakowski et al., 1997). This movement of positive charge can be measured as a membrane depolarization upon inhibition of the Na,K-ATPase with cardiac glycosides or as cardiac glycoside–inhibitable, extracellular K+ (K+o)-dependent steady-state outward current (Na,K pump current) under voltage-clamp conditions where other ionic membrane currents are blocked or inactivated (Cooke et al., 1974; Isenberg and Trautwein, 1974; Eisner and Lederer, 1980; De Weer et al., 1984; Gadsby et al., 1985; Ishizuka et al., 1996; Berlin and Peluffo, 1997). The net movement of positive charge during the enzyme transport cycle also requires that the rate of at least one reaction step is dependent on the membrane potential (Läuger, 1991). The finding that Na,K pump current is membrane potential (VM) dependent in extracellular Na+ (Na+o) and K+o-containing salt solutions confirmed this theoretical requirement and demonstrated that under some conditions the VM-dependent reaction(s) can become rate limiting for steady-state ion transport (Hansen et al., 1981; Läuger, 1991). In fact, Na,K pump current has a bell-shaped dependence on VM in the presence of nonsaturating concentrations of Na+o and K+o (Lafaire and Schwarz, 1986; Bielen et al., 1991; Rakowski et al., 1991; Sagar and Rakowski, 1994; Berlin and Peluffo, 1997). Such a biphasic relationship necessitates the presence of at least two VM-dependent reaction steps in the transport cycle (Läuger, 1991).

One VM-dependent reaction of the Na,K-ATPase was identified with Na+-dependent transport steps by demonstrating that the positive slope of the Na,K pump current-voltage relationship, observed at negative VM in Na+o- and K+o-containing solutions, was lost when Na+ was removed from the extracellular medium (Nakao and Gadsby, 1989; Rakowski et al., 1991; Sagar and Rakowski, 1994). The VM-dependent step underlying this behavior was characterized by trapping the Na,K-ATPase in Na+-transporting conformations (under K+o-free conditions that promote Na+-Na+ exchange) and studying presteady-state kinetics of Na+o-dependent charge-moving reaction steps (Nakao and Gadsby, 1986). Such experiments showed that transport of Na+ to the extracellular medium moves the equivalent of a full charge across two thirds to three quarters of the membrane dielectric (Nakao and Gadsby, 1986; Bühler et al., 1991; Fendler et al., 1993; Rakowski, 1993; Hilgemann, 1994a; Holmgren and Rakowski, 1994; Heyse et al., 1994). Furthermore, the Na+o concentration dependence of this process provided evidence that the VM-dependent step occurs during extracellular Na+ release and binding (Gadsby et al., 1993; Wuddel and Apell, 1995). These results showed that the binding site for at least one Na+ is within the membrane electric field, i.e., the binding site in the E2P enzyme conformation is located in an ion well exposed to the extracellular medium.

In Na+o-free solutions, the Na,K pump current-voltage relationship showed a negative slope in the presence of nonsaturating concentrations of K+o (Bielen et al., 1991; Rakowski et al., 1991; Sagar and Rakowski, 1994; Berlin and Peluffo, 1997; Peluffo et al., 2000). Because Na+o-free solutions prevent Na+o-dependent reactions from influencing the VM dependence of Na,K pump current, this negative slope demonstrated that a second VM-dependent reaction was tied to K+-dependent transport steps. Measurement of transient charge movements dependent on K+o or extracellular Tl+ (Tl+o), a K+ congener, under conditions promoting K+-K+ exchange showed that this reaction moved a charge through approximately one third of the membrane dielectric. The Tl+o concentration dependence of these charge movements also demonstrated that extracellular ion binding was the VM-dependent reaction (Peluffo and Berlin, 1997). Thus, similar to Na+o binding, K+ binds in an extracellularly facing ion well of the Na,K-ATPase.

This extracellularly-facing ion well is most often envisioned conceptually as a narrow access channel in the enzyme, analogous to the pore of an ion channel (De Weer et al., 1988; Rakowski et al., 1997). The VM dependence of ion binding reactions in this scenario occurs because the membrane electric field strength affects the probability that ions occupy the access channel. However, as first pointed out by Hilgemann (1994b), an equally valid mechanism involves rapid, VM-dependent microscopic occlusion reactions in which weakly VM-dependent ion binding/coordination reactions are accompanied by rapid rearrangements of charge in the ion-bound protein with respect to the membrane electric field. Charge rearrangement during microscopic ion occlusion reactions could involve intrinsic protein moieties and/or the bound ion; however, with this mechanism, VM dependence of ion binding reactions arise because electric field strength affects the probability of charge rearrangements during ion binding and release. The problem is that this alternative mechanism of VM-dependent ion binding reactions is experimentally indistinguishable from the access channel model as long as the proportion of the ion-bound, but not -occluded enzyme state remains very small. In fact, experimental data suggest that the fraction of Rb+ (another K+ congener) -bound, but not -occluded enzyme is indeed very low under conditions that promote K+ occlusion by the Na,K-ATPase (González-Lebrero et al., 2002). Thus, it could be very difficult to distinguish between the alternate ion binding mechanisms outlined above in the presence of transported cations such as Na+ and K+. However, if ion transport could be blocked in such a manner that a substantial fraction of the enzyme resides in an ion-bound, but not -occluded state, then these alternative scenarios for VM-dependent ion binding should be experimentally distinguishable.

Organic quaternary amines inhibit ion transport by the Na,K-ATPase in a manner that is competitive with K+o activation of the enzyme (Kropp and Sachs, 1977) and inhibit Rb+ occlusion under conditions that would otherwise promote binding of extracellular ions (Forbush, 1988). Of particular note in the context of the present study, quaternary amines do not appear to become occluded by the enzyme (Forbush, 1988). One interpretation of these data is that organic quaternary amines inhibit ion occlusion by the Na,K-ATPase by binding at or near K+o binding sites in the enzyme. Should this interpretation be correct, these organic amines may allow us to separate binding and occlusion reactions by the enzyme and thereby study the mechanism of VM-dependent ion binding.

This study reports the concentration and VM-dependent properties of Na,K pump current inhibition by two such quaternary amines, TEA and its related analogue, benzyltriethylammonium ion (BTEA). The results show that, while these two compounds inhibit K+o activation of Na,K pump current in a competitive manner, the VM-dependent properties of enzyme inhibition are quite different. The implications of these results for defining the mechanism of VM-dependent extracellular ion binding are then discussed.

Preliminary communications for these data have been published previously (Berlin and Peluffo, 1998; Peluffo et al., 2001).

## Materials And Methods

Ventricular myocytes were enzymatically isolated from rat hearts and voltage-clamped with patch electrodes (1.0–1.5 MΩ) as published previously (Ishizuka et al., 1996). After establishing a whole-cell voltage clamp at a holding potential of −40 mV, cells were superfused in a Na+- and K+-free salt solution (see below) for 5–6 min to allow the contents of the patch electrode solution (see below) to exchange with the cell interior. Na,K pump current was defined as a K+o-activated outward current that was reversibly inhibited in the presence of 1 mM ouabain. During experiments, voltage-clamped cells were exposed to various K+-containing superfusion solutions for short periods, <30 s, every 2 min to ensure that bulk cytosolic Na+ concentration was maintained at or near patch pipette Na+ concentration (115 mM) throughout the experiment. Superfusion solutions containing different K+ concentrations were applied in random order. To determine the VM dependence of Na,K pump current, step changes in VM between −100 and +40 mV were produced from the holding potential every 500 ms for 100 ms.

The time constant for settling of current after a 5-mV depolarization was typically in the 200–300 μs range without series resistance compensation and cell capacitances were in the range of 120–250 pF.

### Solutions

The patch-electrode solution contained (in mM) 85 sodium sulfamate, 20 tetraethylammonium chloride, 10 ATP-Mg2+ salt, 5 pyruvic acid, 5 creatine phosphate–Tris salt, 10 EGTA–Tris salt, and 10 HEPES (pH 7.35 with NaOH). The total concentration of Na+ in this solution was 115 mM. Na+-free superfusion solutions contained the following (in mM): 145 tetramethylammonium (TMA) chloride, 2.3 MgCl2, 0.2 CdCl2, 5.5 dextrose, 10 HEPES (pH 7.4 with Tris). KCl was added to this solution in various concentrations (0.05–5 mM). Our initial experiments were performed with a superfusion solution containing 145 N-methyl-D-glucamine (NMG) instead of TMA; however, TMA blocked contaminating ionic currents more completely than NMG. Furthermore, the K+o dependence of Na,K pump current activation was similar in the presence of NMG and TMA-containing solutions (unpublished data). Tetraethylammonium chloride and benzyltriethylammonium chloride were added to the superfusion solution with equimolar substitution for TMA chloride. All experiments were performed at 37°C. Reagents were obtained from Sigma-Aldrich.

### Statistics and Curve Fitting

Data are presented as mean ± SEM for the indicated number of cells. Current density was calculated by dividing K+o-activated currents by cell capacitance measured as the integral of current elicited with 5-mV pulses. Indicated functions were fit to the data using a nonlinear least squares Gauss-Newton algorithm (Rossi and Garrahan, 1989) or with commercial software (SigmaPlot, SPSS).

## Results

Na,K pump current has been characterized previously as a K+o-activated, ouabain-inhibitable outward current in single cardiac myocytes (Gadsby et al., 1985; Gadsby and Nakao, 1989; Bielen et al., 1991; Stimers et al., 1991; Ishizuka et al., 1996; Berlin and Peluffo, 1997). Fig. 1

shows this outward current in Na+o-free solutions and briefly summarizes the experimental protocol used throughout this study. After establishing a whole-cell voltage clamp, rat ventricular myocytes were initially superfused with a Na+o- and K+o-free salt solution. When the K+o concentration in the superfusion solution was subsequently increased, an outward current was observed (Fig. 1 A) that was maintained throughout the period of K+o exposure except when 1 mM ouabain was also present in the superfusion solution. To determine the VM dependence of Na,K pump current (Fig. 1 B), step changes in VM from the holding potential of −40 mV were produced at the times indicated in Fig. 1 A and K+o-sensitive difference currents were calculated at each VM (Fig. 1 C). When a saturating K+o concentration (5 mM) was used to activate the Na,K pump, the resulting difference current (c–b), showed very little dependence on VM in the range of −100 to +40 mV; however, when nonsaturating K+o concentrations (0.1 mM) were used, the Na,K pump current became smaller at more positive VM (a–b), as reported previously (Rakowski et al., 1991; Sagar and Rakowski, 1994; Berlin and Peluffo, 1997; Peluffo et al., 2000).

The VM dependence of Na,K pump current was determined over a range of K+o concentrations between 0.05 and 5 mM (Fig. 2

A). These data make it obvious that, as K+o is lowered, the current-VM relationship displays a negative slope and an apparent shift to the left along the VM axis. To measure the effect of VM on Na,K pump current, the K+o dependence of current activation was determined by fitting a Hill equation to the data at each VM. The calculated maximal current density (Imax) at each VM is shown in Fig. 2 B. As can be seen in this figure, Imax appeared to show little change with VM. This impression was confirmed by regression analysis that showed the slope of a line through the data as a function of VM was not significantly different than zero. For this reason, Imax was averaged over all test VM to yield a value of 2.40 ± 0.10 pA/pF.

This fitting procedure also yielded the Hill coefficient (γK) for K+o activation of Na,K pump current. Similar to Imax, the value of γK was found to show no trend with VM (Fig. 2 B), so γK at all 15 VM tested were averaged to yield a value of 0.58 ± 0.07. This value is similar to that measured for K+o activation of Na,K pump current under zero extracellular Na+ conditions by Sagar and Rakowski (1994) in oocytes and by Peluffo et al. (2000) in HeLa cells expressing heterologous Na,K-ATPase.

By comparison, the concentration of K+o needed to yield a similar level of current activation increased at more positive VM, as is reflected by an increase in the K+o concentration required for half-maximal activation (K0.5) from 35 ± 4 to 365 ± 53 μM in going from −100 to +40 mV, respectively (Fig. 2 C). Thus, as reported previously (Sagar and Rakowski, 1994; Peluffo et al., 2000), the apparent affinity for K+o activation of the Na,K pump is decreased as the cell interior is made more positive, consistent with the idea that K+ binding occurs in an extracellularly facing ion well.

Under the conditions of the present experiments, high intracellular and zero extracellular Na+, Na,K pump current can be described by a pseudo two-state model that includes one lumped reaction step as well as a second VM- and K+o-dependent reaction (Hansen et al., 1981; Sagar and Rakowski, 1994; Peluffo et al., 2000). Using this model, the following equation,

$K_{0.5}=K_{0.5}^{0}\mathrm{exp}\left(\mathrm{{\lambda}}_{K}U\right)\mathrm{,}$
(1)

can be derived to determine the K0.5 at 0 mV (K00.5) and the portion of the membrane electric field (λK) dissipated during K+o-dependent reactions. The term “U” is the unitless form of voltage, zFVM/RT, where z is equal to 1, the net charge moved during the ion transport cycle. Fitting Eq. 1 to the data in Fig. 2 C yields values of K00.5 and λK of 204 ± 4 μM and 0.39 ± 0.01 (n = 15), respectively. This value of K00.5 and the fact that K+o-dependent reactions dissipate approximately one third of the membrane dielectric are consistent with our previous data (Berlin and Peluffo, 1997; Peluffo et al., 2000).

The two-step fitting procedure described above for the data in Fig. 2 allowed for the determination of four parameters, Imax, K00.5, γK, and λK, that describe the K+o concentration and VM dependence of Na,K pump current under the conditions of these experiments. To determine the utility of this fitting procedure, Eq. 6 in the PP1appendix was solved at each VM and K+o concentration tested using these parameter values. The calculated current densities were then compared with the experimental data. Using the entire set of experimentally measured current densities, the root mean square error of the calculated current densities was found to be 0.0057 pA/pF. Similarly, the sum percent error of the calculated current densities, defined as

$\mathrm{Sum\ percent\ error}=\frac{{{\sum}_{i}}{{\sum}_{j}}\left|Y^{\mathrm{^}}_{ij}{-}Y_{ij}\right|}{{{\sum}_{i}}{{\sum}_{j}}Y_{ij}}{\times}100\mathrm{,}$

where Y^ij and Yij are the calculated and measured current densities, respectively, at the ith K+o concentration and jthVM, was determined to be 3.5%. Both error values show that differences between calculated and measured currents are quite small, an indication that our fitting procedure provides an acceptable description of the data. The accuracy of the fitting procedure can also be inferred by comparing the experimental data to curves determined by solving Eq. 6 (PP1appendix) using the fitted parameters values for Imax, K00.5, γK, and λK. As can be seen in the inset of Fig. 2 C, the calculated curves closely follow the experimentally measured current densities at −100, 0, and +40 mV. Given the outcome of these tests, we used a similar two-step fitting procedure when analyzing subsequent data.

### Effect of Extracellular Tetraethylammonium ions on Na,K Pump Current

Extracellular TEA has been shown to inhibit activity (Zemková et al., 1988), ion transport (Sachs and Conrad, 1968), and current generation by the Na,K-ATPase (Eckstein-Ludwig et al., 1998). For this reason, our initial characterization of the effects of organic quaternary amines began with TEA. Experiments were performed in the same manner as outlined in Fig. 1 A except that immediately after establishing a whole-cell voltage-clamp, the cells were superfused in the Na+o- and K+o-free salt solution that also contained TEA at the indicated concentrations. TEA was substituted in equimolar fashion for TMA, which itself does not affect the Na,K pump (unpublished data). Fig. 3

A shows that 25 mM TEA decreased the 0.1 mM K+o-activated outward current at all VM tested. Similar to control conditions (0 TEA), the current-VM relationship displayed a negative slope.

To describe the inhibitory effects of TEA on Na,K pump current completely, the experiments shown in Fig. 2, measuring K+o and VM dependence of Na,K pump current, were repeated in the presence of various TEA concentrations. The effect of 25 mM TEA on the K+o and VM dependence of Na,K pump current was examined in eight cells. The resulting data, shown in Fig. 3 B, appeared to be quite similar to currents recorded under control conditions (Fig. 2 A) in the sense that, with 5 mM K+o, the current displayed little VM dependence but, at lower K+o concentrations, a negative slope in the current-VM relationship was still obvious. Just as in Fig. 3 A, however, the current densities at nonsaturating K+o concentrations were smaller in the presence of 25 mM TEA.

These data were fit with a Hill equation at each VM to calculate parameter values for Imax, γK, and K0.5. As in Fig. 2, the values for Imax and γK showed no dependence on VM so they were averaged for all 15 VM tested to yield values of 1.99 ± 0.46 pA/pF and 0.61 ± 0.12, respectively. Similar experiments and analyses were performed with 13 cells exposed to 5 mM TEA (not depicted) and the calculated average values for Imax and γK were 2.64 ± 0.31 pA/pF and 0.45 ± 0.08, respectively. These parameter values, along with those determined in the absence of TEA, are summarized in Fig. 3 C.

The calculations above led to two conclusions. First, the values of Imax in the presence of 5 and 25 mM TEA are not significantly different than Imax in the absence of the quaternary amine. In agreement with previous reports, these results suggest that TEA inhibition of Na,K pump current is competitive with K+o activation of ion transport (Sachs and Conrad, 1968). Second, the values of γK were likewise not significantly different in the presence and absence of TEA. Given this lack of effect, the three mean values of γK were averaged and the result (0.55) was used to fix γK during simultaneous fitting of the data in Fig. 4

, as discussed below.

Fig. 4 shows the K0.5 values calculated at each VM and then plotted against VM for each TEA concentration. The K0.5 values in Fig. 2 C are plotted again for purposes of comparison. These data show that TEA produces a concentration-dependent increase in K0.5 at all VM. As in control experiments, the apparent affinity for K+o activation of Na,K pump current was decreased at more positive VM, i.e., K+o activation still appears to be VM-dependent in the presence of TEA.

To determine how TEA affects the VM dependence of Na,K pump activation by K+o, the data in Fig. 4 were fit with an equation derived from a pseudo three-state model for the Na,K pump exposed to an extracellular inhibitor that is competitive with K+o for activation of the enzyme (see PP1appendix):

$K_{0.5}=K_{0.5}^{0}\mathrm{exp}\left(\mathrm{{\lambda}}_{K}U\right){\times}\$
(2)

$\left\{1+\left(\frac{\left[\mathrm{Q}\right]}{K_{\mathrm{Q}}^{0}}\right)^{\mathrm{{\gamma}}_{\mathrm{Q}}}\mathrm{exp}\left({-}\mathrm{{\gamma}}_{\mathrm{Q}}\mathrm{{\lambda}}_{\mathrm{Q}}U\right)\right\}^{\frac{1}{\mathrm{{\gamma}}_{\mathrm{K}}}}\mathrm{,}$

where [Q] and KQ0 are the quaternary amine concentration and the amine concentration for half-maximal inhibition at 0 mV, respectively, and γQ and λQ are the Hill coefficient and the portion of the membrane dielectric dissipated during Na,K pump current inhibition, respectively. This equation has six parameters that can be varied to optimize the fit to the data. However, information regarding one of these parameters, γK, is not available in the plotted data. Even so, as mentioned above and shown in Fig. 3 C, γK is independent of VM and TEA concentration. This finding allows γK to be fixed, in this case, at its average value of 0.55, while the other five parameters are varied to fit Eq. 2 to the data in Fig. 4 as a function of VM and amine concentration.

Solid curves in Fig. 4 are the best-fit functions produced by simultaneously fitting Eq. 2 to the entire displayed dataset. The calculated values of K0.50 and λK in the presence of TEA were 201 ± 6 μM and 0.37 ± 0.02, respectively. These parameter values were not significantly different than those calculated in the absence of TEA with Eq. 1 (in Fig. 2 C). Furthermore, these calculations showed that KQ0 for TEA was 26.6 ± 0.7 mM. The value of λQ, 0.04 ± 0.02, was not significantly different than zero, so that it is clear that TEA binding does not occur in the membrane electric field, i.e., TEA inhibition is VM independent. These results are consistent with those of Eckstein-Ludwig et al. (1998), who also found that TEA block of Na,K pump currents in oocytes is not VM dependent. The results in Fig. 4 are also consistent with the predictions of a pseudo three-state model for the Na,K-ATPase in the presence of a VM-independent blocker that inhibits the enzyme in a competitive manner with K+o (see PP1appendix and Fig. 9).

The value of γQ calculated with Eq. 2 was equal to 0.45 ± 0.02. This value of γQ is significantly less than one, suggesting that negative cooperativity, possibly between multiple TEA molecules, occurs during inhibition of the Na,K-ATPase. This value is also similar to that for γK (0.55).

### Effect of Extracellular Benzyltriethylammonium Ions on Na,K Pump Current

In studying the effects of organic amines on Rb+ occlusion by the Na,K-ATPase, Forbush (1988) noted that quaternary amines containing a benzyl side chain had the highest apparent affinities for slowing release of occluded Rb+ from the enzyme. Therefore, to determine if these benzyl-containing quaternary amines inhibited Na,K pump current in a similar manner to TEA, we examined the effects of a benzylic derivative of TEA, BTEA, on Na,K pump current. At concentrations where TEA had little effect on current, BTEA decreased Na,K pump current amplitude by nearly 50% (Fig. 5

A). Like TEA, current inhibition by BTEA was rapidly reversible at all VM (Fig. 5, A and B). Most interestingly, the VM dependence of current in the presence of BTEA looked quite different than current in the presence of TEA. Inspection of Figs. 3 A and 5 B shows that BTEA appeared to inhibit current more effectively at negative VM, whereas TEA more effectively blocked current at positive VM. Thus, the effect of VM on Na,K pump current in the presence of BTEA appears to be different than that observed with TEA, a VM-independent blocker of the Na,K-ATPase.

To examine the properties of BTEA block more fully, experiments in Figs. 1 and 2 were repeated in the presence of various BTEA concentrations from 0.5 to 5 mM. Fig. 5 C shows the K+o and VM dependence of Na,K pump current measured in the presence of 5 mM BTEA. The notable feature of Fig. 5 B, i.e., the relatively shallow slope of the current-VM relationship in the presence of BTEA, is observed at all K+o examined, quite unlike the current-VM relationship of Na,K pump current in the absence of BTEA (Fig. 2 A). The concentration dependence of BTEA effects on Na,K pump is summarized in Fig. 6

A, which shows current density measured at 0 mV in the presence of different BTEA concentrations and plotted as a function of K+o concentration. BTEA reduced Na,K pump current density in a concentration-dependent manner.

Similar to the analytical procedure shown in Fig. 3, the data at each VM and BTEA concentration were fit with a Hill equation. This fitting procedure showed that both Imax and γK were VM independent at all BTEA concentrations tested (not depicted). As a result, average values of Imax and γK for the 15 VM tested were calculated and plotted in Fig. 6, B and C, respectively. Over the concentration range tested, BTEA had no significant effect on Imax or γK, an indication that this quaternary amine is a competitive inhibitor of Na,K pump current. These average values of Imax and γK were, therefore, averaged again to yield values of 2.30 pA/pF and 0.56, respectively.

Fitting the Na,K pump current data obtained at each VM with Hill equations also showed the VM and BTEA dependence of K0.5 values (Fig. 7)

. BTEA produced a concentration-dependent increase in K0.5 at all VM tested. The unusual feature of the data was the apparent parallel shift of K0.5 to higher values with BTEA, irrespective of VM. This type of behavior is actually predicted for a VM-dependent inhibitor of the Na,K-ATPase (see PP1appendix). By simultaneously fitting Eq. 2 to all data in Fig. 7 with γK fixed at 0.56, K00.5 and λK for current activation, 200 ± 10 μM and 0.35 ± 0.02, were found to be unchanged in the presence of BTEA. On the other hand, the K0Q for block of current activation was calculated to be 4.0 ± 0.3 mM, a value almost sevenfold lower than that for TEA. Thus, consistent with the results of Forbush (1988), the benzylic side chain increased the potency for block of ion transport. These same calculations showed that λQ for BTEA was 0.40 ± 0.05, a demonstration that a sizable fraction of the membrane dielectric was dissipated during BTEA block. In other words, BTEA inhibited the Na,K-ATPase in a VM-dependent manner. In addition, λQ was not significantly different than λK. The implications of this result are presented in the discussion.

To demonstrate that the calculated parameters provided a reasonable description of the data, the values of these parameters were used to solve Eq. 5 (PP1appendix) at each BTEA concentration. The resulting curves (Fig. 6 A) showed that the predicted values were very close to the experimental data points.

Given the average value of γK above, 0.56, Eq. 2 predicts that the value of K0.5 (at any VM) will increase monotonically in a parabolic-like manner with BTEA concentration if γQ equals 1. The inset of Fig. 7 does show that K0.5 (at 0 mV) increases monotonically with BTEA; however, the slope of the relationship between K0.5 and BTEA becomes somewhat less steep at the higher amine concentrations tested. Such a nonlinear relationship between K0.5 and amine concentration is predicted by Eq. 2 if γQ < 1. In fitting Eq. 2 to the data, the value of γQ was found to be 0.56 ± 0.03. By plotting Na,K pump current density versus BTEA concentration at each K+o, γQ was found to be independent of K+o concentration (not depicted). These results indicate that BTEA inhibits the Na,K-ATPase with negative cooperativity.

## Discussion

The main finding of this study is that the organic quaternary amine, BTEA, inhibits Na,K pump current in a VM-dependent manner, whereas tetraethylammonium ion blocks Na,K pump current in a VM-independent manner. To the best of our knowledge, VM-dependent block has not been reported previously for a Na,K-ATPase inhibitor. Therefore, to understand the mechanism of such block, we first need to review the effects of quaternary amines on ion binding and occlusion by the Na,K-ATPase as well as the VM dependence of ion binding reactions themselves. In brief, the basis of our conclusion that quaternary amines produce VM-dependent block resides in the fitting of experimental data to a pseudo three-state model that has two premises: (1) K+o binding to the Na,K-ATPase is VM dependent, and (2) quaternary amine inhibition is competitive with K+o activation. Evidence supporting these premises is reviewed below.

K+o binding is the VM-dependent reaction during ion transport by the Na,K-ATPase: Evidence for the VM dependence of a K+o-dependent reaction step during ion transport by the Na,K-ATPase is shown by experiments, similar to those in Fig. 1, which display a K+o-dependent negative slope in the current-VM relationship of Na,K pump current (Bielen et al., 1991; Rakowski et al., 1991; Sagar and Rakowski, 1994; Berlin and Peluffo, 1997; Peluffo et al., 2000). To attribute this VM dependence of Na,K pump current to K+o-dependent reactions, experiments were performed under conditions that prevent other VM-dependent reaction steps from regulating the kinetics of the ion transport cycle. The concentration of cytosolic Na+ in this study, set by the patch-electrode solution containing 115 mM Na+, is high enough to saturate cytosolic-facing Na+ binding sites of the Na,K-ATPase whose apparent Na+ affinity, under turnover conditions similar to our own, is ∼10 mM (Nakao and Gadsby, 1989). With saturating concentrations, cytosolic Na+ binding reactions are not rate limiting. Therefore, the VM-dependent properties of these reactions cannot be observed. In addition, experiments were also performed in the absence of extracellular Na+, conditions that render extracellular Na+ release reactions by the Na,K-ATPase irreversible so that enzyme turnover will not depend on the VM dependence of these reactions (Läuger, 1991). Our observation that Na,K pump current density measured in the presence of saturating K+o concentrations (5 mM) is almost independent of VM (Figs. 1 C and 2 A) also confirms the point that VM-dependent reaction steps do not control enzyme turnover under these conditions, thereby allowing us to conclude that any VM dependence of intracellular K+-dependent reactions (e.g., Hansen et al., 2002) is not apparent under our experimental conditions. Thus, a K+o-dependent step is the only VM-dependent reaction that can limit enzyme turnover in our experiments.

The mechanism of the K+o- and VM-dependent reaction was identified in our previous work (Peluffo and Berlin, 1997) by trapping the Na,K-ATPase in K+-transporting conformations, i.e., those that promote K+–K+ exchange, and then measuring the Tl+o dependence of presteady-state charge movements during perturbations of VM. The disappearance of these charge movements at saturating Tl+o concentrations demonstrated that the VM-dependent reaction is extracellular Tl+ (or K+) binding.

### Quaternary Amines Competitively Block K+ Transport by the Na,K-ATPase

The competitive nature of quaternary amine inhibition of Na,K pump current is demonstrated by our data showing that maximal Na,K pump current density is unchanged (Figs. 3 C and 6 B) while the apparent affinity for K+o activation of current is significantly decreased (Figs. 4 and 7). Previous studies have also shown comparable effects of extracellular TEA (Sachs and Conrad, 1968) and tetrapropylammonium ion (TPA) on Na+ and K+ transport mediated by the Na,K-ATPase in red blood cells (Kropp and Sachs, 1977) and TEA on Na,K pump currents in oocytes (Eckstein-Ludwig et al., 1998). Thus, we conclude that our results are consistent with previous reports that quaternary amines inhibit K+o-dependent activation of ion transport by the Na,K-ATPase in a competitive manner.

Our own data and previous reports, therefore, suggest that the premises of the pseudo three-state model (see PP1appendix) are valid. As a result, we concluded that BTEA inhibits Na,K pump current in a VM-dependent manner. On the other hand, TEA appears to block Na,K pump current in a VM-independent manner. The occurrence of VM-dependent block and the effect of quaternary amines on ion occlusion by the Na,K-ATPase provide evidence to distinguish between the two alternative mechanisms posited in the literature (Hilgemann, 1994b) to explain VM-dependent ion binding in the Na,K-ATPase. The basis for distinguishing between these alternative mechanisms is discussed below.

The most detailed study of quaternary amine block of ion transport (Kropp and Sachs, 1977) suggested that TPA is a dead-end inhibitor with highest affinity for the Na,K-ATPase without bound K+, slightly decreased affinity when the enzyme is bound with a single K+ and a very much lower affinity for the doubly K+-bound enzyme. This behavior was characterized in Na+o-free conditions, similar to the present study, and a hyperbolic relationship between K+o concentration and ion transport rate was observed in the presence and absence of amine. A linear Dixon plot of ion transport rate versus TPA concentration was interpreted as showing that binding of a single quaternary amine was sufficient to inhibit the Na,K-ATPase. In the present experiments, the Hill coefficient (γK) for K+o activation of Na,K pump current in the absence of TEA or BTEA was 0.58, an indication of negative cooperativity for K+o activation of enzyme turnover. Interestingly, the Hill coefficient (γQ) for quaternary amine inhibition of current had a similar value, whether the amine was TEA or BTEA. Negative cooperativity suggests that more than one amine may be involved in inhibiting Na,K pump current. Furthermore, since the Hill coefficient for TEA and BTEA inhibition of current is independent of K+o, this negative cooperativity is unlikely to be due to a quaternary amine–K+ interaction.

### Effect of Amines on K+ Occlusion by the Na,K-ATPase

Binding and occlusion of extracellular K+ by the Na,K-ATPase appears to be an ordered process in which only one of the two K+ binding sites is accessible from the extracellular medium (Forbush, 1987; González-Lebrero et al., 2002). This postulate rests principally on data showing that, in the presence of Mg2+ and inorganic phosphate (Pi), K+ and K+ congeners slow the release kinetics of ∼50% of the Rb+ occluded by the Na,K-ATPase, without affecting the rapid release of the remaining occluded Rb+ (Glynn et al., 1985; Forbush, 1987). Similar results were obtained whether Rb+ was occluded without other ligands or in the presence of Na+, Mg2+, and micromolar ATP concentrations (González-Lebrero et al., 2002). Forbush (1987) interpreted these data as showing that Rb+ bound to one K+ site, the “fast” site, that could rapidly release Rb+ to the extracellular medium and to a second K+ site, the “slow” site, that could only be occupied by Rb+ that had first bound to the fast site. As a result, when the first K+ binds at the fast site, the second K+ ion cannot bind until the first bound K+ shifts to the second slow site. Ordered binding of Rb+ to the fast and slow K+ sites also led to ordered release from these sites. The conditions of these experiments, i.e., including Mg2+ and Pi during Rb+ release, suggested that release was to the extracellular side of the enzyme.

The accessibility of the fast K+ site to the extracellular medium also appears to be restricted by a flickering or leaky gate (Forbush, 1987; González-Lebrero et al., 2002). The strongest data supporting this concept is that the affinity with which extracellular Rb+ blocks release of bound Rb+ is similar to the affinity for Rb+ occlusion by the Na,K-ATPase (Forbush, 1987). Since the kinetics of extracellular K+ binding reactions are very rapid (Peluffo and Berlin, 1997), such a similarity in Rb+ blocking and occlusion affinities could only occur if access to K+ binding sites is rate limiting. In other words, the gate is poised strongly toward the closed position. This gating process might be analogous to the rapid opening and closing kinetics observed in ion channels (Hille, 2001) and with palytoxin-modified Na,K-ATPase (Kim et al., 1995; Artigas and Gadsby, 2003).

Organic quaternary amines, and in fact a variety of amine-containing compounds, are interesting because, like extracellular K+ congeners, they can block the release of Rb+ from the slow K+ site without affecting release of Rb+ from the fast K+ binding site (Forbush, 1988). Quaternary amines, however, differ from K+ in one important respect. The amine concentrations required to block Rb+ occlusion are far higher than those required to block release of occluded Rb+ (Forbush, 1988). These data have been interpreted as showing that extracellular quaternary amines have access to the fast K+ binding site in the enzyme, but that they do not become occluded like K+. Forbush (1988) envisioned the action of quaternary amines as getting a “foot in the door”, thereby preventing the closing of the occlusion gate. Thus, quaternary amines appear to bind to sites in the Na,K-ATPase without becoming occluded.

### Quaternary Amine Binding Sites in the Na,K-ATPase

The experiments of Forbush (1988) showing that quaternary amines slow the release of 86Rb+ from Na,K-ATPase, do not necessarily demonstrate that the amines bind to the fast K+ binding site in the enzyme. In a similar manner, competitive inhibition of enzyme activation is not sufficient to demonstrate that amines and K+ bind at the same site (Segel, 1975).

Our data show that inhibition of Na,K pump current by TEA or BTEA displays a Hill coefficient significantly less than 1. One implication of this result is that the quaternary amines are capable of interacting at two binding sites. Certainly, our finding that TEA inhibits Na,K pump in a VM-independent manner while BTEA produces VM-dependent block shows that the amines can act at sites with different locations in the membrane electric field; however, the relationship between electrical and physical distances is unknown. Even so, inhibition of Na,K pump current by BTEA dissipates 40% of the membrane electric field, and activation of Na,K pump current by extracellular K+ dissipates a similar fraction of the membrane electric field (35–39%) in our experiments. Given that BTEA is a competitive inhibitor of the enzyme (Fig. 6), the simplest explanation for our data is that this amine is blocking at or very near a K+ binding site located within the membrane dielectric.

With the Hill coefficient for current block by BTEA being less than one, we can also speculate that BTEA is capable of interacting with a second site, possibly the same site located outside of the membrane electric field at which TEA inhibits Na,K pump current. However, if BTEA interacts at two sites in the Na,K-ATPase, an interesting problem arises as to why TEA does not also produce VM-dependent block of the enzyme. In fact, we cannot rule out this possibility because TEA may interact with the K+ site located in the electric field with much lower affinity than BTEA. In this regard, we performed a few experiments using 100 mM extracellular TEA (unpublished data), and the data gave some indication that VM-dependent block occurred, specifically, a small decrease in the slope of the K0.5-VM relationship in the presence of 100 mM TEA. We did not investigate this point because any VM-dependent effects of TEA do not change our mechanistic interpretation of VM-dependent inhibition of the Na,K-ATPase, and we were concerned that high TEA concentrations were having additional effects on the cells under study. In any case, if the amines are interacting at two sites in the Na,K-ATPase, one located in the membrane electric field, the other outside of the field, this result does not itself demonstrate that the amines are binding at the slow and fast K+ sites identified by Forbush (1987), since we cannot conclude whether an amine binding site located outside of the membrane electric field is a K+ binding site; however, such a possibility is not incompatible with our data.

### Implications of VM-dependent Block by BTEA for the Mechanism of VM-dependent K+o Binding by the Na,K-ATPase

As mentioned above, our previous work has demonstrated that a VM-dependent step during K+ transport by the Na,K-ATPase is extracellular K+ binding, and that this reaction is followed by a slower electroneutral reaction (Peluffo and Berlin, 1997). The Albers-Post reaction model for the Na,K-ATPase (Glynn, 1985) would suggest this slow electroneutral reaction is probably occlusion and enzyme dephosphorylation. Scheme I

summarizes this reaction sequence without specifying the stoichiometry of K+ binding or the phosphorylation

state of the enzyme. Given Rb+ occlusion data suggesting the existence of a flickering gate, the underlying reason for slow occlusion kinetics is that the gate opens with low probability (Forbush, 1987; González-Lebrero et al., 2002). Data showing rapid gating of the palytoxin-modified enzyme (Kim et al., 1995; Artigas and Gadsby, 2003), analogous to the opening and closing of the gate in an ion channel, might suggest that Scheme I could be modified so that rapid ion binding is followed by a rapidly reversible gating reaction and then slower ion occlusion (Scheme II)

. Hilgemann (1994b) predicted a similar reaction sequence and suggested that the gating step (to form E{K}) would be a microscopic occlusion reaction that precedes the formation of a stable ion-occluded enzyme intermediate. With such a reaction scheme, Hilgemann (1994b) also predicted that as long as the fraction of K+-bound but not -occluded enzyme (EK conformation) is small, it would not be practical to distinguish whether the VM-dependent reaction is ion binding or rapid gating of the ion-bound enzyme.

The first issue then is whether the concentration of EK in Scheme II is likely to be small during K+ binding and occlusion. In Rb+ occlusion conditions, González-Lebrero et al. (2002) calculated the equilibrium constants for ion binding and occlusion to show that the reaction equilibrium is displaced almost entirely toward the occluded state (E(K) conformation), a point made quite obvious by the ability to occlude Rb+ in the presence of Na+, Mg2+, and micromolar concentrations of ATP (Glynn and Richards, 1982). Therefore, at least in the forward direction, fast gating must very much favor the closed gate conformation.

In the reverse direction, there is no direct information about the equilibrium position of the ion binding reaction under the conditions of our experiments. Indirectly, one could infer that the bound conformation would rapidly decay to the unbound state because the K0.5 for equilibrium Rb+ occlusion in the absence of other ligands is much smaller than the Rb+ concentration that produces half-maximal rate of occlusion (González-Lebrero et al., 2002). In Scheme II, this difference in K0.5 values would suggest that either the formation of the Rb+-bound but not -occluded state and/or the closed gate conformation of the enzyme is disfavored. However, as pointed out above, the gating reaction is strongly poised in the closed direction (González-Lebrero et al., 2002), so that only the binding reaction can realistically be disfavored. Thus, a reasonable conclusion is that the K+-bound but not -occluded conformation of the enzyme exists only in very small quantities.

Organic quaternary amines make it possible to distinguish whether the ion binding or gating reactions are VM dependent. Forbush (1988) showed that these amines bind to the Na,K-ATPase but are not occluded, and he interpreted these data as showing that closing of the flickering gate is blocked. If this is the case, the enzyme can accumulate to a significant degree in the amine-bound (but not -occluded) conformation. Under these conditions, it should be possible to distinguish whether binding or gating is the VM-dependent reaction. If gating is VM dependent then amine block of enzyme cycling should be VM independent. Conversely, if binding is VM dependent, one could anticipate that amine block could also be VM dependent. The present data clearly demonstrate that the latter is possible. In addition, we found that the fraction of the membrane dielectric dissipated during BTEA inhibition is similar to that dissipated during K+o activation of enzyme cycling. These data suggest that the binding step is the major VM-dependent reaction while conformational changes associated with ion binding move little charge in the membrane electric field. This VM dependence of ion binding will occur as K+ moves from the extracellular medium to its binding sites in the enzyme, implying that a high-field access channel exists in the Na,K-ATPase. Thus, these data provide strong evidence for an ion channel–like structure in the Na,K-ATPase.

### Summary

In this study, we have demonstrated that the organic quaternary amine, BTEA, is a competitive, VM-dependent blocker for K+o activation of Na,K pump current under conditions where K+o binding is the only VM-dependent reaction controlling enzyme turnover. Given previous data concerning amine inhibition of ion transport and occlusion by the Na,K-ATPase as well as our understanding of VM-dependent K+o binding by this enzyme, the present results indicate that BTEA is binding at a K+ site located in the membrane electric field. Furthermore, these data demonstrate that this binding site must be located in an access channel–like structure in the enzyme. Thus, quaternary amines should provide a valuable tool to probe the structure of K+ binding sites and the access channel of the Na,K-ATPase.

## Appendix

In the absence of quaternary amines, Na,K pump current measured in Na+o-free solution and saturating cytosolic Na+ concentrations can be described with a pseudo two-state model in which a single reaction step is VM dependent (Sagar and Rakowski, 1994). To model the effect of quaternary amines on Na,K pump current, the following properties of quaternary amines are assumed: (a) inhibition is competitive with K+o activation, (b) inhibition prevents the enzyme from supporting ion transport, i.e., no current arises from inhibited enzymes, (c) the amine binds with unknown stoichiometry, and (d) inhibition can be VM dependent. With these assumptions, Scheme III

describes possible reactions of the Na,K-ATPase where the enzyme is assumed to occur in three forms, a K+-bound form (EK), an ion-free form equivalent to E2P (Eo), and an amine-bound form (EQ). Stoichiometry of K+ binding to the enzyme is not shown in this scheme.

Voltage dependence is assigned arbitrarily to the binding steps to Eo (k1 for K+o and k3 for Q) rather than dissociation steps (k−1 and k−3, respectively). If the backward running of the enzyme under our conditions is assumed to be negligible, i.e., k−2 ∼0, Scheme III can be solved for the concentration of EK under steady-state conditions to be as in Eq. 3

$\left[E\mathrm{K}\right]=N\left(1+\frac{\left(k_{{-}1}+k_{2}\right)}{k_{1}\left[\mathrm{K}^{\mathrm{+}}\right]^{\mathrm{{\gamma}}_{\mathrm{K}}}\mathrm{exp}\left({-}\mathrm{{\gamma}}_{\mathrm{K}}\mathrm{{\lambda}}_{\mathrm{K}}U\right)}+\ \right.$
(3)

\begin{eqnarray*}&&\\&&\frac{\left(k_{{-}1}+k_{2}\right)k_{3}\left[\mathrm{Q}\right]^{\mathrm{{\gamma}}_{\mathrm{Q}}}\mathrm{exp}\left({-}\mathrm{{\gamma}}_{\mathrm{Q}}\mathrm{{\lambda}}_{\mathrm{Q}}U\right)}{k_{{-}3}k_{1}\left[\mathrm{K}^{\mathrm{+}}\right]^{\mathrm{{\gamma}}_{\mathrm{K}}}\mathrm{exp}\left({-}\mathrm{{\gamma}}_{\mathrm{K}}\mathrm{{\lambda}}_{\mathrm{K}}U\right)}\right)^{{-}1}\mathrm{,}\end{eqnarray*}

where N is the total concentration of Na,K-ATPase proteins involved in ion transport, i.e., N = [Eo] + [EK] + [EQ], and [K+] is the K+o concentration. Assuming that Na,K pump current (I) is equal to zFk2[EK], where z = 1 charge per cycle, Eq. 3 can be substituted into this expression to yield:

$I=I_{\mathrm{max}}\left(1+\left(\frac{K_{0.5}^{0}}{\left[\mathrm{K}^{\mathrm{+}}\right]}\right)^{\mathrm{{\gamma}}_{\mathrm{K}}}\mathrm{exp}\left(\mathrm{{\gamma}}_{\mathrm{K}}\mathrm{{\lambda}}_{\mathrm{K}}U\right)\ {\times}\ \right.$
(4)

$\left(1+\left(\frac{\left[\mathrm{Q}\ \right]}{K_{\mathrm{Q}}^{0}}\right)^{\mathrm{{\gamma}}_{\mathrm{Q}}}\mathrm{exp}\left({-}\mathrm{{\gamma}}_{\mathrm{Q}}\mathrm{{\lambda}}_{\mathrm{Q}}U\right)\right)\right)^{{-}1}\mathrm{,}$

where

$K_{0.5}^{0}=\left(\frac{k_{{-}1}+k_{2}}{k_{1}}\right)^{\frac{1}{\mathrm{{\gamma}}_{\mathrm{K}}}}$
(4a)

$K_{\mathrm{Q}}^{0}=\left(\frac{k_{{-}3}}{k_{3}}\right)^{\frac{1}{\mathrm{{\gamma}}_{\mathrm{Q}}}}$
(4b)

$I_{\mathrm{max}}=zFk_{2}N\mathrm{.}$
(4c)

Na,K pump current (I) in Eq. 4 depends on [K+], the amine concentration ([Q]) and VM, all experimentally controllable parameters. Fig. 8

shows the results of a simulation in which the amine is assumed to act in a VM-independent fashion, i.e., λQ = 0. Simulated current is activated at constant [K+] (0.2 mM) in the absence and presence of the inhibitory quaternary amine. Note the similarity to Fig. 3 A, inhibition of Na pump current by TEA. Like the experimental data, the VM-independent inhibitor appears to block simulated Na,K pump current more effectively at positive VM. However, when the simulation is run with a VM-dependent blocker (λQ = 0.4 and 0.75), the VM dependence of the current progressively changes so that the inhibitor appears to block current more effectively at negative VM, similar to the data shown in Fig. 5 B.

Eq. 4 can also be expressed in the more familiar format of a Hill equation:

$I=I_{\mathrm{max}}\left(1+\left(\frac{K_{0.5}}{\left[\mathrm{K}^{\mathrm{+}}\right]}\right)^{\mathrm{{\gamma}}_{\mathrm{K}}}\right)^{{-}1}\mathrm{,}$
(5)

where K0.5 is expressed as in Eq. 2. Simulations for the effect of VM-independent and -dependent blockers on the K0.5-VM relationship are shown in Fig. 9

. If inhibition of Na pump current is VM independent (λQ = 0), the K0.5 for K+o will simply be scaled up from the curve obtained in the absence of inhibitor. On the other hand, with VM-dependent inhibition, the curves progressively flatten out and change curvature as the ion-well depth for the inhibitor is increased (λQ = 0.4 and 0.75). Again, the simulations with λQ = 0 and 0.4 are qualitatively similar to the VM dependence of K0.5 values observed in the presence of TEA (Fig. 4) and BTEA (Fig. 7), respectively.

In the absence of organic amines, Scheme III reduces to a pseudo two-state model. In this case, Eq. 4 simplifies to

$I=I_{\mathrm{max}}\left(1+\left(\frac{K_{0.5}^{0}}{\left[\mathrm{K}^{\mathrm{+}}\right]}\right)^{\mathrm{{\gamma}}_{\mathrm{K}}}\mathrm{exp}\left(\mathrm{{\gamma}}_{\mathrm{K}}\mathrm{{\lambda}}_{\mathrm{K}}U\right)\right)^{{-}1}\mathrm{,}$
(6)

and the expression for K0.5 becomes Eq. 1, which is used to fit the data in Fig. 2 C.

## Acknowledgments

The authors wish to thank the technical assistance of Palak Raval Nelson and Marguerite Schmid, and Dr. Dawn A. Lott for helpful discussions on validating our fitting procedures.

This work was supported by the National Institutes of Health (J.R. Berlin) and the American Heart Association (R.D. Peluffo).

Olaf S. Andersen served as editor.

## References

References
2003
. Na+/K+-pump ligands modulate gating of palytoxin-induced ion channels.
100
:
501
–505.
Berlin, J.R., and R.D. Peluffo.
1997
. Mechanism of electrogenic reaction steps during K+ transport by the Na,K-ATPase.
834
:
251
–259.
Berlin, J.R., and R.D. Peluffo.
1998
. Inhibition of Na,K pump current in cardiac myocytes by organic quaternary amines.
Biophys. J.
74
:
A338
.
Bielen, F.V., H.G. Glitsch, and F. Verdonck.
1991
. Dependence of Na+ pump current on external monovalent cations and membrane potential in rabbit cardiac Purkinje cells.
J. Physiol.
442
:
169
–189.
Bühler, R., W. Sturmer, H.-J. Apell, and P. Läuger.
1991
. Charge translocation by the Na+,K+-pump: I. Kinetics of local field changes studied by time-resolved fluorescence measurements.
J. Membr. Biol.
121
:
141
–161.
Cooke, I.M., G. LeBlanc, and L. Tauc.
1974
. Sodium pump stoichiometry in Aplysia neurons from simultaneous current and tracer measurements.
Nature.
251
:
254
–256.
De Weer, P., R.F. Rakowski, and D.C. Gadsby.
1984
. Current generated by backward-running electrogenic Na pump in squid giant axons.
Nature.
309
:
450
–452.
De Weer, P., D.C. Gadsby, and R.F. Rakowski.
1988
. Voltage dependence of the Na-K pump.
Annu. Rev. Physiol.
50
:
225
–241.
Eckstein-Ludwig, U., J. Rettinger, L.A. Vasilets, and W. Schwarz.
1998
. Voltage-dependent inhibition of the Na+,K+ pump by tetraethylammonium.
Biochim. Biophys. Acta.
1372
:
289
–300.
Eisner, D.A., and W.J. Lederer.
1980
. Characterization of the electrogenic sodium pump in cardiac Purkinje fibers.
J. Physiol.
303
:
441
–474.
Fendler, K., S. Jaruschewski, A. Hobbs, W. Albers, and J.P. Froehlich.
1993
. Pre-steady-state charge translocation in Na,K-ATPase from eel electric organ.
J. Gen. Physiol.
102
:
631
–666.
Forbush, B.
1987
. Rapid release of 42K or 86Rb from two distinct transport sites on the Na,K-pump in the presence of Pi or vanadate.
J. Biol. Chem.
262
:
11116
–11127.
Forbush, B.
1988
. The interaction of amines with the occluded state of the Na,K-pump.
J. Biol. Chem.
263
:
7979
–7988.
1989
. Steady-state current-voltage relationship of the Na/K pump in guinea pig ventricular myocytes.
J. Gen. Physiol.
94
:
511
–537.
Gadsby, D.C., J. Kimura, and A. Noma.
1985
. Voltage dependence of Na/K pump current in isolated heart cells.
Nature.
315
:
63
–65.
Gadsby, D.C., R.F. Rakowski, and P. De Weer.
1993
. Extracellular access to the Na,K pump: pathway similar to ion channel.
Science.
260
:
100
–103.
Glynn, I.M. 1985. The Na+, K+-transporting adenosine triphosphatase. The Enzymes of Biological Membranes, Vol. 3. A.N. Martonosi, editor. Plenum Press, New York. 35–114.
Glynn, I.M., and D.E. Richards.
1982
. Occlusion of rubidium ions by the sodium-potassium pump: its implications for the mechanism of potassium transport.
J. Physiol.
330
:
17
–43.
Glynn, I.M., and S.J.D. Karlish.
1990
. Occluded cations in active transport.
Annu. Rev. Biochem.
59
:
171
–205.
Glynn, I.M., J.L. Howland, and D.E. Richards.
1985
. Evidence for the ordered release of rubidium ions occluded within the Na,K-ATPase of mammalian kidney.
J. Physiol.
368
:
453
–469.
González-Lebrero, R.M., S.B. Kaufman, M.R. Montes, J.G. Nørby, P.J. Garrahan, and R.C. Rossi.
2002
. The occlusion of Rb+ in the Na+/K+-ATPase.
J. Biol. Chem.
277
:
5910
–5921.
Hansen, U., D. Gradmann, D. Sanders, and C.L. Slayman.
1981
. Interpretation of current-voltage relationships for “active” ion transport systems: I. Steady-state reaction-kinetic analysis of Class-I mechanisms.
J. Membr. Biol.
63
:
165
–190.
Hansen, P.S., K.A. Buhagiar, B.Y. Kong, R.J. Clarke, D.F. Gray, and H.H. Rasmussen.
2002
. Dependence of Na+-K+ pump current-voltage relationship on intracellular Na+, K+, and Cs+ in rabbit cardiac myocytes.
Am. J. Physiol.
283
:
C1511
–C1521.
Heyse, S., I. Wuddel, H.-J. Apell, and W. Stürmer.
1994
. Partial reactions of the Na,K-ATPase: determination of rate constants.
J. Gen. Physiol.
104
:
197
–240.
Hilgemann, D.W.
1994
a. Channel-like function of the Na,K pump probed at microsecond resolution in giant membrane patches.
Science.
263
:
1429
–1432.
Hilgemann, D.W. 1994b. Flexibility and constraint in the interpretation of Na+/K+ pump electrogenicity: What is an access channel? The Sodium Pump. E. Bamberg and W. Schoner, editors. Steinkopff/Darmstadt. New York. 507–516.
Hille, B. 2001. Ion Channels of Excitable Membranes. 3rd ed. Sinauer Associates, Sunderland, MA.
Holmgren, M., and R.F. Rakowski.
1994
. Pre-steady-state transient currents mediated by the Na/K pump in internally perfused Xenopus oocytes.
Biophys. J.
66
:
912
–922.
Isenberg, G., and W. Trautwein.
1974
. The effect of dihydroouabain and lithium ions on the outward current in cardiac Purkinje fibers.
Pflugers Arch.
350
:
41
–54.
Ishizuka, N., A.J. Fielding, and J.R. Berlin.
1996
. Na pump current can be separated into ouabain-sensitive and -insensitive components in single rat ventricular myocytes.
Jap. J. Physiol.
46
:
215
–223.
Kim, S.Y., K.A. Marx, and C.H. Wu.
1995
. Involvement of the Na,K-ATPase in the induction of ion channels by palytoxin.
Naunyn-Schmiedeberg's Arch Pharmacol.
351
:
542
–554.
Kropp, D.L., and J.R. Sachs.
1977
. Kinetics of the inhibition of the Na-K pump by tetrapropylammonium chloride.
J. Physiol.
264
:
471
–487.
Lafaire, A.V., and W. Schwarz.
1986
. Voltage dependence of the rheogenic Na+/K+ ATPase in the membrane of oocytes of Xenopus laevis.
J. Membr. Biol.
91
:
43
–51.
Läuger, P. 1991. Electrogenic Ion Pumps. Sinauer Associates, Sunderland, MA.
1986
. Voltage dependence of Na translocation by the Na/K pump.
Nature.
323
:
628
–630.
1989
. [Na] and [K] dependence of the Na/K pump current-voltage relationship in guinea pig ventricular myocytes.
J. Gen. Physiol.
94
:
539
–565.
Peluffo, R.D., and J.R. Berlin.
1997
. Electrogenic K+ transport by the Na+-K+ pump in rat cardiac ventricular myocytes.
J. Physiol. (Lond.)
.
501
:
33
–40.
Peluffo, R.D., J.M. Argüello, and J.R. Berlin.
2000
. The role of Na,K-ATPase α subunit serine 775 and glutamate 779 in determining the extracellular K+ and membrane potential-dependent properties of the Na,K-pump.
J. Gen. Physiol.
116
:
47
–59.
Peluffo, R.D., Y. Hara, and J.R. Berlin.
2001
. Voltage dependence of inhibition of the cardiac Na,K-pump forward cycling by quaternary amines.
Biophys. J.
80
:
37a
.
Rakowski, R.F.
1993
. Charge movement by the Na/K pump in Xenopus oocytes.
J. Gen. Physiol.
101
:
117
–144.
Rakowski, R.F., D.C. Gadsby, and P. De Weer.
1989
. Stoichiometry and voltage dependence of the sodium pump in voltage-clamped, internally dialyzed squid giant axon.
J. Gen. Physiol.
93
:
903
–941.
Rakowski, R.F., L.A. Vasilets, J. LaTona, and W. Schwarz.
1991
. A negative slope in the current-voltage relationship of the Na+/K+ pump in Xenopus oocytes produced by reduction of external.
J. Membr. Biol.
121
:
177
–187.
Rakowski, R.F., D.C. Gadsby, and P. De Weer.
1997
. Voltage dependence of the Na/K pump.
J. Membr. Biol.
155
:
105
–112.
Rossi, R.C., and P.J. Garrahan.
1989
. Steady-state kinetic analysis of the Na+/K+-ATPase. The effects of adenosine 5′-[β,γ-methylene]triphosphate on substrate kinetics.
Biochim. Biophys. Acta.
981
:
85
–94.
1968
. Effect of tetraethylammonium on the active cation transport system of the red blood cell.
Am. J. Physiol.
215
:
795
–798.
Sagar, A., and R.F. Rakowski.
1994
. Access channel model for the voltage dependence of the forward-running Na+/K+ pump.
J. Gen. Physiol.
103
:
869
–894.
Segel, I.H. 1975. Enzyme Kinetics. Wiley & Sons, New York.
Stimers, J.R., S. Liu, and M. Lieberman.
1991
. Apparent affinity of the Na/K pump for ouabain in cultured chick cardiac myocytes.
J. Gen. Physiol.
98
:
815
–833.
Vasilets, L.A., and W. Schwarz.
1993
. Structure-function relationships of cation binding in the Na+/K+-ATPase.
Biochim. Biophys. Acta.
1154
:
201
–222.
Wuddel, I., and H.-J. Apell.
1995
. Electrogenicity of the sodium transport pathway in the Na,K-ATPase probed by charge-pulse experiments.
Biophys. J.
69
:
909
–921.
Zemková, H., J. Teisinger, and F. Vyskocil.
1988
. Inhibition of the electrogenic Na,K pump and Na,K-ATPase activity by tetraethylammonium, tetrabutylammonium, and Apamin.
J. Neurosci. Res.
19
:
497
–503.

Abbreviations used in this paper: BTEA, benzyltriethylammonium ion; NMG, N-methyl-D-glucamine; TMA, tetramethylammonium ion; TPA, tetrapropylammonium ion.


text:
# Chapter's title page before every chapter, and every chapter with its title

I am doing a research work and I googled half an hour and found nothing about this. Is there some way to get a title page (without any style) for every chapter? I think the "without style" thing is done with \thispagestyle{empty}

-
By title page you mean something like the LaTeX \part in book document style? –  m0nhawk Jan 7 '13 at 18:56
@m0nhawk: what? –  dot dot Jan 7 '13 at 20:09
I mean something like this. Is this what you want? –  m0nhawk Jan 8 '13 at 10:33
@m0nhawk: yes but without numbering. –  dot dot Jan 12 '13 at 1:22
If I understand you right (in fact I have difficulties in understanding, what you want, but this may be a lack in my knowledge of English language) a \thispagestyle{empty}\cleardoublepage directly after \chapter{…} should do, what you want. If yes, there are ways for automatic application of these macros … –  Speravir Jan 12 '13 at 21:46

Something like this with the help of titlesec package:

\documentclass[oneside]{book}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage{lipsum}

\usepackage{titlesec}

\titleclass{\chapter}{page}
\assignpagestyle{\chapter}{empty}
\titleformat{\chapter}
[display]
{\centering\Huge\bfseries}
{\chaptername\ \thechapter}
{0pt}
{\huge}
[\clearpage]

\begin{document}
\chapter{First}
\lipsum[10-20]
\chapter{Second}
\lipsum[10-20]
\end{document}

Additional formatting can be done via modifying the \titleformat options, see #3.1 in titlesec package manual.

And the result:

Chapter page:

And text page:

-


text:
Oliver Marsh
193 posts / 1 project
Olster1.github.io
Attach assets to exe

In a handmade chat I remember Casey say you can attach your assets to the end of your exe to turn your program into a single file.

The only way I can work out how do this is append the assets to the end of the exe, but instead of having the meta data (like asset offsets etc.) straight after where the original exe ended, putting it at the end. So I can get the file size of my bundled exe when I run my program & move backward from the end of the file.

Is this how you would do it without knowing the size of your original exe prior to building it?

Hope I explained it satisfactory 😅

Mārtiņš Možeiko
2381 posts / 2 projects
Attach assets to exe
Edited by Mārtiņš Možeiko on

That is one way how to do that. Sometimes strict corporate anti-virus software may not like that, because technically those kind of exe files have invalid format.

"Proper" way is to put them in resources.

Create whatever.rc file and put all your assets there in following format:

1 RCDATA "image.png"
2 RCDATA "sounds.wav"
3 RCDATA ...

void LoadAsset(int id)
{
HRSRC res = FindResource(GetModuleHandle(0), MAKEINTRESOURCE(id), RT_RCDATA);
void* data = LockResource(handle);
DWORD size = SizeofResource(0, res);
// ... use data & size
}

You can even share those 1/2/3 numbers with your c code by having them as #define's in some .h file and including it in .rc and .c files.

And if you choose to use clang compiler, then there is even better way - which works on Windows and Linux the same, using .incbin inline assembly directive: https://gist.github.com/mmozeiko/ed9655cf50341553d282 This way you can have all your assets referenced from source code.

Regardless of which option you choose, don't forget that exe/dll files on Windows have 2GB size limitation.

Oliver Marsh
193 posts / 1 project
Olster1.github.io
Attach assets to exe

Thanks Martin, that helps a lot. Oh didn't realise the 2gb limit which might make it not worth it.

3 posts
Attach assets to exe
Edited by Adamarla on Reason: Missed detail in previous post

Could your INCBIN macro be used to include separate glsl files as char* literals? I guess the difference would be appending a 0 byte. A separate file would make going to errors by line number easier, improve syntax highlighting and indentation.

I suppose for msvc it would need to preprocess to another file for inclusion.

Mārtiņš Možeiko
2381 posts / 2 projects
Attach assets to exe
Edited by Mārtiņš Možeiko on

incbin can include as many files as you want. Just use different name for variable name. Currently it automatically appends 0 byte at end (that's what .byte 0 means there).

Not sure what you mean by separate files. When GLSL error happens, the shader compiler does not know filename where string is coming from, it will not show .c/.cpp file name. It will show line number in original glsl file. It's your job to map it to correct file name when you get error message.

As for msvc - yeah, it does not support this kind of inline assembly. What I've done before is to make custom build step for .c file that contains these incbins, and call clang-cl.exe on them - that produces same .obj as cl.exe that I can link together. Then simply have extern's to access them for other TU's.

Neo Ar
165 posts / 1 project
riscy.tv host
Attach assets to exe

What is the reasoning for the 16-byte alignment? I don't know as much about x86; I thought we just want 4-byte alignment for data and 16-byte alignment was just a stack abi convention? In noid I used 4-byte alignment for this (https://gitlab.com/riscy-business/noid/-/blob/v1.0.0-alpha.5.1/ototo_kun.s) but if there is a perf reason for a different alignment I'll change it. Also should .balign be preferred over .align for any reason here / any other recommendations to improve how I did it for noid?

Mārtiņš Možeiko
2381 posts / 2 projects
Attach assets to exe
Edited by Mārtiņš Možeiko on
.align number is platform specific. For some targets it is amount of bytes, in some it is log2 of amount (basically zero bit count in LSB). .balign is always in bytes for GNU compatible assemblers.


text:
# arcsin (x) + arcsin(y) = arcsin (x $$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$)

We will learn how to prove the property of the inverse trigonometric function arcsin (x) + arcsin(y) = arcsin (x $$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$)

Proof:

Let, sin$$^{-1}$$ x = α and sin$$^{-1}$$ y = β

From sin$$^{-1}$$ x = α we get,

x = sin α

and from sin$$^{-1}$$ y = β we get,

y = sin β

Now, sin (α + β) = sin α cos β + cos α sin β

sin (α + β) = sin α $$\sqrt{1 - sin^{2} β}$$ + $$\sqrt{1 - sin^{2} α}$$ sin β

sin (α + β) = x ∙ $$\sqrt{1 - y^{2}}$$ + $$\sqrt{1 - x^{2}}$$ ∙ y

Therefore, α + β = sin$$^{-1}$$ (x $$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$)

or, sin$$^{-1}$$ x + sin$$^{-1}$$ y = sin$$^{-1}$$ (x $$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$).       Proved.

Note: If x > 0, y > 0 and x$$^{2}$$ + y$$^{2}$$ > 1, then the sin$$^{-1}$$ x + sin$$^{-1}$$ y may be an angle more than π/2 while sin$$^{-1}$$ (x $$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$), is an angle between – π/2 and π/2.

Therefore, sin$$^{-1}$$ x + sin$$^{-1}$$ y = π - sin$$^{-1}$$ (x $$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$)

1. Prove that sin$$^{-1}$$ $$\frac{3}{5}$$ + sin$$^{-1}$$ $$\frac{8}{17}$$ = sin$$^{-1}$$ $$\frac{77}{85}$$

Solution:

L. H. S. = sin$$^{-1}$$ $$\frac{3}{5}$$ + sin$$^{-1}$$ $$\frac{8}{17}$$

Now, we will apply the formula sin$$^{-1}$$ x + sin$$^{-1}$$ y = sin$$^{-1}$$ (x $$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$)

= sin$$^{-1}$$ ($$\frac{3}{5}$$ $$\sqrt{1 - (\frac{8}{17})^{2}}$$ + $$\frac{8}{17}$$$$\sqrt{1 - (\frac{3}{5})^{2}}$$)

= sin$$^{-1}$$ ($$\frac{3}{5}$$ ×  $$\frac{15}{17}$$ + $$\frac{8}{17}$$ ×  $$\frac{4}{5}$$)

= sin$$^{-1}$$  $$\frac{77}{85}$$ = R. H. S.                  Proved.

2. Show that, sin$$^{-1}$$ $$\frac{4}{5}$$ + sin$$^{-1}$$ $$\frac{5}{13}$$ + sin$$^{-1}$$ $$\frac{16}{65}$$ = $$\frac{π}{2}$$.

Solution:

L. H. S. = (sin$$^{-1}$$$$\frac{4}{5}$$ + sin$$^{-1}$$$$\frac{5}{13}$$) + sin$$^{-1}$$$$\frac{16}{65}$$

Now, we will apply the formula sin$$^{-1}$$ x + sin$$^{-1}$$ y = sin$$^{-1}$$ (x$$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$)

= sin$$^{-1}$$ ($$\frac{4}{5}$$ $$\sqrt{1 - (\frac{5}{13})^{2}}$$ + $$\frac{5}{13}$$$$\sqrt{1 - (\frac{4}{5})^{2}}$$ + sin$$^{-1}$$$$\frac{16}{65}$$

= sin$$^{-1}$$ ($$\frac{4}{5}$$ ×  $$\frac{12}{13}$$ + $$\frac{5}{13}$$ ×  $$\frac{3}{5}$$) + sin$$^{-1}$$$$\frac{16}{65}$$

= sin$$^{-1}$$ $$\frac{63}{65}$$ + sin$$^{-1}$$$$\frac{16}{65}$$

= sin$$^{-1}$$ $$\frac{63}{65}$$ + cos$$^{-1}$$$$\frac{63}{65}$$, [Since, sin$$^{-1}$$ $$\frac{16}{65}$$ = cos$$^{-1}$$ $$\frac{63}{65}$$]

= $$\frac{π}{2}$$, [Since, sin$$^{-1}$$ x + cos$$^{-1}$$ x = $$\frac{π}{2}$$] = R. H. S.            Proved.

Note: sin$$^{-1}$$ = arcsin (x)

Inverse Trigonometric Functions


text:
# Cournot Nash Equilibrium Between Two Firms [migrated]

Suppose we have two firms with specialized, but similar products. Suppose market demand for the two products is: $$p_1(q_1,q_2)=a-bq_1-dq_2$$ $$p_2(q_1,q_2)=a-bq_2-dq_1$$ where $d \in (-b,b)$. Suppose that both firms have cost $c(q)=q$

What does $d$ mean intuitively? Is the Cournot Nash Equilibrium for this $$q_1=\frac{2ba - ad + dc'(q_2)-c'(q_1)2b} {1 - d^2}$$ $$q_2=\frac{2ba - ad + dc'(q_1)-c'(q_2)2b} {1 - d^2}$$ Thanks a ton.

-